From e173566e0d2dfa84ac1c4d3f6c31f51369e1cd68 Mon Sep 17 00:00:00 2001
From: "github-actions[bot]"
 <41898282+github-actions[bot]@users.noreply.github.com>
Date: Mon, 19 Feb 2024 16:13:34 +0000
Subject: [PATCH] Deploy to GitHub pages

---
 .nojekyll                                     |     0
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 theories/html/Goedel.PRrepresentable.html     |    79 +
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 theories/html/Goedel.fixPoint.html            |   101 +
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 theories/html/additions.BinaryStrat.html      |    69 +
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 theories/html/hydras.Epsilon0.F_omega.html    |   217 +
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 .../html/hydras.Hydra.Hydra_Termination.html  |   173 +
 .../html/hydras.Hydra.Hydra_Theorems.html     |   240 +
 theories/html/hydras.Hydra.KP_example.html    |   152 +
 theories/html/hydras.Hydra.O2H.html           |   256 +
 theories/html/hydras.Hydra.Omega2_Small.html  |   200 +
 theories/html/hydras.Hydra.Omega_Small.html   |   147 +
 theories/html/hydras.MoreAck.Ack.html         |   420 +
 theories/html/hydras.MoreAck.AckNotPR.html    |   351 +
 theories/html/hydras.MoreAck.BadSubst.html    |    98 +
 theories/html/hydras.MoreAck.FolExamples.html |   588 +
 .../html/hydras.MoreAck.Iterate_compat.html   |    54 +
 .../html/hydras.MoreAck.LNN_Examples.html     |   170 +
 .../html/hydras.MoreAck.PrimRecExamples.html  |   481 +
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 ...s.OrdinalNotations.Example_3PlusOmega.html |    70 +
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 ....OrdinalNotations.ON_Omega_plus_omega.html |   221 +
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 theories/html/hydras.Prelude.Comparable.html  |   296 +
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 theories/html/hydras.Prelude.Exp2.html        |    56 +
 .../html/hydras.Prelude.First_toggle.html     |    75 +
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 .../html/hydras.Prelude.MoreDecidable.html    |    51 +
 theories/html/hydras.Prelude.MoreLibHyps.html |    59 +
 theories/html/hydras.Prelude.MoreLists.html   |   417 +
 theories/html/hydras.Prelude.MoreOrders.html  |   115 +
 theories/html/hydras.Prelude.MoreVectors.html |   236 +
 theories/html/hydras.Prelude.More_Arith.html  |   177 +
 .../html/hydras.Prelude.OrdNotations.html     |    46 +
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 .../html/hydras.Prelude.STDPP_compat.html     |    75 +
 .../html/hydras.Prelude.Simple_LexProd.html   |    97 +
 theories/html/hydras.Prelude.Sort_spec.html   |   214 +
 theories/html/hydras.Prelude.WfVariant.html   |    79 +
 .../html/hydras.Prelude.ssrnat_extracts.html  |    61 +
 theories/html/hydras.Schutte.AP.html          |   330 +
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 theories/html/hydras.Schutte.CNF.html         |   258 +
 .../html/hydras.Schutte.Correctness_E0.html   |   175 +
 theories/html/hydras.Schutte.Countable.html   |   155 +
 theories/html/hydras.Schutte.Critical.html    |   320 +
 theories/html/hydras.Schutte.GRelations.html  |   155 +
 theories/html/hydras.Schutte.Lub.html         |    63 +
 .../html/hydras.Schutte.MoreEpsilonIota.html  |   163 +
 .../hydras.Schutte.Ordering_Functions.html    |   519 +
 theories/html/hydras.Schutte.PartialFun.html  |   266 +
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 .../html/hydras.Schutte.Schutte_basics.html   |   824 +
 theories/html/hydras.Schutte.Well_Orders.html |   173 +
 theories/html/hydras.rpo.closure.html         |    73 +
 theories/html/hydras.rpo.decidable_set.html   |    48 +
 theories/html/hydras.rpo.dickson.html         |   190 +
 theories/html/hydras.rpo.list_permut.html     |   356 +
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 theories/html/hydras.rpo.term.html            |   916 +
 .../html/hydras.solutions_exercises.F_3.html  |   200 +
 ...ydras.solutions_exercises.FibonacciPR.html |   207 +
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 .../hydras.solutions_exercises.MinPR.html     |    65 +
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 ...as.solutions_exercises.MorePRExamples.html |   123 +
 ...hydras.solutions_exercises.MultisetWf.html |   190 +
 ...hydras.solutions_exercises.OnCodeList.html |    50 +
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 ...dras.solutions_exercises.ge_omega_iff.html |    43 +
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 ...s.solutions_exercises.predSuccUnicity.html |   109 +
 ...exercises.schutte_cnf_counter_example.html |    65 +
 theories/html/index.html                      | 58440 ++++++++++++++++
 theories/html/toc.html                        |  2100 +
 196 files changed, 114045 insertions(+)
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 create mode 100644 theories/html/additions.Strategies.html
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 create mode 100644 theories/html/additions.fib.html
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diff --git a/theories/html/Goedel.PRrepresentable.html b/theories/html/Goedel.PRrepresentable.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.PRrepresentable</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.PRrepresentable</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+    Original version by Russel O'Connor 
+
+<div class="paragraph"> </div>
+
+    This library is dedicated to the proof of the theorem:
+    "Every primitive recursive function is representable in <span class="inlinecode"><span class="id" title="var">NN</span></span>"
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> Let  <span class="inlinecode"><span class="id" title="var">f</span></span> be any <span class="inlinecode"><span class="id" title="var">n</span></span>-ary arithmetic function, and <span class="inlinecode"><span class="id" title="var">p</span></span> a proof that this function is primitive recursive.
+      Out of <span class="inlinecode"><span class="id" title="var">p</span></span>, we can associate a <span class="inlinecode"><span class="id" title="var">NN</span></span>-formula which correctly expresses <span class="inlinecode"><span class="id" title="var">f</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> The main difficulty is to associate a <span class="inlinecode"><span class="id" title="var">NN</span></span>-formula to the case where <span class="inlinecode"><span class="id" title="var">f</span></span> can be defined by a primitive 
+        recursive scheme. 
+        This is made possible with the help of Goedel's <span class="inlinecode"><span class="id" title="keyword">beta</span></span>-function, studied in the 
+         <a=href="https://github.com/thery/coqprime/blob/master/src/Coqprime/N/ChineseRem.v">ChineseRem</a>
+        library. This function allows 
+         to represent a sequence of computation steps through a first-order formula (theorem <span class="inlinecode"><span class="id" title="var">ChineseRem.BetaTheorem</span></span>).
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="primRecRepresentable" class="idref" href="#primRecRepresentable"><span class="id" title="lemma">primRecRepresentable</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:473" class="idref" href="#n:473"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:474" class="idref" href="#f:474"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a>) (<a id="p:475" class="idref" href="#p:475"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a>),<br/>
+&nbsp;<a class="idref" href="Goedel.PRrepresentable.html#Representable"><span class="id" title="definition">Representable</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a> (<a class="idref" href="Goedel.PRrepresentable.html#primRecFormula"><span class="id" title="definition">primRecFormula</span></a> <a class="idref" href="Goedel.PRrepresentable.html#n:473"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#fun2PR"><span class="id" title="definition">fun2PR</span></a> <a class="idref" href="Goedel.PRrepresentable.html#f:474"><span class="id" title="variable">f</span></a>)).<br/>
+
+<br/>
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.codeSysPrf</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.codeSysPrf</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+TO DO: Define abbreviations and re-indent !!! 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#"><span class="id" title="library">checkPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.PRrepresentable.html#"><span class="id" title="library">PRrepresentable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.expressible.html#"><span class="id" title="library">expressible</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PA.html#"><span class="id" title="library">PA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NNtheory.html#"><span class="id" title="library">NNtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeList.html#"><span class="id" title="library">codeList</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.wellFormed.html#"><span class="id" title="library">wellFormed</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.prLogic.html#"><span class="id" title="library">prLogic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="code_SysPrf" class="idref" href="#code_SysPrf"><span class="id" title="section">code_SysPrf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.L" class="idref" href="#code_SysPrf.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.codeArityF" class="idref" href="#code_SysPrf.codeArityF"><span class="id" title="variable">codeArityF</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.codeArityR" class="idref" href="#code_SysPrf.codeArityR"><span class="id" title="variable">codeArityR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityFIsPR:7" class="idref" href="#codeArityFIsPR:7"><span class="id" title="binder">codeArityFIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityF"><span class="id" title="variable">codeArityF</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.codeArityFIsCorrect1" class="idref" href="#code_SysPrf.codeArityFIsCorrect1"><span class="id" title="variable">codeArityFIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityF"><span class="id" title="variable">codeArityF</span></a> (<a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:8"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:8"><span class="id" title="variable">f</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.codeArityFIsCorrect2" class="idref" href="#code_SysPrf.codeArityFIsCorrect2"><span class="id" title="variable">codeArityFIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:11" class="idref" href="#f:11"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:11"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:10"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityRIsPR:13" class="idref" href="#codeArityRIsPR:13"><span class="id" title="binder">codeArityRIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityR"><span class="id" title="variable">codeArityR</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.codeArityRIsCorrect1" class="idref" href="#code_SysPrf.codeArityRIsCorrect1"><span class="id" title="variable">codeArityRIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="r:14" class="idref" href="#r:14"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityR"><span class="id" title="variable">codeArityR</span></a> (<a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="Goedel.codeSysPrf.html#r:14"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#r:14"><span class="id" title="variable">r</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.codeArityRIsCorrect2" class="idref" href="#code_SysPrf.codeArityRIsCorrect2"><span class="id" title="variable">codeArityRIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="r:17" class="idref" href="#r:17"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="Goedel.codeSysPrf.html#r:17"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:16"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="module">LNN</span></a> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="module">NN</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="code_SysPrf.LNN" class="idref" href="#code_SysPrf.LNN"><span class="id" title="section">LNN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.LNN.T" class="idref" href="#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="code_SysPrf.LNN.TextendsNN" class="idref" href="#code_SysPrf.LNN.TextendsNN"><span class="id" title="variable">TextendsNN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.LNN.U" class="idref" href="#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.LNN.fU" class="idref" href="#code_SysPrf.LNN.fU"><span class="id" title="variable">fU</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>. 
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="code_SysPrf.LNN.v0" class="idref" href="#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="code_SysPrf.LNN.freeVarfU" class="idref" href="#code_SysPrf.LNN.freeVarfU"><span class="id" title="variable">freeVarfU</span></a> : <span class="id" title="keyword">∀</span> <a id="v:24" class="idref" href="#v:24"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:24"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.fU"><span class="id" title="variable">fU</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:24"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.LNN.expressU1" class="idref" href="#code_SysPrf.LNN.expressU1"><span class="id" title="variable">expressU1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:26" class="idref" href="#f:26"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:26"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.fU"><span class="id" title="variable">fU</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:26"><span class="id" title="variable">f</span></a>))).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="code_SysPrf.LNN.expressU2" class="idref" href="#code_SysPrf.LNN.expressU2"><span class="id" title="variable">expressU2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:28" class="idref" href="#f:28"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:28"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.fU"><span class="id" title="variable">fU</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:28"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrf" class="idref" href="#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="nv:30" class="idref" href="#nv:30"><span class="id" title="binder">nv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Goedel.PRrepresentable.html#primRecFormula"><span class="id" title="definition">primRecFormula</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfIsPR"><span class="id" title="instance">checkPrfIsPR</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.cL"><span class="id" title="variable">cL</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityR"><span class="id" title="variable">codeArityR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityFIsPR"><span class="id" title="variable">codeArityFIsPR</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.codeArityRIsPR"><span class="id" title="variable">codeArityRIsPR</span></a>))) 0 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>))) 2 (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.LT"><span class="id" title="definition">LT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>)) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Goedel.PRrepresentable.html#primRecFormula"><span class="id" title="definition">primRecFormula</span></a> 2 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#codeInIsPR"><span class="id" title="instance">codeInIsPR</span></a>)) 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>))) 1 (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>)) 0 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.fU"><span class="id" title="variable">fU</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="Goedel.codeSysPrf.html#nv:30"><span class="id" title="variable">nv</span></a>))))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfCorrect1" class="idref" href="#codeSysPrfCorrect1"><span class="id" title="lemma">codeSysPrfCorrect1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:31" class="idref" href="#f:31"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>)) (<a id="p:33" class="idref" href="#p:33"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:31"><span class="id" title="variable">f</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:34" class="idref" href="#g:34"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:34"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:34"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:31"><span class="id" title="variable">f</span></a>))) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:31"><span class="id" title="variable">f</span></a> <a class="idref" href="Goedel.codeSysPrf.html#p:33"><span class="id" title="variable">p</span></a>))).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfCorrect2" class="idref" href="#codeSysPrfCorrect2"><span class="id" title="lemma">codeSysPrfCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:57" class="idref" href="#f:57"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="A:58" class="idref" href="#A:58"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:59" class="idref" href="#g:59"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:59"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:59"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:60" class="idref" href="#p:60"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:57"><span class="id" title="variable">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:57"><span class="id" title="variable">f</span></a>))) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:57"><span class="id" title="variable">f</span></a> <a class="idref" href="Goedel.codeSysPrf.html#p:60"><span class="id" title="variable">p</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfCorrect3" class="idref" href="#codeSysPrfCorrect3"><span class="id" title="lemma">codeSysPrfCorrect3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:75" class="idref" href="#f:75"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="A:77" class="idref" href="#A:77"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>)) (<a id="p:78" class="idref" href="#p:78"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:77"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:75"><span class="id" title="variable">f</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#n:76"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:77"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:75"><span class="id" title="variable">f</span></a> <a class="idref" href="Goedel.codeSysPrf.html#p:78"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:75"><span class="id" title="variable">f</span></a>))) 1 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:76"><span class="id" title="variable">n</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarCodeSysPrf" class="idref" href="#freeVarCodeSysPrf"><span class="id" title="lemma">freeVarCodeSysPrf</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="v:79" class="idref" href="#v:79"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:79"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:79"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPf" class="idref" href="#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarCodeSysPf" class="idref" href="#freeVarCodeSysPf"><span class="id" title="lemma">freeVarCodeSysPf</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="v:80" class="idref" href="#v:80"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:80"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:80"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPfCorrect" class="idref" href="#codeSysPfCorrect"><span class="id" title="lemma">codeSysPfCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:81" class="idref" href="#f:81"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:81"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:81"><span class="id" title="variable">f</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNot" class="idref" href="#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> 0 (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 2))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="Goedel.PRrepresentable.html#primRecFormula"><span class="id" title="definition">primRecFormula</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Ackermann.prLogic.html#codeNotIsPR"><span class="id" title="instance">codeNotIsPR</span></a>)) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 2)) 1 (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarCodeSysPrfN" class="idref" href="#freeVarCodeSysPrfN"><span class="id" title="lemma">freeVarCodeSysPrfN</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="v:82" class="idref" href="#v:82"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:82"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.codeSysPrf.html#v:82"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfNCorrect1" class="idref" href="#codeSysPrfNCorrect1"><span class="id" title="lemma">codeSysPrfNCorrect1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:83" class="idref" href="#f:83"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="A:84" class="idref" href="#A:84"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="p:85" class="idref" href="#p:85"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:84"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:83"><span class="id" title="variable">f</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:86" class="idref" href="#g:86"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:86"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:84"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:86"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:83"><span class="id" title="variable">f</span></a>))) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:84"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:83"><span class="id" title="variable">f</span></a>) <a class="idref" href="Goedel.codeSysPrf.html#p:85"><span class="id" title="variable">p</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfNCorrect2" class="idref" href="#codeSysPrfNCorrect2"><span class="id" title="lemma">codeSysPrfNCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:87" class="idref" href="#f:87"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:89" class="idref" href="#g:89"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:89"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U"><span class="id" title="variable">U</span></a> <a class="idref" href="Goedel.codeSysPrf.html#g:89"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:90" class="idref" href="#p:90"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:88"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:87"><span class="id" title="variable">f</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:87"><span class="id" title="variable">f</span></a>))) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:88"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:87"><span class="id" title="variable">f</span></a>) <a class="idref" href="Goedel.codeSysPrf.html#p:90"><span class="id" title="variable">p</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSysPrfNCorrect3" class="idref" href="#codeSysPrfNCorrect3"><span class="id" title="lemma">codeSysPrfNCorrect3</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:91" class="idref" href="#f:91"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="n:92" class="idref" href="#n:92"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="A:93" class="idref" href="#A:93"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a>) (<a id="p:94" class="idref" href="#p:94"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.L"><span class="id" title="variable">L</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:93"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:91"><span class="id" title="variable">f</span></a>)),<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#n:92"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#A:93"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:91"><span class="id" title="variable">f</span></a>) <a class="idref" href="Goedel.codeSysPrf.html#p:94"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.codeSysPrf.html#f:91"><span class="id" title="variable">f</span></a>))) 1 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="Goedel.codeSysPrf.html#n:92"><span class="id" title="variable">n</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf.LNN"><span class="id" title="section">LNN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.codeSysPrf.html#code_SysPrf"><span class="id" title="section">code_SysPrf</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/Goedel.fixPoint.html b/theories/html/Goedel.fixPoint.html
new file mode 100644
index 00000000..7c4600d9
--- /dev/null
+++ b/theories/html/Goedel.fixPoint.html
@@ -0,0 +1,101 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.fixPoint</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.fixPoint</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#"><span class="id" title="library">codeSubFormula</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="library">codeNatToTerm</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.PRrepresentable.html#"><span class="id" title="library">PRrepresentable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.expressible.html#"><span class="id" title="library">expressible</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="subStar" class="idref" href="#subStar"><span class="id" title="definition">subStar</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="v:2" class="idref" href="#v:2"><span class="id" title="binder">v</span></a> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> <a class="idref" href="Goedel.fixPoint.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="Goedel.fixPoint.html#v:2"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm"><span class="id" title="definition">codeNatToTerm</span></a> <a class="idref" href="Goedel.fixPoint.html#n:3"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="subStarIsPR" class="idref" href="#subStarIsPR"><span class="id" title="instance">subStarIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="Goedel.fixPoint.html#subStar"><span class="id" title="definition">subStar</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="LNN_FixPoint" class="idref" href="#LNN_FixPoint"><span class="id" title="section">LNN_FixPoint</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="LNN_FixPoint.codeFormula" class="idref" href="#LNN_FixPoint.codeFormula"><span class="id" title="variable">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FixPointLNN" class="idref" href="#FixPointLNN"><span class="id" title="lemma">FixPointLNN</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:22" class="idref" href="#A:22"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:23" class="idref" href="#v:23"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="B:24" class="idref" href="#B:24"><span class="id" title="binder">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Goedel.fixPoint.html#B:24"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.fixPoint.html#A:22"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.fixPoint.html#v:23"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> (<a class="idref" href="Goedel.fixPoint.html#LNN_FixPoint.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="Goedel.fixPoint.html#B:24"><span class="id" title="variable">B</span></a>)))%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:25" class="idref" href="#x:25"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.fixPoint.html#x:25"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.fixPoint.html#B:24"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.fixPoint.html#x:25"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="Goedel.fixPoint.html#v:23"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.fixPoint.html#A:22"><span class="id" title="variable">A</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.fixPoint.html#LNN_FixPoint"><span class="id" title="section">LNN_FixPoint</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PA.html#"><span class="id" title="library">PA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN2PA.html#"><span class="id" title="library">NN2PA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="LNT_FixPoint" class="idref" href="#LNT_FixPoint"><span class="id" title="section">LNT_FixPoint</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="LNT_FixPoint.codeFormula" class="idref" href="#LNT_FixPoint.codeFormula"><span class="id" title="variable">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FixPointLNT" class="idref" href="#FixPointLNT"><span class="id" title="lemma">FixPointLNT</span></a>  (<a id="A:26" class="idref" href="#A:26"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:27" class="idref" href="#v:27"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="B:28" class="idref" href="#B:28"><span class="id" title="binder">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="Goedel.fixPoint.html#B:28"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.fixPoint.html#A:26"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.fixPoint.html#v:27"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNT"><span class="id" title="definition">natToTermLNT</span></a> (<a class="idref" href="Goedel.fixPoint.html#LNT_FixPoint.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="Goedel.fixPoint.html#B:28"><span class="id" title="variable">B</span></a>)))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.fixPoint.html#x:29"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.fixPoint.html#B:28"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.fixPoint.html#x:29"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="Goedel.fixPoint.html#v:27"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.fixPoint.html#A:26"><span class="id" title="variable">A</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.fixPoint.html#LNT_FixPoint"><span class="id" title="section">LNT_FixPoint</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/Goedel.goedel1.html b/theories/html/Goedel.goedel1.html
new file mode 100644
index 00000000..2727755b
--- /dev/null
+++ b/theories/html/Goedel.goedel1.html
@@ -0,0 +1,139 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.goedel1</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.goedel1</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.fixPoint.html#"><span class="id" title="library">fixPoint</span></a> <a class="idref" href="Goedel.codeSysPrf.html#"><span class="id" title="library">codeSysPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.wConsistent.html#"><span class="id" title="library">wConsistent</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#"><span class="id" title="library">checkPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="module">codeNatToTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFNN" class="idref" href="#codeFNN"><span class="id" title="definition">codeFNN</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>) .<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="reflection" class="idref" href="#reflection"><span class="id" title="abbreviation">reflection</span></a> <span class="id" title="var">f</span> := (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.goedel1.html#codeFNN"><span class="id" title="definition">codeFNN</span></a> <span class="id" title="var">f</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Goedel's_1st_Incompleteness" class="idref" href="#Goedel's_1st_Incompleteness"><span class="id" title="section">Goedel's_1st_Incompleteness</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Goedel's_1st_Incompleteness.T" class="idref" href="#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Goedel's_1st_Incompleteness.extendsNN" class="idref" href="#Goedel's_1st_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Goedel's_1st_Incompleteness.repT" class="idref" href="#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Goedel's_1st_Incompleteness.v0" class="idref" href="#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Goedel's_1st_Incompleteness.freeVarRepT" class="idref" href="#Goedel's_1st_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> : <span class="id" title="keyword">∀</span> <a id="v:5" class="idref" href="#v:5"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.goedel1.html#v:5"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.goedel1.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Goedel's_1st_Incompleteness.expressT1" class="idref" href="#Goedel's_1st_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#f:7"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="Goedel.goedel1.html#reflection"><span class="id" title="abbreviation">reflection</span></a> <a class="idref" href="Goedel.goedel1.html#f:7"><span class="id" title="variable">f</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Goedel's_1st_Incompleteness.expressT2" class="idref" href="#Goedel's_1st_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="Goedel.goedel1.html#reflection"><span class="id" title="abbreviation">reflection</span></a> <a class="idref" href="Goedel.goedel1.html#f:9"><span class="id" title="variable">f</span></a>)<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrf" class="idref" href="#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPf" class="idref" href="#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="G" class="idref" href="#G"><span class="id" title="definition">G</span></a> := <span class="id" title="keyword">let</span> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>,<span class="id" title="var">_</span>) := <a class="idref" href="Goedel.fixPoint.html#FixPointLNN"><span class="id" title="lemma">FixPointLNN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.goedel1.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a>) 0 <span class="id" title="tactic">in</span> <a class="idref" href="Goedel.goedel1.html#a:11"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPfCorrect" class="idref" href="#codeSysPfCorrect"><span class="id" title="definition">codeSysPfCorrect</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPfCorrect"><span class="id" title="lemma">codeSysPfCorrect</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect2" class="idref" href="#codeSysPrfCorrect2"><span class="id" title="definition">codeSysPrfCorrect2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect2"><span class="id" title="lemma">codeSysPrfCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect3" class="idref" href="#codeSysPrfCorrect3"><span class="id" title="definition">codeSysPrfCorrect3</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect3"><span class="id" title="lemma">codeSysPrfCorrect3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect2"><span class="id" title="lemma">codeArityLNNRIsCorrect2</span></a>  <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarG" class="idref" href="#freeVarG"><span class="id" title="lemma">freeVarG</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#closed"><span class="id" title="definition">closed</span></a> <a class="idref" href="Goedel.goedel1.html#G"><span class="id" title="definition">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FirstIncompletenessA" class="idref" href="#FirstIncompletenessA"><span class="id" title="lemma">FirstIncompletenessA</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#G"><span class="id" title="definition">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FirstIncompletenessB" class="idref" href="#FirstIncompletenessB"><span class="id" title="lemma">FirstIncompletenessB</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wConsistent.html#wConsistent"><span class="id" title="definition">wConsistent</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.goedel1.html#G"><span class="id" title="definition">G</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Goedel'sIncompleteness1st" class="idref" href="#Goedel'sIncompleteness1st"><span class="id" title="lemma">Goedel'sIncompleteness1st</span></a> :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.wConsistent.html#wConsistent"><span class="id" title="definition">wConsistent</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#independent"><span class="id" title="definition">independent</span></a> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel1.html#f:24"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#closed"><span class="id" title="definition">closed</span></a> <a class="idref" href="Goedel.goedel1.html#f:24"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.goedel1.html#Goedel's_1st_Incompleteness"><span class="id" title="section">Goedel's_1st_Incompleteness</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.goedel2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.goedel2</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.fixPoint.html#"><span class="id" title="library">fixPoint</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN2PA.html#"><span class="id" title="library">NN2PA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.codeSysPrf.html#"><span class="id" title="library">codeSysPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PAtheory.html#"><span class="id" title="library">PAtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#"><span class="id" title="library">checkPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="library">codeNatToTerm</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.rosserPA.html#"><span class="id" title="library">rosserPA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Goedel's_2nd_Incompleteness" class="idref" href="#Goedel's_2nd_Incompleteness"><span class="id" title="section">Goedel's_2nd_Incompleteness</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Goedel's_2nd_Incompleteness.T" class="idref" href="#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Goedel's_2nd_Incompleteness.extendsPA" class="idref" href="#Goedel's_2nd_Incompleteness.extendsPA"><span class="id" title="variable">extendsPA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Goedel's_2nd_Incompleteness.repT" class="idref" href="#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Goedel's_2nd_Incompleteness.v0" class="idref" href="#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Goedel's_2nd_Incompleteness.freeVarRepT" class="idref" href="#Goedel's_2nd_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> : <span class="id" title="keyword">∀</span> <a id="v:5" class="idref" href="#v:5"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.goedel2.html#v:5"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.goedel2.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Goedel's_2nd_Incompleteness.expressT1" class="idref" href="#Goedel's_2nd_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#f:7"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.goedel2.html#f:7"><span class="id" title="variable">f</span></a>))).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Goedel's_2nd_Incompleteness.expressT2" class="idref" href="#Goedel's_2nd_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.goedel2.html#f:9"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="codeSysPf" class="idref" href="#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Goedel.codeSysPrf.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Goedel's_2nd_Incompleteness.Goedel1PA" class="idref" href="#Goedel's_2nd_Incompleteness.Goedel1PA"><span class="id" title="section">Goedel1PA</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="T'" class="idref" href="#T'"><span class="id" title="definition">T'</span></a> := <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="extendsNN" class="idref" href="#extendsNN"><span class="id" title="definition">extendsNN</span></a> := <a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Tprf2T'prf" class="idref" href="#Tprf2T'prf"><span class="id" title="definition">Tprf2T'prf</span></a> := <a class="idref" href="Goedel.rosserPA.html#Tprf2T'prf"><span class="id" title="lemma">Tprf2T'prf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="expressT'1" class="idref" href="#expressT'1"><span class="id" title="definition">expressT'1</span></a> := <a class="idref" href="Goedel.rosserPA.html#expressT'1"><span class="id" title="lemma">expressT'1</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="expressT'2" class="idref" href="#expressT'2"><span class="id" title="definition">expressT'2</span></a> := <a class="idref" href="Goedel.rosserPA.html#expressT'2"><span class="id" title="lemma">expressT'2</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="freeVarRepT'" class="idref" href="#freeVarRepT'"><span class="id" title="definition">freeVarRepT'</span></a> := <a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="codeSysPfCorrect" class="idref" href="#codeSysPfCorrect"><span class="id" title="definition">codeSysPfCorrect</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPfCorrect"><span class="id" title="lemma">codeSysPfCorrect</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> <a class="idref" href="Goedel.goedel2.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.goedel2.html#extendsNN"><span class="id" title="definition">extendsNN</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.goedel2.html#freeVarRepT'"><span class="id" title="definition">freeVarRepT'</span></a> <a class="idref" href="Goedel.goedel2.html#expressT'1"><span class="id" title="definition">expressT'1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="G" class="idref" href="#G"><span class="id" title="definition">G</span></a> := <span class="id" title="keyword">let</span> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>,<span class="id" title="var">_</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.fixPoint.html#FixPointLNT"><span class="id" title="lemma">FixPointLNT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="Goedel.goedel2.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a>)) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="Goedel.goedel2.html#a:11"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="box" class="idref" href="#box"><span class="id" title="definition">box</span></a> (<a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="Goedel.goedel2.html#codeSysPf"><span class="id" title="definition">codeSysPf</span></a>) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.goedel2.html#f:12"><span class="id" title="variable">f</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="GS" class="idref" href="#GS"><span class="id" title="lemma">GS</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="Goedel.goedel2.html#G"><span class="id" title="definition">G</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#G"><span class="id" title="definition">G</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HBL1" class="idref" href="#HBL1"><span class="id" title="lemma">HBL1</span></a> : <span class="id" title="keyword">∀</span> <a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a>, <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#f:13"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="FirstIncompletenessA" class="idref" href="#FirstIncompletenessA"><span class="id" title="lemma">FirstIncompletenessA</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#G"><span class="id" title="definition">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.Goedel1PA"><span class="id" title="section">Goedel1PA</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="F" class="idref" href="#F"><span class="id" title="definition">F</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<span class="id" title="var">L</span>:= <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Con" class="idref" href="#Con"><span class="id" title="definition">Con</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#F"><span class="id" title="definition">F</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Goedel's_2nd_Incompleteness.HBL2" class="idref" href="#Goedel's_2nd_Incompleteness.HBL2"><span class="id" title="variable">HBL2</span></a> : <span class="id" title="keyword">∀</span> <a id="f:22" class="idref" href="#f:22"><span class="id" title="binder">f</span></a>, <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#f:22"><span class="id" title="variable">f</span></a>) (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#f:22"><span class="id" title="variable">f</span></a>))).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Goedel's_2nd_Incompleteness.HBL3" class="idref" href="#Goedel's_2nd_Incompleteness.HBL3"><span class="id" title="variable">HBL3</span></a> : <span class="id" title="keyword">∀</span> <a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="Goedel.goedel2.html#f:24"><span class="id" title="variable">f</span></a> <a class="idref" href="Goedel.goedel2.html#g:25"><span class="id" title="variable">g</span></a>)) (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#f:24"><span class="id" title="variable">f</span></a>) (<a class="idref" href="Goedel.goedel2.html#box"><span class="id" title="definition">box</span></a> <a class="idref" href="Goedel.goedel2.html#g:25"><span class="id" title="variable">g</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="SecondIncompletness" class="idref" href="#SecondIncompletness"><span class="id" title="lemma">SecondIncompletness</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.goedel2.html#Con"><span class="id" title="definition">Con</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness"><span class="id" title="section">Goedel's_2nd_Incompleteness</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.rosser</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.rosser</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.fixPoint.html#"><span class="id" title="library">fixPoint</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.codeSysPrf.html#"><span class="id" title="library">codeSysPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NNtheory.html#"><span class="id" title="library">NNtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.PRrepresentable.html#"><span class="id" title="library">PRrepresentable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.expressible.html#"><span class="id" title="library">expressible</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#"><span class="id" title="library">checkPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="library">codeNatToTerm</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="module">LNN</span></a> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="module">NN</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Rosser's_Incompleteness" class="idref" href="#Rosser's_Incompleteness"><span class="id" title="section">Rosser's_Incompleteness</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFormula" class="idref" href="#codeFormula"><span class="id" title="definition">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.T" class="idref" href="#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Rosser's_Incompleteness.extendsNN" class="idref" href="#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.repT" class="idref" href="#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.v0" class="idref" href="#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.freeVarRepT" class="idref" href="#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> : <span class="id" title="keyword">∀</span> <a id="v:5" class="idref" href="#v:5"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#v:5"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.rosser.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.expressT1" class="idref" href="#Rosser's_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:7"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosser.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosser.html#f:7"><span class="id" title="variable">f</span></a>))).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.expressT2" class="idref" href="#Rosser's_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosser.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosser.html#f:9"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrf" class="idref" href="#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect1" class="idref" href="#codeSysPrfCorrect1"><span class="id" title="definition">codeSysPrfCorrect1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect1"><span class="id" title="lemma">codeSysPrfCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect2" class="idref" href="#codeSysPrfCorrect2"><span class="id" title="definition">codeSysPrfCorrect2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect2"><span class="id" title="lemma">codeSysPrfCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect3" class="idref" href="#codeSysPrfCorrect3"><span class="id" title="definition">codeSysPrfCorrect3</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect3"><span class="id" title="lemma">codeSysPrfCorrect3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>   <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect2"><span class="id" title="lemma">codeArityLNNRIsCorrect2</span></a>  <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codePrf" class="idref" href="#codePrf"><span class="id" title="definition">codePrf</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNot" class="idref" href="#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="freeVarCodeSysPrfN" class="idref" href="#freeVarCodeSysPrfN"><span class="id" title="definition">freeVarCodeSysPrfN</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#freeVarCodeSysPrfN"><span class="id" title="lemma">freeVarCodeSysPrfN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect1" class="idref" href="#codeSysPrfNCorrect1"><span class="id" title="definition">codeSysPrfNCorrect1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect1"><span class="id" title="lemma">codeSysPrfNCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect2" class="idref" href="#codeSysPrfNCorrect2"><span class="id" title="definition">codeSysPrfNCorrect2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect2"><span class="id" title="lemma">codeSysPrfNCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect3" class="idref" href="#codeSysPrfNCorrect3"><span class="id" title="definition">codeSysPrfNCorrect3</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect3"><span class="id" title="lemma">codeSysPrfNCorrect3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect2"><span class="id" title="lemma">codeArityLNNRIsCorrect2</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN"><span class="id" title="variable">extendsNN</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decideAxioms" class="idref" href="#decideAxioms"><span class="id" title="lemma">decideAxioms</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:11"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:11"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formulas"><span class="id" title="definition">Formulas</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:13" class="idref" href="#g:13"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#g:13"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosser.html#x:12"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#g:13"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:14" class="idref" href="#g:14"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#g:14"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosser.html#x:12"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#g:14"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="searchProof" class="idref" href="#searchProof"><span class="id" title="lemma">searchProof</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:15"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a> <a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formulas"><span class="id" title="definition">Formulas</span></a>) (<a id="p:19" class="idref" href="#p:19"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosser.html#A:18"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.rosser.html#a:16"><span class="id" title="variable">a</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="B:20" class="idref" href="#B:20"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formulas"><span class="id" title="definition">Formulas</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="q:21" class="idref" href="#q:21"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosser.html#B:20"><span class="id" title="variable">B</span></a> <a class="idref" href="Goedel.rosser.html#b:17"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosser.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#q:21"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="Goedel.rosser.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#p:19"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="Goedel.rosser.html#B:20"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:22"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="B:23" class="idref" href="#B:23"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formulas"><span class="id" title="definition">Formulas</span></a>) (<a id="q:24" class="idref" href="#q:24"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosser.html#B:23"><span class="id" title="variable">B</span></a> <a class="idref" href="Goedel.rosser.html#b:17"><span class="id" title="variable">b</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosser.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#q:24"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="Goedel.rosser.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#p:19"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosser.html#B:23"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#g:25"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+To prove the strong contructive result we need the decidability of T 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Rosser'sIncompleteness" class="idref" href="#Rosser'sIncompleteness"><span class="id" title="lemma">Rosser'sIncompleteness</span></a> :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#x:26"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:28" class="idref" href="#v:28"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#v:28"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.rosser.html#f:27"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.rosser.html#f:27"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.rosser.html#Rosser's_Incompleteness"><span class="id" title="section">Rosser's_Incompleteness</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="RepresentsInSelf" class="idref" href="#RepresentsInSelf"><span class="id" title="definition">RepresentsInSelf</span></a> (<a id="T:65" class="idref" href="#T:65"><span class="id" title="binder">T</span></a>:<a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="rep:66" class="idref" href="#rep:66"><span class="id" title="binder">rep</span></a>:<a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="v:67" class="idref" href="#v:67"><span class="id" title="binder">v</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:68" class="idref" href="#x:68"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosser.html#x:68"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.rosser.html#rep:66"><span class="id" title="variable">rep</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.rosser.html#x:68"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.rosser.html#v:67"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:69" class="idref" href="#f:69"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="Goedel.rosser.html#T:65"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:69"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#T:65"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.rosser.html#rep:66"><span class="id" title="variable">rep</span></a> <a class="idref" href="Goedel.rosser.html#v:67"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosser.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosser.html#f:69"><span class="id" title="variable">f</span></a>)))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:70" class="idref" href="#f:70"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="Goedel.rosser.html#T:65"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#T:65"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.rosser.html#rep:66"><span class="id" title="variable">rep</span></a> <a class="idref" href="Goedel.rosser.html#v:67"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosser.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosser.html#f:70"><span class="id" title="variable">f</span></a>))))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="DecidableSet" class="idref" href="#DecidableSet"><span class="id" title="definition">DecidableSet</span></a> (<a id="A:71" class="idref" href="#A:71"><span class="id" title="binder">A</span></a>:<span class="id" title="var">_</span>)(<a id="s:72" class="idref" href="#s:72"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="Goedel.rosser.html#A:71"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="x:73" class="idref" href="#x:73"><span class="id" title="binder">x</span></a> : <a class="idref" href="Goedel.rosser.html#A:71"><span class="id" title="variable">A</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="Goedel.rosser.html#A:71"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.rosser.html#s:72"><span class="id" title="variable">s</span></a> <a class="idref" href="Goedel.rosser.html#x:73"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="Goedel.rosser.html#A:71"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.rosser.html#s:72"><span class="id" title="variable">s</span></a> <a class="idref" href="Goedel.rosser.html#x:73"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Incompleteness" class="idref" href="#Incompleteness"><span class="id" title="lemma">Incompleteness</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="T:74" class="idref" href="#T:74"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosser.html#RepresentsInSelf"><span class="id" title="definition">RepresentsInSelf</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosser.html#DecidableSet"><span class="id" title="definition">DecidableSet</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:75" class="idref" href="#f:75"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#Sentence"><span class="id" title="definition">Sentence</span></a> <a class="idref" href="Goedel.rosser.html#f:75"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosser.html#f:75"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.rosser.html#f:75"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosser.html#T:74"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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\ No newline at end of file
diff --git a/theories/html/Goedel.rosserPA.html b/theories/html/Goedel.rosserPA.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/Goedel.rosserPA.html
@@ -0,0 +1,222 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Goedel.rosserPA</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library Goedel.rosserPA</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NNtheory.html#"><span class="id" title="library">NNtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN2PA.html#"><span class="id" title="library">NN2PA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.fixPoint.html#"><span class="id" title="library">fixPoint</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.codeSysPrf.html#"><span class="id" title="library">codeSysPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PAtheory.html#"><span class="id" title="library">PAtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Goedel</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Goedel.PRrepresentable.html#"><span class="id" title="library">PRrepresentable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.expressible.html#"><span class="id" title="library">expressible</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#"><span class="id" title="library">checkPrf</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="library">codeNatToTerm</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Rosser's_Incompleteness" class="idref" href="#Rosser's_Incompleteness"><span class="id" title="section">Rosser's_Incompleteness</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="codeFormula" class="idref" href="#codeFormula"><span class="id" title="definition">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.T" class="idref" href="#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="T'" class="idref" href="#T'"><span class="id" title="definition">T'</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Union"><span class="id" title="inductive">Union</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:2" class="idref" href="#x:2"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#x:2"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extendsNN" class="idref" href="#extendsNN"><span class="id" title="lemma">extendsNN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Rosser's_Incompleteness.extendsPA" class="idref" href="#Rosser's_Incompleteness.extendsPA"><span class="id" title="variable">extendsPA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.repT" class="idref" href="#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Rosser's_Incompleteness.v0" class="idref" href="#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.freeVarRepT" class="idref" href="#Rosser's_Incompleteness.freeVarRepT"><span class="id" title="variable">freeVarRepT</span></a> : <span class="id" title="keyword">∀</span> <a id="v:6" class="idref" href="#v:6"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#v:6"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.rosserPA.html#v:6"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.expressT1" class="idref" href="#Rosser's_Incompleteness.expressT1"><span class="id" title="variable">expressT1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:8"><span class="id" title="variable">f</span></a>))).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Rosser's_Incompleteness.expressT2" class="idref" href="#Rosser's_Incompleteness.expressT2"><span class="id" title="variable">expressT2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:10"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarRepT'" class="idref" href="#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="v:12" class="idref" href="#v:12"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#v:12"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Goedel.rosserPA.html#v:12"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Tprf2T'prf" class="idref" href="#Tprf2T'prf"><span class="id" title="lemma">Tprf2T'prf</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:13"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="expressT'1" class="idref" href="#expressT'1"><span class="id" title="lemma">expressT'1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:20" class="idref" href="#f:20"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:20"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:20"><span class="id" title="variable">f</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="expressT'2" class="idref" href="#expressT'2"><span class="id" title="lemma">expressT'2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> (<a class="idref" href="Goedel.rosserPA.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:21"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrf" class="idref" href="#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrf"><span class="id" title="definition">codeSysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect1" class="idref" href="#codeSysPrfCorrect1"><span class="id" title="definition">codeSysPrfCorrect1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect1"><span class="id" title="lemma">codeSysPrfCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> <a class="idref" href="Goedel.rosserPA.html#expressT'1"><span class="id" title="lemma">expressT'1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect2" class="idref" href="#codeSysPrfCorrect2"><span class="id" title="definition">codeSysPrfCorrect2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect2"><span class="id" title="lemma">codeSysPrfCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> <a class="idref" href="Goedel.rosserPA.html#expressT'2"><span class="id" title="lemma">expressT'2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfCorrect3" class="idref" href="#codeSysPrfCorrect3"><span class="id" title="definition">codeSysPrfCorrect3</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfCorrect3"><span class="id" title="lemma">codeSysPrfCorrect3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect2"><span class="id" title="lemma">codeArityLNTRIsCorrect2</span></a>  <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codePrf" class="idref" href="#codePrf"><span class="id" title="definition">codePrf</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNot" class="idref" href="#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNot"><span class="id" title="definition">codeSysPrfNot</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="freeVarCodeSysPrfN" class="idref" href="#freeVarCodeSysPrfN"><span class="id" title="definition">freeVarCodeSysPrfN</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#freeVarCodeSysPrfN"><span class="id" title="lemma">freeVarCodeSysPrfN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect1" class="idref" href="#codeSysPrfNCorrect1"><span class="id" title="definition">codeSysPrfNCorrect1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect1"><span class="id" title="lemma">codeSysPrfNCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> <a class="idref" href="Goedel.rosserPA.html#expressT'1"><span class="id" title="lemma">expressT'1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect2" class="idref" href="#codeSysPrfNCorrect2"><span class="id" title="definition">codeSysPrfNCorrect2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect2"><span class="id" title="lemma">codeSysPrfNCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a> <a class="idref" href="Goedel.rosserPA.html#expressT'2"><span class="id" title="lemma">expressT'2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSysPrfNCorrect3" class="idref" href="#codeSysPrfNCorrect3"><span class="id" title="definition">codeSysPrfNCorrect3</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="Goedel.codeSysPrf.html#codeSysPrfNCorrect3"><span class="id" title="lemma">codeSysPrfNCorrect3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect2"><span class="id" title="lemma">codeArityLNTRIsCorrect2</span></a>  <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#extendsNN"><span class="id" title="lemma">extendsNN</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT"><span class="id" title="variable">repT</span></a>) <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0"><span class="id" title="variable">v0</span></a> <a class="idref" href="Goedel.rosserPA.html#freeVarRepT'"><span class="id" title="lemma">freeVarRepT'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decideAxioms" class="idref" href="#decideAxioms"><span class="id" title="lemma">decideAxioms</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:22"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formulas"><span class="id" title="definition">Formulas</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:24" class="idref" href="#g:24"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#g:24"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosserPA.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#g:24"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosserPA.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#g:25"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="searchProof" class="idref" href="#searchProof"><span class="id" title="lemma">searchProof</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:26"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="b:28" class="idref" href="#b:28"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="A:29" class="idref" href="#A:29"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formulas"><span class="id" title="definition">Formulas</span></a>) (<a id="p:30" class="idref" href="#p:30"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.rosserPA.html#A:29"><span class="id" title="variable">A</span></a> <a class="idref" href="Goedel.rosserPA.html#a:27"><span class="id" title="variable">a</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="B:31" class="idref" href="#B:31"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formulas"><span class="id" title="definition">Formulas</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="q:32" class="idref" href="#q:32"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.rosserPA.html#B:31"><span class="id" title="variable">B</span></a> <a class="idref" href="Goedel.rosserPA.html#b:28"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#q:32"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="Goedel.rosserPA.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#p:30"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:33" class="idref" href="#x:33"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#x:33"><span class="id" title="variable">x</span></a> <a class="idref" href="Goedel.rosserPA.html#B:31"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:33"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="B:34" class="idref" href="#B:34"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formulas"><span class="id" title="definition">Formulas</span></a>) (<a id="q:35" class="idref" href="#q:35"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.rosserPA.html#B:34"><span class="id" title="variable">B</span></a> <a class="idref" href="Goedel.rosserPA.html#b:28"><span class="id" title="variable">b</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Goedel.rosserPA.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#q:35"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="Goedel.rosserPA.html#codePrf"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#p:30"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:36" class="idref" href="#g:36"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#g:36"><span class="id" title="variable">g</span></a> <a class="idref" href="Goedel.rosserPA.html#B:34"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#g:36"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T'prf2Tprf" class="idref" href="#T'prf2Tprf"><span class="id" title="lemma">T'prf2Tprf</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="Goedel.rosserPA.html#T'"><span class="id" title="definition">T'</span></a> <a class="idref" href="Goedel.rosserPA.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="Goedel.rosserPA.html#f:37"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Rosser'sIncompleteness" class="idref" href="#Rosser'sIncompleteness"><span class="id" title="lemma">Rosser'sIncompleteness</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:56" class="idref" href="#x:56"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:56"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#x:56"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:57" class="idref" href="#f:57"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:58" class="idref" href="#v:58"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="Goedel.rosserPA.html#v:58"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="Goedel.rosserPA.html#f:57"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> <a class="idref" href="Goedel.rosserPA.html#f:57"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.rosserPA.html#f:57"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness.T"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="Goedel.rosserPA.html#Rosser's_Incompleteness"><span class="id" title="section">Rosser's_Incompleteness</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.codePA.html#"><span class="id" title="library">codePA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PAconsistent.html#"><span class="id" title="library">PAconsistent</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="PAIncomplete" class="idref" href="#PAIncomplete"><span class="id" title="lemma">PAIncomplete</span></a> :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:109" class="idref" href="#f:109"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#Sentence"><span class="id" title="definition">Sentence</span></a> <a class="idref" href="Goedel.rosserPA.html#f:109"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="Goedel.rosserPA.html#f:109"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="Goedel.rosserPA.html#f:109"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.AM.html b/theories/html/additions.AM.html
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+++ b/theories/html/additions.AM.html
@@ -0,0 +1,528 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.AM</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.AM</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Abstract machine for following an Euclidean addition chain  adapted from the old contrib coq-additions 
+<div class="paragraph"> </div>
+
+ Work in progress 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.PArith.html#"><span class="id" title="library">PArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="module">Morphisms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a> <a class="idref" href="additions.Strategies.html#"><span class="id" title="library">Strategies</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="module">Monoid_def</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#"><span class="id" title="library">Euclidean_Chains</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Dichotomy.html#"><span class="id" title="library">Dichotomy</span></a> <a class="idref" href="additions.BinaryStrat.html#"><span class="id" title="library">BinaryStrat</span></a>.<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+basic instructions 
+</div>
+<div class="code">
+<span class="id" title="keyword">Inductive</span> <a id="instr" class="idref" href="#instr"><span class="id" title="inductive">instr</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+| <a id="MUL" class="idref" href="#MUL"><span class="id" title="constructor">MUL</span></a> : <a class="idref" href="additions.AM.html#instr:1"><span class="id" title="inductive">instr</span></a><br/>
+| <a id="SQR" class="idref" href="#SQR"><span class="id" title="constructor">SQR</span></a> : <a class="idref" href="additions.AM.html#instr:1"><span class="id" title="inductive">instr</span></a><br/>
+| <a id="PUSH" class="idref" href="#PUSH"><span class="id" title="constructor">PUSH</span></a> : <a class="idref" href="additions.AM.html#instr:1"><span class="id" title="inductive">instr</span></a><br/>
+| <a id="SWAP" class="idref" href="#SWAP"><span class="id" title="constructor">SWAP</span></a> : <a class="idref" href="additions.AM.html#instr:1"><span class="id" title="inductive">instr</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="code" class="idref" href="#code"><span class="id" title="definition">code</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="additions.AM.html#instr"><span class="id" title="inductive">instr</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="mults_squares" class="idref" href="#mults_squares"><span class="id" title="definition">mults_squares</span></a>  <a id="c:3" class="idref" href="#c:3"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="count:9" class="idref" href="#count:9"><span class="id" title="binder">count</span></a> (<a id="c:4" class="idref" href="#c:4"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) <a id="mults:5" class="idref" href="#mults:5"><span class="id" title="binder">mults</span></a> <a id="squares:6" class="idref" href="#squares:6"><span class="id" title="binder">squares</span></a>  :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#c:4"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#mults:5"><span class="id" title="variable">mults</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#squares:6"><span class="id" title="variable">squares</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">c'</span> | <a class="idref" href="additions.AM.html#SWAP"><span class="id" title="constructor">SWAP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">c'</span> ⇒ <a class="idref" href="additions.AM.html#count:7"><span class="id" title="variable">count</span></a> <span class="id" title="var">c'</span> <a class="idref" href="additions.AM.html#mults:5"><span class="id" title="variable">mults</span></a> <a class="idref" href="additions.AM.html#squares:6"><span class="id" title="variable">squares</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">c'</span> ⇒ <a class="idref" href="additions.AM.html#count:7"><span class="id" title="variable">count</span></a> <span class="id" title="var">c'</span> <a class="idref" href="additions.AM.html#mults:5"><span class="id" title="variable">mults</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.AM.html#squares:6"><span class="id" title="variable">squares</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">c'</span> ⇒ <a class="idref" href="additions.AM.html#count:7"><span class="id" title="variable">count</span></a> <span class="id" title="var">c'</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.AM.html#mults:5"><span class="id" title="variable">mults</span></a>) <a class="idref" href="additions.AM.html#squares:6"><span class="id" title="variable">squares</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>  <span class="id" title="tactic">in</span> <a class="idref" href="additions.AM.html#count:9"><span class="id" title="variable">count</span></a> <a class="idref" href="additions.AM.html#c:3"><span class="id" title="variable">c</span></a> 0%<span class="id" title="var">nat</span> 0%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Semantics" class="idref" href="#Semantics"><span class="id" title="section">Semantics</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Semantics.A" class="idref" href="#Semantics.A"><span class="id" title="variable">A</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Semantics.mul" class="idref" href="#Semantics.mul"><span class="id" title="variable">mul</span></a> : <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Semantics.one" class="idref" href="#Semantics.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="stack" class="idref" href="#stack"><span class="id" title="definition">stack</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="config" class="idref" href="#config"><span class="id" title="definition">config</span></a> := (<a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a>)%<span class="id" title="keyword">type</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="exec" class="idref" href="#exec"><span class="id" title="definition">exec</span></a> (<a id="c:13" class="idref" href="#c:13"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) (<a id="x:14" class="idref" href="#x:14"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.AM.html#Semantics.A"><span class="id" title="variable">A</span></a>) (<a id="s:15" class="idref" href="#s:15"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.AM.html#stack"><span class="id" title="definition">stack</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="additions.AM.html#config"><span class="id" title="definition">config</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#c:13"><span class="id" title="variable">c</span></a>, <a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">c</span>, <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s</span> ⇒ <a class="idref" href="additions.AM.html#exec:16"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c:13"><span class="id" title="variable">c</span></a> (<a class="idref" href="additions.AM.html#Semantics.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span>) <a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">c</span>, <span class="id" title="var">s</span> ⇒ <a class="idref" href="additions.AM.html#exec:16"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c:13"><span class="id" title="variable">c</span></a> (<a class="idref" href="additions.AM.html#Semantics.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">c</span>, <span class="id" title="var">s</span> ⇒ <a class="idref" href="additions.AM.html#exec:16"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c:13"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a> (<a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.AM.html#SWAP"><span class="id" title="constructor">SWAP</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">c</span>, <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s</span> ⇒ <a class="idref" href="additions.AM.html#exec:16"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c:13"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:14"><span class="id" title="variable">x</span></a> (<span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#s:15"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>,<span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="exec_app" class="idref" href="#exec_app"><span class="id" title="lemma">exec_app</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="c:19" class="idref" href="#c:19"><span class="id" title="binder">c</span></a> <a id="c':20" class="idref" href="#c':20"><span class="id" title="binder">c'</span></a> : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) <a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a> <a id="s:22" class="idref" href="#s:22"><span class="id" title="binder">s</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> (<a class="idref" href="additions.AM.html#c:19"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="additions.AM.html#c':20"><span class="id" title="variable">c'</span></a>) <a class="idref" href="additions.AM.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:22"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c:19"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:22"><span class="id" title="variable">s</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#c':20"><span class="id" title="variable">c'</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Main well-formed chains 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="F1" class="idref" href="#F1"><span class="id" title="definition">F1</span></a> : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+raises x to its cube 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F3" class="idref" href="#F3"><span class="id" title="definition">F3</span></a> := <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+chain for raising x to its (2 ^ q)th power 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="F2q_of_nat" class="idref" href="#F2q_of_nat"><span class="id" title="definition">F2q_of_nat</span></a> <a id="q:23" class="idref" href="#q:23"><span class="id" title="binder">q</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#q:23"><span class="id" title="variable">q</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0%<span class="id" title="var">nat</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.AM.html#F2q_of_nat:24"><span class="id" title="definition">F2q_of_nat</span></a>  <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F2q" class="idref" href="#F2q"><span class="id" title="definition">F2q</span></a> (<a id="p:26" class="idref" href="#p:26"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.AM.html#F2q_of_nat"><span class="id" title="definition">F2q_of_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.AM.html#p:26"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+for computing x^(pq+r) passing by  x^p 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="KFF" class="idref" href="#KFF"><span class="id" title="definition">KFF</span></a> (<a id="kpr:27" class="idref" href="#kpr:27"><span class="id" title="binder">kpr</span></a> <a id="mq:28" class="idref" href="#mq:28"><span class="id" title="binder">mq</span></a>:<a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.AM.html#kpr:27"><span class="id" title="variable">kpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++(</span></a><a class="idref" href="additions.AM.html#mq:28"><span class="id" title="variable">mq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+for computing x^p and x^(pq) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="FFK" class="idref" href="#FFK"><span class="id" title="definition">FFK</span></a> (<a id="mp:29" class="idref" href="#mp:29"><span class="id" title="binder">mp</span></a> <a id="mq:30" class="idref" href="#mq:30"><span class="id" title="binder">mq</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) := <a class="idref" href="additions.AM.html#mp:29"><span class="id" title="variable">mp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.AM.html#mq:30"><span class="id" title="variable">mq</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+for computing x^p then x^(pq + r)  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="KFK" class="idref" href="#KFK"><span class="id" title="definition">KFK</span></a> (<a id="kpr:31" class="idref" href="#kpr:31"><span class="id" title="binder">kpr</span></a> <a id="mq:32" class="idref" href="#mq:32"><span class="id" title="binder">mq</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.AM.html#kpr:31"><span class="id" title="variable">kpr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#SWAP"><span class="id" title="constructor">SWAP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#mq:32"><span class="id" title="variable">mq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="FK" class="idref" href="#FK"><span class="id" title="definition">FK</span></a> (<a id="fn:33" class="idref" href="#fn:33"><span class="id" title="binder">fn</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) := <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#fn:33"><span class="id" title="variable">fn</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#Semantics"><span class="id" title="section">Semantics</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_apply" class="idref" href="#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a id="c:34" class="idref" href="#c:34"><span class="id" title="binder">c</span></a> {<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="op:36" class="idref" href="#op:36"><span class="id" title="binder">op</span></a>:<a class="idref" href="additions.AM.html#A:35"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="additions.AM.html#A:35"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="additions.AM.html#A:35"><span class="id" title="variable">A</span></a>}{<a id="one:37" class="idref" href="#one:37"><span class="id" title="binder">one</span></a>:<a class="idref" href="additions.AM.html#A:35"><span class="id" title="variable">A</span></a>}{<a id="equ:38" class="idref" href="#equ:38"><span class="id" title="binder">equ</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.AM.html#A:35"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M:39" class="idref" href="#M:39"><span class="id" title="binder">M</span></a>: <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#op:36"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#one:37"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.AM.html#equ:38"><span class="id" title="variable">equ</span></a>) <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a><br/>
+&nbsp;&nbsp;:= <a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <span class="id" title="var">_</span> <a class="idref" href="additions.AM.html#op:36"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#c:34"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Example code for 7 via 3 
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span>  <a id="M7_3" class="idref" href="#M7_3"><span class="id" title="definition">M7_3</span></a> := <a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#PUSH"><span class="id" title="constructor">PUSH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#SQR"><span class="id" title="constructor">SQR</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#SWAP"><span class="id" title="constructor">SWAP</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#MUL"><span class="id" title="constructor">MUL</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Example code for 31 via 7 
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="C31_7" class="idref" href="#C31_7"><span class="id" title="definition">C31_7</span></a> := <a class="idref" href="additions.AM.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.AM.html#M7_3"><span class="id" title="definition">M7_3</span></a> (<a class="idref" href="additions.AM.html#F2q"><span class="id" title="definition">F2q</span></a> 2).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+24, 3 
+<div class="paragraph"> </div>
+
+ For 99 and 24 
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="K99_24" class="idref" href="#K99_24"><span class="id" title="definition">K99_24</span></a> := <a class="idref" href="additions.AM.html#KFK"><span class="id" title="definition">KFK</span></a> (<a class="idref" href="additions.AM.html#FFK"><span class="id" title="definition">FFK</span></a> <a class="idref" href="additions.AM.html#F3"><span class="id" title="definition">F3</span></a> (<a class="idref" href="additions.AM.html#F2q"><span class="id" title="definition">F2q</span></a> 3)) (<a class="idref" href="additions.AM.html#F2q"><span class="id" title="definition">F2q</span></a> 2).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Specification and generation of correct chains 
+<div class="paragraph"> </div>
+
+ We have to build equivalences between configurations 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="stack_equiv" class="idref" href="#stack_equiv"><span class="id" title="inductive">stack_equiv</span></a> {<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a>}`{<a id="M:45" class="idref" href="#M:45"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:41"><span class="id" title="variable">A</span></a> <a id="op:42" class="idref" href="#op:42"><span class="id" title="binder">op</span></a> <a id="one:43" class="idref" href="#one:43"><span class="id" title="binder">one</span></a> <a id="equ:44" class="idref" href="#equ:44"><span class="id" title="binder">equ</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span><br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;<a id="stack_equiv0" class="idref" href="#stack_equiv0"><span class="id" title="constructor">stack_equiv0</span></a> : <a class="idref" href="additions.AM.html#stack_equiv:46"><span class="id" title="inductive">stack_equiv</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <a id="stack_equivn" class="idref" href="#stack_equivn"><span class="id" title="constructor">stack_equivn</span></a>: <span class="id" title="keyword">∀</span> <a id="x:48" class="idref" href="#x:48"><span class="id" title="binder">x</span></a> <a id="y:49" class="idref" href="#y:49"><span class="id" title="binder">y</span></a> <a id="s:50" class="idref" href="#s:50"><span class="id" title="binder">s</span></a> <a id="s':51" class="idref" href="#s':51"><span class="id" title="binder">s'</span></a>,  <a class="idref" href="additions.AM.html#x:48"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.AM.html#y:49"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.AM.html#stack_equiv:46"><span class="id" title="inductive">stack_equiv</span></a> <a class="idref" href="additions.AM.html#s:50"><span class="id" title="variable">s</span></a> <a class="idref" href="additions.AM.html#s':51"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#stack_equiv:46"><span class="id" title="inductive">stack_equiv</span></a> (<a class="idref" href="additions.AM.html#x:48"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#s:50"><span class="id" title="variable">s</span></a>) (<a class="idref" href="additions.AM.html#y:49"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.AM.html#s':51"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="config_equiv" class="idref" href="#config_equiv"><span class="id" title="definition">config_equiv</span></a>  {<a id="A:52" class="idref" href="#A:52"><span class="id" title="binder">A</span></a>}`{<a id="M:56" class="idref" href="#M:56"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:52"><span class="id" title="variable">A</span></a> <a id="op:53" class="idref" href="#op:53"><span class="id" title="binder">op</span></a> <a id="one:54" class="idref" href="#one:54"><span class="id" title="binder">one</span></a> <a id="equ:55" class="idref" href="#equ:55"><span class="id" title="binder">equ</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="c:57" class="idref" href="#c:57"><span class="id" title="binder">c</span></a> <a id="c':58" class="idref" href="#c':58"><span class="id" title="binder">c'</span></a> : <a class="idref" href="additions.AM.html#config"><span class="id" title="definition">config</span></a> <a class="idref" href="additions.AM.html#A:52"><span class="id" title="variable">A</span></a>): <span class="id" title="keyword">Prop</span> := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="additions.AM.html#c:57"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="additions.AM.html#c':58"><span class="id" title="variable">c'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="additions.AM.html#stack_equiv"><span class="id" title="inductive">stack_equiv</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="additions.AM.html#c:57"><span class="id" title="variable">c</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="additions.AM.html#c':58"><span class="id" title="variable">c'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="result_equiv" class="idref" href="#result_equiv"><span class="id" title="inductive">result_equiv</span></a>`{<a id="M:63" class="idref" href="#M:63"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:59" class="idref" href="#A:59"><span class="id" title="binder">A</span></a> <a id="op:60" class="idref" href="#op:60"><span class="id" title="binder">op</span></a> <a id="one:61" class="idref" href="#one:61"><span class="id" title="binder">one</span></a> <a id="equ:62" class="idref" href="#equ:62"><span class="id" title="binder">equ</span></a>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="additions.AM.html#config"><span class="id" title="definition">config</span></a> <span class="id" title="var">A</span>)):= <br/>
+&nbsp;&nbsp;<a id="result_equiv_fail" class="idref" href="#result_equiv_fail"><span class="id" title="constructor">result_equiv_fail</span></a> : <a class="idref" href="additions.AM.html#result_equiv:64"><span class="id" title="inductive">result_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+| <a id="result_equiv_success" class="idref" href="#result_equiv_success"><span class="id" title="constructor">result_equiv_success</span></a> : <span class="id" title="keyword">∀</span> <a id="x:66" class="idref" href="#x:66"><span class="id" title="binder">x</span></a> <a id="s:67" class="idref" href="#s:67"><span class="id" title="binder">s</span></a> <a id="y:68" class="idref" href="#y:68"><span class="id" title="binder">y</span></a> <a id="s':69" class="idref" href="#s':69"><span class="id" title="binder">s'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#config_equiv"><span class="id" title="definition">config_equiv</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#x:66"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.AM.html#s:67"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#y:68"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#s':69"><span class="id" title="variable">s'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv:64"><span class="id" title="inductive">result_equiv</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#x:66"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.AM.html#s:67"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#y:68"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.AM.html#s':69"><span class="id" title="variable">s'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fchain_correct" class="idref" href="#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a id="p:70" class="idref" href="#p:70"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="c:71" class="idref" href="#c:71"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>) :=<br/>
+&nbsp;&nbsp;(<span class="id" title="keyword">∀</span>  <a id="A:72" class="idref" href="#A:72"><span class="id" title="binder">A</span></a> <a id="op:73" class="idref" href="#op:73"><span class="id" title="binder">op</span></a> <a id="one:74" class="idref" href="#one:74"><span class="id" title="binder">one</span></a> <a id="equ:75" class="idref" href="#equ:75"><span class="id" title="binder">equ</span></a> (<a id="M:76" class="idref" href="#M:76"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:73"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#one:74"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.AM.html#equ:75"><span class="id" title="variable">equ</span></a>) (<a id="x:77" class="idref" href="#x:77"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.AM.html#A:72"><span class="id" title="variable">A</span></a>) <a id="s:78" class="idref" href="#s:78"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.AM.html#M:76"><span class="id" title="variable">M</span></a>) (<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:73"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#c:71"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:78"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pow.Pos_bpow</span></a> <a class="idref" href="additions.AM.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#p:70"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#s:78"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>)) .<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kchain_correct" class="idref" href="#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  <a id="n:79" class="idref" href="#n:79"><span class="id" title="binder">n</span></a> <a id="p:80" class="idref" href="#p:80"><span class="id" title="binder">p</span></a> <a id="c:81" class="idref" href="#c:81"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span>   <a id="A:82" class="idref" href="#A:82"><span class="id" title="binder">A</span></a> <a id="op:83" class="idref" href="#op:83"><span class="id" title="binder">op</span></a> <a id="one:84" class="idref" href="#one:84"><span class="id" title="binder">one</span></a> <a id="equ:85" class="idref" href="#equ:85"><span class="id" title="binder">equ</span></a> (<a id="M:86" class="idref" href="#M:86"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:82"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:83"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#one:84"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.AM.html#equ:85"><span class="id" title="variable">equ</span></a>) (<a id="x:87" class="idref" href="#x:87"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.AM.html#A:82"><span class="id" title="variable">A</span></a>) <a id="s:88" class="idref" href="#s:88"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a> (<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:82"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:83"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#c:81"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:87"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:88"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pow.Pos_bpow</span></a>  <a class="idref" href="additions.AM.html#x:87"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#n:79"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pow.Pos_bpow</span></a>  <a class="idref" href="additions.AM.html#x:87"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#p:80"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="additions.AM.html#s:88"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="correctness_statement" class="idref" href="#correctness_statement"><span class="id" title="definition">correctness_statement</span></a> (<a id="s:89" class="idref" href="#s:89"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#s:89"><span class="id" title="variable">s</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">p</span> ⇒ <span class="id" title="keyword">fun</span> <a id="c:91" class="idref" href="#c:91"><span class="id" title="binder">c</span></a> ⇒ <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <span class="id" title="var">p</span> <a class="idref" href="additions.AM.html#c:91"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">d</span>   ⇒ <span class="id" title="keyword">fun</span> <a id="c:92" class="idref" href="#c:92"><span class="id" title="binder">c</span></a> ⇒ <a class="idref" href="additions.AM.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>) <span class="id" title="var">p</span> <a class="idref" href="additions.AM.html#c:92"><span class="id" title="variable">c</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Stack_equiv_refl" class="idref" href="#Stack_equiv_refl"><span class="id" title="instance">Stack_equiv_refl</span></a> {<a id="A:93" class="idref" href="#A:93"><span class="id" title="binder">A</span></a>}`{<a id="M:97" class="idref" href="#M:97"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:93"><span class="id" title="variable">A</span></a> <a id="op:94" class="idref" href="#op:94"><span class="id" title="binder">op</span></a> <a id="one:95" class="idref" href="#one:95"><span class="id" title="binder">one</span></a> <a id="equ:96" class="idref" href="#equ:96"><span class="id" title="binder">equ</span></a>} :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a> <a class="idref" href="additions.AM.html#stack_equiv"><span class="id" title="inductive">stack_equiv</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Stack_equiv_equiv" class="idref" href="#Stack_equiv_equiv"><span class="id" title="instance">Stack_equiv_equiv</span></a> {<a id="A:98" class="idref" href="#A:98"><span class="id" title="binder">A</span></a>}`{<a id="M:102" class="idref" href="#M:102"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:98"><span class="id" title="variable">A</span></a> <a id="op:99" class="idref" href="#op:99"><span class="id" title="binder">op</span></a> <a id="one:100" class="idref" href="#one:100"><span class="id" title="binder">one</span></a> <a id="equ:101" class="idref" href="#equ:101"><span class="id" title="binder">equ</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="additions.AM.html#stack_equiv"><span class="id" title="inductive">stack_equiv</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="result_equiv_equiv" class="idref" href="#result_equiv_equiv"><span class="id" title="instance">result_equiv_equiv</span></a> `{<a id="M:107" class="idref" href="#M:107"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:103" class="idref" href="#A:103"><span class="id" title="binder">A</span></a> <a id="op:104" class="idref" href="#op:104"><span class="id" title="binder">op</span></a> <a id="one:105" class="idref" href="#one:105"><span class="id" title="binder">one</span></a> <a id="equ:106" class="idref" href="#equ:106"><span class="id" title="binder">equ</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exec_equiv" class="idref" href="#exec_equiv"><span class="id" title="lemma">exec_equiv</span></a> `{<a id="M:112" class="idref" href="#M:112"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:108" class="idref" href="#A:108"><span class="id" title="binder">A</span></a> <a id="op:109" class="idref" href="#op:109"><span class="id" title="binder">op</span></a> <a id="one:110" class="idref" href="#one:110"><span class="id" title="binder">one</span></a> <a id="equ:111" class="idref" href="#equ:111"><span class="id" title="binder">equ</span></a>} :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="c:113" class="idref" href="#c:113"><span class="id" title="binder">c</span></a> <a id="x:114" class="idref" href="#x:114"><span class="id" title="binder">x</span></a> <a id="s:115" class="idref" href="#s:115"><span class="id" title="binder">s</span></a> <a id="y:116" class="idref" href="#y:116"><span class="id" title="binder">y</span></a> <a id="s':117" class="idref" href="#s':117"><span class="id" title="binder">s'</span></a> , <a class="idref" href="additions.AM.html#config_equiv"><span class="id" title="definition">config_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#x:114"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#s:115"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#y:116"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#s':117"><span class="id" title="variable">s'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a> (<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:108"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#c:113"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#x:114"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:115"><span class="id" title="variable">s</span></a>) (<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:108"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.AM.html#c:113"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.AM.html#y:116"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.AM.html#s':117"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="exec_Proper" class="idref" href="#exec_Proper"><span class="id" title="instance">exec_Proper</span></a> `{<a id="M:122" class="idref" href="#M:122"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:118" class="idref" href="#A:118"><span class="id" title="binder">A</span></a> <a id="op:119" class="idref" href="#op:119"><span class="id" title="binder">op</span></a> <a id="one:120" class="idref" href="#one:120"><span class="id" title="binder">one</span></a> <a id="equ:121" class="idref" href="#equ:121"><span class="id" title="binder">equ</span></a>} :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.AM.html#stack_equiv"><span class="id" title="inductive">stack_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a>) (@<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:118"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:119"><span class="id" title="variable">op</span></a>) .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F1_correct" class="idref" href="#F1_correct"><span class="id" title="lemma">F1_correct</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a>  1  <a class="idref" href="additions.AM.html#F1"><span class="id" title="definition">F1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+F3 is correct 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F3_correct" class="idref" href="#F3_correct"><span class="id" title="lemma">F3_correct</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a>  3 <a class="idref" href="additions.AM.html#F3"><span class="id" title="definition">F3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.Pow.html#"><span class="id" title="module">Pow</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F2q_correct_0" class="idref" href="#F2q_correct_0"><span class="id" title="lemma">F2q_correct_0</span></a>: <span class="id" title="keyword">∀</span> `{<a id="M:127" class="idref" href="#M:127"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:123" class="idref" href="#A:123"><span class="id" title="binder">A</span></a> <a id="op:124" class="idref" href="#op:124"><span class="id" title="binder">op</span></a> <a id="one:125" class="idref" href="#one:125"><span class="id" title="binder">one</span></a> <a id="equ:126" class="idref" href="#equ:126"><span class="id" title="binder">equ</span></a>} (<a id="n:128" class="idref" href="#n:128"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="x:129" class="idref" href="#x:129"><span class="id" title="binder">x</span></a> <a id="s:130" class="idref" href="#s:130"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a> (<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:123"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#op:124"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.AM.html#F2q_of_nat"><span class="id" title="definition">F2q_of_nat</span></a> <a class="idref" href="additions.AM.html#n:128"><span class="id" title="variable">n</span></a>) <a class="idref" href="additions.AM.html#x:129"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#s:130"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">Pow.power</span></a> <a class="idref" href="additions.AM.html#x:129"><span class="id" title="variable">x</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.AM.html#n:128"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.AM.html#s:130"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F2q_correct_1" class="idref" href="#F2q_correct_1"><span class="id" title="lemma">F2q_correct_1</span></a> (<a id="n:131" class="idref" href="#n:131"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.AM.html#n:131"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.AM.html#F2q_of_nat"><span class="id" title="definition">F2q_of_nat</span></a> <a class="idref" href="additions.AM.html#n:131"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Pos_to_nat_diff_0" class="idref" href="#Pos_to_nat_diff_0"><span class="id" title="lemma">Pos_to_nat_diff_0</span></a> <a id="n:132" class="idref" href="#n:132"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.AM.html#n:132"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="F2q_correct" class="idref" href="#F2q_correct"><span class="id" title="lemma">F2q_correct</span></a> (<a id="n:133" class="idref" href="#n:133"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)  : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#72c1e614b36b07dc51a712bc261ee813"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.AM.html#n:133"><span class="id" title="variable">n</span></a>) (<a class="idref" href="additions.AM.html#F2q"><span class="id" title="definition">F2q</span></a> <a class="idref" href="additions.AM.html#n:133"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="CompositionProofs" class="idref" href="#CompositionProofs"><span class="id" title="section">CompositionProofs</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="CompositionProofs.FK" class="idref" href="#CompositionProofs.FK"><span class="id" title="section">FK</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="CompositionProofs.FK.n" class="idref" href="#CompositionProofs.FK.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="CompositionProofs.FK.cn" class="idref" href="#CompositionProofs.FK.cn"><span class="id" title="variable">cn</span></a> : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.FK.Hn" class="idref" href="#CompositionProofs.FK.Hn"><span class="id" title="variable">Hn</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FK.n"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FK.cn"><span class="id" title="variable">cn</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="FK_correct" class="idref" href="#FK_correct"><span class="id" title="lemma">FK_correct</span></a> : <a class="idref" href="additions.AM.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FK.n"><span class="id" title="variable">n</span></a> 1 (<a class="idref" href="additions.AM.html#FK"><span class="id" title="definition">FK</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FK.cn"><span class="id" title="variable">cn</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#CompositionProofs.FK"><span class="id" title="section">FK</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="CompositionProofs.App" class="idref" href="#CompositionProofs.App"><span class="id" title="section">App</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.App.n" class="idref" href="#CompositionProofs.App.n"><span class="id" title="variable">n</span></a> <a id="CompositionProofs.App.p" class="idref" href="#CompositionProofs.App.p"><span class="id" title="variable">p</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.App.cn" class="idref" href="#CompositionProofs.App.cn"><span class="id" title="variable">cn</span></a> <a id="CompositionProofs.App.cp" class="idref" href="#CompositionProofs.App.cp"><span class="id" title="variable">cp</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.App.Hn" class="idref" href="#CompositionProofs.App.Hn"><span class="id" title="variable">Hn</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.n"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.cn"><span class="id" title="variable">cn</span></a> .<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.App.Hp" class="idref" href="#CompositionProofs.App.Hp"><span class="id" title="variable">Hp</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.p"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.cp"><span class="id" title="variable">cp</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="correct_app" class="idref" href="#correct_app"><span class="id" title="lemma">correct_app</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="additions.AM.html#CompositionProofs.App.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="additions.AM.html#CompositionProofs.App.cn"><span class="id" title="variable">cn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.App.cp"><span class="id" title="variable">cp</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#CompositionProofs.App"><span class="id" title="section">App</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="CompositionProofs.FFK" class="idref" href="#CompositionProofs.FFK"><span class="id" title="section">FFK</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.FFK.p" class="idref" href="#CompositionProofs.FFK.p"><span class="id" title="variable">p</span></a> <a id="CompositionProofs.FFK.q" class="idref" href="#CompositionProofs.FFK.q"><span class="id" title="variable">q</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.FFK.cp" class="idref" href="#CompositionProofs.FFK.cp"><span class="id" title="variable">cp</span></a> <a id="CompositionProofs.FFK.cq" class="idref" href="#CompositionProofs.FFK.cq"><span class="id" title="variable">cq</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.FFK.Hp" class="idref" href="#CompositionProofs.FFK.Hp"><span class="id" title="variable">Hp</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.p"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.cp"><span class="id" title="variable">cp</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.FFK.Hq" class="idref" href="#CompositionProofs.FFK.Hq"><span class="id" title="variable">Hq</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.q"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.cq"><span class="id" title="variable">cq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="FFK_correct" class="idref" href="#FFK_correct"><span class="id" title="lemma">FFK_correct</span></a> : <a class="idref" href="additions.AM.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> (<a class="idref" href="additions.AM.html#CompositionProofs.FFK.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.q"><span class="id" title="variable">q</span></a>) <a class="idref" href="additions.AM.html#CompositionProofs.FFK.p"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.AM.html#FFK"><span class="id" title="definition">FFK</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.cp"><span class="id" title="variable">cp</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.FFK.cq"><span class="id" title="variable">cq</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#CompositionProofs.FFK"><span class="id" title="section">FFK</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="CompositionProofs.KFK" class="idref" href="#CompositionProofs.KFK"><span class="id" title="section">KFK</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.KFK.p" class="idref" href="#CompositionProofs.KFK.p"><span class="id" title="variable">p</span></a> <a id="CompositionProofs.KFK.q" class="idref" href="#CompositionProofs.KFK.q"><span class="id" title="variable">q</span></a> <a id="CompositionProofs.KFK.r" class="idref" href="#CompositionProofs.KFK.r"><span class="id" title="variable">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="CompositionProofs.KFK.kpr" class="idref" href="#CompositionProofs.KFK.kpr"><span class="id" title="variable">kpr</span></a> <a id="CompositionProofs.KFK.mq" class="idref" href="#CompositionProofs.KFK.mq"><span class="id" title="variable">mq</span></a> : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.KFK.Hpr" class="idref" href="#CompositionProofs.KFK.Hpr"><span class="id" title="variable">Hpr</span></a> : <a class="idref" href="additions.AM.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.p"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.r"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.kpr"><span class="id" title="variable">kpr</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="CompositionProofs.KFK.Hq" class="idref" href="#CompositionProofs.KFK.Hq"><span class="id" title="variable">Hq</span></a> : <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.q"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.mq"><span class="id" title="variable">mq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="KFK_correct" class="idref" href="#KFK_correct"><span class="id" title="lemma">KFK_correct</span></a> : <a class="idref" href="additions.AM.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> (<a class="idref" href="additions.AM.html#CompositionProofs.KFK.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.r"><span class="id" title="variable">r</span></a>)  <a class="idref" href="additions.AM.html#CompositionProofs.KFK.p"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.AM.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.kpr"><span class="id" title="variable">kpr</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.mq"><span class="id" title="variable">mq</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="KFF_correct" class="idref" href="#KFF_correct"><span class="id" title="lemma">KFF_correct</span></a>: <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="additions.AM.html#CompositionProofs.KFK.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.r"><span class="id" title="variable">r</span></a>)  (<a class="idref" href="additions.AM.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.kpr"><span class="id" title="variable">kpr</span></a> <a class="idref" href="additions.AM.html#CompositionProofs.KFK.mq"><span class="id" title="variable">mq</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#CompositionProofs.KFK"><span class="id" title="section">KFK</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#CompositionProofs"><span class="id" title="section">CompositionProofs</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab338"></a><h1 class="section">Euclidean chain generation</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="OK" class="idref" href="#OK"><span class="id" title="definition">OK</span></a> (<a id="s:156" class="idref" href="#s:156"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) <br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">fun</span> <a id="c:157" class="idref" href="#c:157"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> ⇒ <a class="idref" href="additions.AM.html#correctness_statement"><span class="id" title="definition">correctness_statement</span></a> <a class="idref" href="additions.AM.html#s:156"><span class="id" title="variable">s</span></a> <a class="idref" href="additions.AM.html#c:157"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Gamma" class="idref" href="#Gamma"><span class="id" title="section">Gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Gamma.gamma" class="idref" href="#Gamma.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="Hgamma:159" class="idref" href="#Hgamma:159"><span class="id" title="binder">Hgamma</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.AM.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="var">chain_gen</span>  (<a id="s:169" class="idref" href="#s:169"><span class="id" title="binder">s</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) {<span class="id" title="keyword">measure</span> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a>}<br/>
+&nbsp;&nbsp;:  <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#s:160"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">i</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">i</span> 1 <span class="id" title="keyword">then</span> <a class="idref" href="additions.AM.html#F1"><span class="id" title="definition">F1</span></a> <span class="id" title="keyword">else</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">i</span> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="additions.AM.html#F3"><span class="id" title="definition">F3</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.AM.html#F2q"><span class="id" title="definition">F2q</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <span class="id" title="var">i</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> (<a class="idref" href="additions.AM.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.AM.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  ⇒ <a class="idref" href="additions.AM.html#KFF"><span class="id" title="definition">KFF</span></a> (<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.AM.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">d</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">p</span> 1 <span class="id" title="keyword">then</span> <a class="idref" href="additions.AM.html#FK"><span class="id" title="definition">FK</span></a> (<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>)  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="additions.AM.html#FFK"><span class="id" title="definition">FFK</span></a> (<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">p</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  ⇒ <a class="idref" href="additions.AM.html#KFK"><span class="id" title="definition">KFK</span></a> (<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.AM.html#chain_gen:161"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="make_chain" class="idref" href="#make_chain"><span class="id" title="definition">make_chain</span></a> (<a id="n:227" class="idref" href="#n:227"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.AM.html#code"><span class="id" title="definition">code</span></a> :=<br/>
+&nbsp;(<a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.AM.html#n:227"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Proof of correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="chain_gen_OK" class="idref" href="#chain_gen_OK"><span class="id" title="lemma">chain_gen_OK</span></a> : <span class="id" title="keyword">∀</span> <a id="s:228" class="idref" href="#s:228"><span class="id" title="binder">s</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>, <a class="idref" href="additions.AM.html#OK"><span class="id" title="definition">OK</span></a>  <a class="idref" href="additions.AM.html#s:228"><span class="id" title="variable">s</span></a> (<a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a>  <a class="idref" href="additions.AM.html#s:228"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Gamma.All_OK" class="idref" href="#Gamma.All_OK"><span class="id" title="section">All_OK</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Gamma.All_OK.n" class="idref" href="#Gamma.All_OK.n"><span class="id" title="variable">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Gamma.All_OK.c" class="idref" href="#Gamma.All_OK.c"><span class="id" title="variable">c</span></a> := <a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.AM.html#Gamma.All_OK.n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L0" class="idref" href="#L0"><span class="id" title="lemma">L0</span></a>: <a class="idref" href="additions.AM.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.AM.html#Gamma.All_OK.n"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.AM.html#Gamma.All_OK.c"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#Gamma.All_OK"><span class="id" title="section">All_OK</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="AM_power" class="idref" href="#AM_power"><span class="id" title="definition">AM_power</span></a> {<a id="A:230" class="idref" href="#A:230"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="M:234" class="idref" href="#M:234"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:230"><span class="id" title="variable">A</span></a> <a id="E_op:231" class="idref" href="#E_op:231"><span class="id" title="binder">E_op</span></a> <a id="E_one:232" class="idref" href="#E_one:232"><span class="id" title="binder">E_one</span></a> <a id="E_eq:233" class="idref" href="#E_eq:233"><span class="id" title="binder">E_eq</span></a>) (<a id="x:235" class="idref" href="#x:235"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.AM.html#A:230"><span class="id" title="variable">A</span></a>) (<a id="n:236" class="idref" href="#n:236"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.AM.html#exec"><span class="id" title="definition">exec</span></a> <a class="idref" href="additions.AM.html#A:230"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.AM.html#E_op:231"><span class="id" title="variable">E_op</span></a> (<a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.AM.html#n:236"><span class="id" title="variable">n</span></a>))  <a class="idref" href="additions.AM.html#x:235"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AM_power_Ok" class="idref" href="#AM_power_Ok"><span class="id" title="lemma">AM_power_Ok</span></a> {<a id="A:237" class="idref" href="#A:237"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="M:241" class="idref" href="#M:241"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.AM.html#A:237"><span class="id" title="variable">A</span></a> <a id="E_op:238" class="idref" href="#E_op:238"><span class="id" title="binder">E_op</span></a> <a id="E_one:239" class="idref" href="#E_one:239"><span class="id" title="binder">E_one</span></a> <a id="E_eq:240" class="idref" href="#E_eq:240"><span class="id" title="binder">E_eq</span></a>) (<a id="x:242" class="idref" href="#x:242"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.AM.html#A:237"><span class="id" title="variable">A</span></a>) (<a id="n:243" class="idref" href="#n:243"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.AM.html#result_equiv"><span class="id" title="inductive">result_equiv</span></a> (<a class="idref" href="additions.AM.html#AM_power"><span class="id" title="definition">AM_power</span></a> <a class="idref" href="additions.AM.html#M:241"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.AM.html#x:242"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#n:243"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.AM.html#x:242"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.AM.html#n:243"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.AM.html#Gamma"><span class="id" title="section">Gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> <span class="id" title="var">gamma</span> {<span class="id" title="var">Hgamma</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F197887" class="idref" href="#F197887"><span class="id" title="definition">F197887</span></a> := <a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a>  <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> 197887).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Addition_Chains.html b/theories/html/additions.Addition_Chains.html
new file mode 100644
index 00000000..e2a878c8
--- /dev/null
+++ b/theories/html/additions.Addition_Chains.html
@@ -0,0 +1,1074 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Addition_Chains</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Addition_Chains</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab339"></a><h1 class="section">Addition Chains</h1>
+
+Pierre Casteran, LaBRI, University of Bordeaux
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Param</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Param</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a>.<br/>
+<span class="id" title="keyword">Generalizable Variables</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">Aeq</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="2596316c6f2ff161cb03e25fd8914390" class="idref" href="#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">&quot;</span></a>==" := <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">Monoid_def.equiv</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70) : <span class="id" title="var">type_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab340"></a><h2 class="section">Computations composed of multiplications over type  A</h2>
+
+  (in Continuation Passing Style) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="computation" class="idref" href="#computation"><span class="id" title="inductive">computation</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}  : <span class="id" title="keyword">Type</span> :=<br/>
+| <a id="Return" class="idref" href="#Return"><span class="id" title="constructor">Return</span></a> (<a id="a:4" class="idref" href="#a:4"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+| <a id="Mult" class="idref" href="#Mult"><span class="id" title="constructor">Mult</span></a> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="k:7" class="idref" href="#k:7"><span class="id" title="binder">k</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#computation:2"><span class="id" title="inductive">computation</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab341"></a><h3 class="section">Monadic Notation for computations</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="5870a17092f4a8078b9692b4c6687556" class="idref" href="#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&quot;</span></a>z '&lt;---'  x 'times' y ';' e2 " :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Addition_Chains.html#Mult"><span class="id" title="constructor">Mult</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>  (<span class="id" title="keyword">fun</span> <a id="z:8" class="idref" href="#z:8"><span class="id" title="binder">z</span></a> ⇒ <span class="id" title="var">e2</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="comp128" class="idref" href="#comp128"><span class="id" title="definition">comp128</span></a> : <a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a>  :=<br/>
+&nbsp;&nbsp;<a id="x:9" class="idref" href="#x:9"><span class="id" title="binder">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> 2 <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> 2<a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="y:10" class="idref" href="#y:10"><span class="id" title="binder">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> 2<a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="z:11" class="idref" href="#z:11"><span class="id" title="binder">z</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="t:12" class="idref" href="#t:12"><span class="id" title="binder">t</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> 2 <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#z:11"><span class="id" title="variable">z</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#t:12"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab342"></a><h3 class="section">Definition</h3>
+
+
+<div class="paragraph"> </div>
+
+An <i>addition chain</i> (in short a <i>chain</i>) is a function that maps
+ any type  and any value  of type  into a computation on .
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain" class="idref" href="#chain"><span class="id" title="definition">chain</span></a> := <span class="id" title="keyword">∀</span> <a id="A:13" class="idref" href="#A:13"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>, <a class="idref" href="additions.Addition_Chains.html#A:13"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#A:13"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The chain associated with the empty computation 
+   (raising to the first power) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="C1" class="idref" href="#C1"><span class="id" title="definition">C1</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=  (@<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="C3" class="idref" href="#C3"><span class="id" title="definition">C3</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="A:14" class="idref" href="#A:14"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:14"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="x2:16" class="idref" href="#x2:16"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:15"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="x3:17" class="idref" href="#x3:17"><span class="id" title="binder">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:16"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:15"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:17"><span class="id" title="variable">x3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Chain associated with the 7-th power 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="C7" class="idref" href="#C7"><span class="id" title="definition">C7</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:18"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;<a id="x2:20" class="idref" href="#x2:20"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:19"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;<a id="x3:21" class="idref" href="#x3:21"><span class="id" title="binder">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:20"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:19"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;<a id="x6:22" class="idref" href="#x6:22"><span class="id" title="binder">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:21"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:21"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;<a id="x7:23" class="idref" href="#x7:23"><span class="id" title="binder">x7</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x6:22"><span class="id" title="variable">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:19"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#x7:23"><span class="id" title="variable">x7</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab343"></a><h3 class="section">Our Favorite example</h3>
+
+
+<div class="paragraph"> </div>
+
+  The chain below is intented to compute 87-th power in any EMonoid.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span>  <a id="C87" class="idref" href="#C87"><span class="id" title="definition">C87</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> <a id="A:24" class="idref" href="#A:24"><span class="id" title="binder">A</span></a> (<a id="x:25" class="idref" href="#x:25"><span class="id" title="binder">x</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:24"><span class="id" title="variable">A</span></a>)=&gt;<br/>
+&nbsp;&nbsp;<a id="x2:26" class="idref" href="#x2:26"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x3:27" class="idref" href="#x3:27"><span class="id" title="binder">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:26"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x6:28" class="idref" href="#x6:28"><span class="id" title="binder">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:27"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:27"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x7:29" class="idref" href="#x7:29"><span class="id" title="binder">x7</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x6:28"><span class="id" title="variable">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x10:30" class="idref" href="#x10:30"><span class="id" title="binder">x10</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x7:29"><span class="id" title="variable">x7</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x3:27"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x20:31" class="idref" href="#x20:31"><span class="id" title="binder">x20</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x10:30"><span class="id" title="variable">x10</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x10:30"><span class="id" title="variable">x10</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x40:32" class="idref" href="#x40:32"><span class="id" title="binder">x40</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x20:31"><span class="id" title="variable">x20</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x20:31"><span class="id" title="variable">x20</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x80:33" class="idref" href="#x80:33"><span class="id" title="binder">x80</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x40:32"><span class="id" title="variable">x40</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x40:32"><span class="id" title="variable">x40</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a>  <br/>
+&nbsp;&nbsp;<a id="x87:34" class="idref" href="#x87:34"><span class="id" title="binder">x87</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x80:33"><span class="id" title="variable">x80</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x7:29"><span class="id" title="variable">x7</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#x87:34"><span class="id" title="variable">x87</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab344"></a><h3 class="section">Chain length (number of mutiplications)</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="computation_length" class="idref" href="#computation_length"><span class="id" title="definition">computation_length</span></a> {<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a>} (<a id="a:36" class="idref" href="#a:36"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:35"><span class="id" title="variable">A</span></a>) (<a id="m:37" class="idref" href="#m:37"><span class="id" title="binder">m</span></a> : @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#A:35"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#m:37"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#Mult"><span class="id" title="constructor">Mult</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">k</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="additions.Addition_Chains.html#computation_length:38"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="additions.Addition_Chains.html#a:36"><span class="id" title="variable">a</span></a> (<span class="id" title="var">k</span> <a class="idref" href="additions.Addition_Chains.html#a:36"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ 0%<span class="id" title="var">nat</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_length" class="idref" href="#chain_length"><span class="id" title="definition">chain_length</span></a> (<a id="c:40" class="idref" href="#c:40"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) := <a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:40"><span class="id" title="variable">c</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab345"></a><h3 class="section">Execution of chains</h3>
+
+
+<div class="paragraph"> </div>
+
+Chains are designed for effectively computing powers. 
+
+<div class="paragraph"> </div>
+
+First, we define recursively the <i>evaluation</i>  of a computation, then 
+the <i>execution</i> of a chain.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="computation_execute" class="idref" href="#computation_execute"><span class="id" title="definition">computation_execute</span></a>  {<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="op:42" class="idref" href="#op:42"><span class="id" title="binder">op</span></a>: <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Addition_Chains.html#A:41"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="c:43" class="idref" href="#c:43"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#c:43"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">x</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#Mult"><span class="id" title="constructor">Mult</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">k</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#computation_execute:44"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Addition_Chains.html#op:42"><span class="id" title="variable">op</span></a> (<span class="id" title="var">k</span> (<span class="id" title="var">x</span> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <span class="id" title="var">y</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="computation_eval" class="idref" href="#computation_eval"><span class="id" title="definition">computation_eval</span></a> `{<a id="M:50" class="idref" href="#M:50"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a> <a id="E_op:47" class="idref" href="#E_op:47"><span class="id" title="binder">E_op</span></a> <a id="E_one:48" class="idref" href="#E_one:48"><span class="id" title="binder">E_one</span></a> <a id="E_eq:49" class="idref" href="#E_eq:49"><span class="id" title="binder">E_eq</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="c:51" class="idref" href="#c:51"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a>) : <a class="idref" href="additions.Addition_Chains.html#A:46"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Addition_Chains.html#E_op:47"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Addition_Chains.html#c:51"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_execute" class="idref" href="#chain_execute"><span class="id" title="definition">chain_execute</span></a> (<a id="c:52" class="idref" href="#c:52"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) {<a id="A:53" class="idref" href="#A:53"><span class="id" title="binder">A</span></a>} <a id="op:54" class="idref" href="#op:54"><span class="id" title="binder">op</span></a>  (<a id="a:55" class="idref" href="#a:55"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:53"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Addition_Chains.html#op:54"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:52"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#A:53"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#a:55"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_apply" class="idref" href="#chain_apply"><span class="id" title="definition">chain_apply</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="c:56" class="idref" href="#c:56"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) `{<a id="M:61" class="idref" href="#M:61"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:57" class="idref" href="#A:57"><span class="id" title="binder">A</span></a> <a id="E_op:58" class="idref" href="#E_op:58"><span class="id" title="binder">E_op</span></a> <a id="E_one:59" class="idref" href="#E_one:59"><span class="id" title="binder">E_one</span></a> <a id="E_eq:60" class="idref" href="#E_eq:60"><span class="id" title="binder">E_eq</span></a>} <a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:57"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:56"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#A:57"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#a:62"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="computation_eval_rw" class="idref" href="#computation_eval_rw"><span class="id" title="lemma">computation_eval_rw</span></a> `{<a id="M:67" class="idref" href="#M:67"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:63" class="idref" href="#A:63"><span class="id" title="binder">A</span></a> <a id="E_op:64" class="idref" href="#E_op:64"><span class="id" title="binder">E_op</span></a> <a id="E_one:65" class="idref" href="#E_one:65"><span class="id" title="binder">E_one</span></a> <a id="E_eq:66" class="idref" href="#E_eq:66"><span class="id" title="binder">E_eq</span></a>} <a id="c:68" class="idref" href="#c:68"><span class="id" title="binder">c</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> <a class="idref" href="additions.Addition_Chains.html#c:68"><span class="id" title="variable">c</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Addition_Chains.html#E_op:64"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Addition_Chains.html#c:68"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab346"></a><h2 class="section">Chain correctness</h2>
+
+
+<div class="paragraph"> </div>
+
+In this section, we define formally the correctness of a given chain 
+ with respect to  some given exponent,
+and more generally the correctness of a chain generator, i.e.  a function that,
+given any positive integer , returns a chain that is correct with respect to  .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_correct_nat" class="idref" href="#chain_correct_nat"><span class="id" title="definition">chain_correct_nat</span></a> (<a id="n:69" class="idref" href="#n:69"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="c:70" class="idref" href="#c:70"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) := <a class="idref" href="additions.Addition_Chains.html#n:69"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+<span class="id" title="keyword">∀</span> `(<a id="M:75" class="idref" href="#M:75"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a id="A:71" class="idref" href="#A:71"><span class="id" title="binder">A</span></a> <a id="E_op:72" class="idref" href="#E_op:72"><span class="id" title="binder">E_op</span></a> <a id="E_one:73" class="idref" href="#E_one:73"><span class="id" title="binder">E_one</span></a> <a id="E_eq:74" class="idref" href="#E_eq:74"><span class="id" title="binder">E_eq</span></a>) (<a id="x:76" class="idref" href="#x:76"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:71"><span class="id" title="variable">A</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a class="idref" href="additions.Addition_Chains.html#c:70"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#x:76"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>   <a class="idref" href="additions.Addition_Chains.html#x:76"><span class="id" title="variable">x</span></a>  <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a>  <a class="idref" href="additions.Addition_Chains.html#n:69"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_correct" class="idref" href="#chain_correct"><span class="id" title="definition">chain_correct</span></a> (<a id="p:77" class="idref" href="#p:77"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="c:78" class="idref" href="#c:78"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_correct_nat"><span class="id" title="definition">chain_correct_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#p:77"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.Addition_Chains.html#c:78"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="optimal" class="idref" href="#optimal"><span class="id" title="definition">optimal</span></a> (<a id="p:79" class="idref" href="#p:79"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="c:80" class="idref" href="#c:80"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="c':81" class="idref" href="#c':81"><span class="id" title="binder">c'</span></a>, <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Addition_Chains.html#p:79"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Addition_Chains.html#c':81"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Addition_Chains.html#chain_length"><span class="id" title="definition">chain_length</span></a> <a class="idref" href="additions.Addition_Chains.html#c:80"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Addition_Chains.html#chain_length"><span class="id" title="definition">chain_length</span></a> <a class="idref" href="additions.Addition_Chains.html#c':81"><span class="id" title="variable">c'</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A slow tactic for proving a chain's correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">slow_chain_correct_tac</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[ |- <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> ?<span class="id" title="var">p</span> ?<span class="id" title="var">c</span> ] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">A</span> := <span class="id" title="tactic">fresh</span> "A" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">op</span> := <span class="id" title="tactic">fresh</span> "op" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">one</span> := <span class="id" title="tactic">fresh</span> "one" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">eqv</span> := <span class="id" title="tactic">fresh</span> "eqv" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">fresh</span> "M" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">x</span> := <span class="id" title="tactic">fresh</span> "x"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span>  <span class="id" title="tactic">split</span>;[<span class="id" title="tactic">discriminate</span> | <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <span class="id" title="var">c</span>, <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a>, <a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a>; <span class="id" title="tactic">simpl</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">eq</span> <span class="id" title="var">M</span> <span class="id" title="var">x</span>; <span class="id" title="var">monoid_simpl</span> <span class="id" title="var">M</span>; <span class="id" title="tactic">reflexivity</span>]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="C7_ok" class="idref" href="#C7_ok"><span class="id" title="definition">C7_ok</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 7 <a class="idref" href="additions.Addition_Chains.html#C7"><span class="id" title="definition">C7</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The following proof takes a very long time. Happily, C87's correctness 
+will be proved  more  efficiently, using  reflection or parametricity. 
+
+<div class="paragraph"> </div>
+
+ Remove the comment if you can wait ... 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab347"></a><h1 class="section">Correctness Proof by Reflection</h1>
+
+  See chap 16 of Coq'Art 
+</div>
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Binary trees of multiplications over A 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Monoid_Exp" class="idref" href="#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> (<a id="A:82" class="idref" href="#A:82"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;<a id="Mul_node" class="idref" href="#Mul_node"><span class="id" title="constructor">Mul_node</span></a> (<a id="t:85" class="idref" href="#t:85"><span class="id" title="binder">t</span></a> <a id="t':86" class="idref" href="#t':86"><span class="id" title="binder">t'</span></a> : <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp:83"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:82"><span class="id" title="variable">A</span></a>) | <a id="One_node" class="idref" href="#One_node"><span class="id" title="constructor">One_node</span></a> | <a id="A_node" class="idref" href="#A_node"><span class="id" title="constructor">A_node</span></a> (<a id="a:87" class="idref" href="#a:87"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:82"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a> {<span class="id" title="var">A</span>} .<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Addition_Chains.html#A_node"><span class="id" title="constructor">A_node</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> .<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Linearization functions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="flatten_aux" class="idref" href="#flatten_aux"><span class="id" title="definition">flatten_aux</span></a> {<a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="t:89" class="idref" href="#t:89"><span class="id" title="binder">t</span></a> <a id="fin:90" class="idref" href="#fin:90"><span class="id" title="binder">fin</span></a> : <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:88"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:88"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#t:89"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a>  <span class="id" title="var">t</span> <span class="id" title="var">t'</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#flatten_aux:91"><span class="id" title="definition">flatten_aux</span></a> <a class="idref" href="additions.Addition_Chains.html#t:89"><span class="id" title="variable">t</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten_aux:91"><span class="id" title="definition">flatten_aux</span></a> <span class="id" title="var">t'</span> <a class="idref" href="additions.Addition_Chains.html#fin:90"><span class="id" title="variable">fin</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a>  ⇒ <a class="idref" href="additions.Addition_Chains.html#fin:90"><span class="id" title="variable">fin</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">x</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a>  <span class="id" title="var">x</span> <a class="idref" href="additions.Addition_Chains.html#fin:90"><span class="id" title="variable">fin</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="flatten" class="idref" href="#flatten"><span class="id" title="definition">flatten</span></a> {<a id="A:93" class="idref" href="#A:93"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="t:94" class="idref" href="#t:94"><span class="id" title="binder">t</span></a>: <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:93"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:93"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#t:94"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a> <span class="id" title="var">t</span> <span class="id" title="var">t'</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#flatten_aux"><span class="id" title="definition">flatten_aux</span></a> <a class="idref" href="additions.Addition_Chains.html#t:94"><span class="id" title="variable">t</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten:95"><span class="id" title="definition">flatten</span></a> <span class="id" title="var">t'</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a> ⇒ <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">X</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a> <span class="id" title="var">X</span> <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Interpretation function 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="tactic">eval</span> {<a id="A:129" class="idref" href="#A:129"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="op:130" class="idref" href="#op:130"><span class="id" title="binder">op</span></a> <a id="one:131" class="idref" href="#one:131"><span class="id" title="binder">one</span></a> <a id="eqv:132" class="idref" href="#eqv:132"><span class="id" title="binder">eqv</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M:133" class="idref" href="#M:133"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Addition_Chains.html#A:101"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Addition_Chains.html#one:103"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Addition_Chains.html#eqv:104"><span class="id" title="variable">eqv</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="t:134" class="idref" href="#t:134"><span class="id" title="binder">t</span></a>: <a class="idref" href="additions.Addition_Chains.html#Monoid_Exp"><span class="id" title="inductive">Monoid_Exp</span></a> <a class="idref" href="additions.Addition_Chains.html#A:101"><span class="id" title="variable">A</span></a>) : <a class="idref" href="additions.Addition_Chains.html#A:101"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#t:106"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> ⇒ (<a class="idref" href="additions.Addition_Chains.html#eval:107"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:105"><span class="id" title="variable">M</span></a> <span class="id" title="var">t1</span> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#eval:107"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:105"><span class="id" title="variable">M</span></a> <span class="id" title="var">t2</span>)%<span class="id" title="var">M</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a> ⇒ <a class="idref" href="additions.Addition_Chains.html#one:103"><span class="id" title="variable">one</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Addition_Chains.html#A_node"><span class="id" title="constructor">A_node</span></a> <span class="id" title="var">a</span> ⇒ <span class="id" title="var">a</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="flatten_aux_valid" class="idref" href="#flatten_aux_valid"><span class="id" title="lemma">flatten_aux_valid</span></a> `(<a id="M:163" class="idref" href="#M:163"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:159" class="idref" href="#A:159"><span class="id" title="binder">A</span></a> <a id="op:160" class="idref" href="#op:160"><span class="id" title="binder">op</span></a> <a id="one:161" class="idref" href="#one:161"><span class="id" title="binder">one</span></a> <a id="eqv:162" class="idref" href="#eqv:162"><span class="id" title="binder">eqv</span></a>):<br/>
+<span class="id" title="keyword">∀</span> <a id="t:164" class="idref" href="#t:164"><span class="id" title="binder">t</span></a> <a id="t':165" class="idref" href="#t':165"><span class="id" title="binder">t'</span></a>, (<a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:163"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Addition_Chains.html#t:164"><span class="id" title="variable">t</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:163"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Addition_Chains.html#t':165"><span class="id" title="variable">t'</span></a>)%<span class="id" title="var">M</span> <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:163"><span class="id" title="variable">M</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten_aux"><span class="id" title="definition">flatten_aux</span></a> <a class="idref" href="additions.Addition_Chains.html#t:164"><span class="id" title="variable">t</span></a> <a class="idref" href="additions.Addition_Chains.html#t':165"><span class="id" title="variable">t'</span></a>)<a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="flatten_valid" class="idref" href="#flatten_valid"><span class="id" title="lemma">flatten_valid</span></a> `(<a id="M:170" class="idref" href="#M:170"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:166" class="idref" href="#A:166"><span class="id" title="binder">A</span></a> <a id="op:167" class="idref" href="#op:167"><span class="id" title="binder">op</span></a> <a id="one:168" class="idref" href="#one:168"><span class="id" title="binder">one</span></a> <a id="eqv:169" class="idref" href="#eqv:169"><span class="id" title="binder">eqv</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:171" class="idref" href="#t:171"><span class="id" title="binder">t</span></a> , <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:170"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Addition_Chains.html#t:171"><span class="id" title="variable">t</span></a> <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:170"><span class="id" title="variable">M</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="additions.Addition_Chains.html#t:171"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="flatten_valid_2" class="idref" href="#flatten_valid_2"><span class="id" title="lemma">flatten_valid_2</span></a> `(<a id="M:176" class="idref" href="#M:176"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:172" class="idref" href="#A:172"><span class="id" title="binder">A</span></a> <a id="op:173" class="idref" href="#op:173"><span class="id" title="binder">op</span></a> <a id="one:174" class="idref" href="#one:174"><span class="id" title="binder">one</span></a> <a id="eqv:175" class="idref" href="#eqv:175"><span class="id" title="binder">eqv</span></a>):<br/>
+<span class="id" title="keyword">∀</span> <a id="t:177" class="idref" href="#t:177"><span class="id" title="binder">t</span></a> <a id="t':178" class="idref" href="#t':178"><span class="id" title="binder">t'</span></a> , <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a>  <a class="idref" href="additions.Addition_Chains.html#M:176"><span class="id" title="variable">M</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="additions.Addition_Chains.html#t:177"><span class="id" title="variable">t</span></a>) <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:176"><span class="id" title="variable">M</span></a> (<a class="idref" href="additions.Addition_Chains.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="additions.Addition_Chains.html#t':178"><span class="id" title="variable">t'</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:176"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Addition_Chains.html#t:177"><span class="id" title="variable">t</span></a> <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="additions.Addition_Chains.html#M:176"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Addition_Chains.html#t':178"><span class="id" title="variable">t'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+"Quote" tactic 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">model</span> <span class="id" title="var">A</span>  <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">v</span> :=<br/>
+<span class="id" title="keyword">match</span> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span> <br/>
+| (?<span class="id" title="var">x</span>  <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> ?<span class="id" title="var">y</span>)%<span class="id" title="var">M</span> ⇒ <span class="id" title="keyword">let</span> <span class="id" title="var">r1</span> := <span class="id" title="var">model</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <span class="id" title="var">r2</span> :=(<span class="id" title="var">model</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">y</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span>  <span class="id" title="keyword">constr</span>:(@<a class="idref" href="additions.Addition_Chains.html#Mul_node"><span class="id" title="constructor">Mul_node</span></a> <span class="id" title="var">A</span> <span class="id" title="var">r1</span> <span class="id" title="var">r2</span>)<br/>
+| <span class="id" title="var">one</span> ⇒ <span class="id" title="keyword">constr</span>:(@<a class="idref" href="additions.Addition_Chains.html#One_node"><span class="id" title="constructor">One_node</span></a> <span class="id" title="var">A</span>)<br/>
+| ?<span class="id" title="var">x</span> ⇒ <span class="id" title="keyword">constr</span>:(@<a class="idref" href="additions.Addition_Chains.html#A_node"><span class="id" title="constructor">A_node</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span>)<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_eq_A</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">E_eq</span> <span class="id" title="var">M</span>  :=<br/>
+<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+| [ |- <span class="id" title="var">E_eq</span>  ?<span class="id" title="var">X</span> ?<span class="id" title="var">Y</span> ] ⇒<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">tX</span> := <span class="id" title="var">model</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">X</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">tY</span> := <span class="id" title="var">model</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">Y</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">change</span> (<span class="id" title="var">E_eq</span> (<a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <span class="id" title="var">M</span> <span class="id" title="var">tX</span>) (<a class="idref" href="additions.Addition_Chains.html#eval"><span class="id" title="definition">eval</span></a> <span class="id" title="var">M</span> <span class="id" title="var">tY</span>)))<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">reflection_correct_tac</span> :=<br/>
+<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+[ |- <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">c</span> ] ⇒<br/>
+&nbsp;<span class="id" title="tactic">split</span>; [<span class="id" title="tactic">try</span> <span class="id" title="tactic">discriminate</span> |<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">A</span> := <span class="id" title="tactic">fresh</span> "A"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">op</span> := <span class="id" title="tactic">fresh</span> "op"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">one</span> := <span class="id" title="tactic">fresh</span> "one" <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">E_eq</span> := <span class="id" title="tactic">fresh</span> "eq" <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">fresh</span> "M"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">x</span> := <span class="id" title="tactic">fresh</span> "x" <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span>  (<span class="id" title="tactic">try</span> <span class="id" title="tactic">unfold</span> <span class="id" title="var">c</span>); <span class="id" title="tactic">unfold</span> <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>  <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">E_eq</span> <span class="id" title="var">M</span> <span class="id" title="var">x</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a>;<span class="id" title="tactic">simpl</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">monoid_eq_A</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">E_eq</span> <span class="id" title="var">M</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="additions.Addition_Chains.html#flatten_valid_2"><span class="id" title="lemma">flatten_valid_2</span></a>;<span class="id" title="tactic">try</span> <span class="id" title="tactic">reflexivity</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;]<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="C87_ok" class="idref" href="#C87_ok"><span class="id" title="definition">C87_ok</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 87 <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a>.  
+<br/>
+<span class="id" title="keyword">Example</span> <a id="C7_ok'" class="idref" href="#C7_ok'"><span class="id" title="definition">C7_ok'</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 7 <a class="idref" href="additions.Addition_Chains.html#C7"><span class="id" title="definition">C7</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab348"></a><h1 class="section">Correctness and parametricity</h1>
+
+
+<div class="paragraph"> </div>
+
+In this section, we show some tools for proving automatically the
+correctness of a given chain, and try to avoid spending time
+while proceeding to a lot of setoid rewritings 
+
+<div class="paragraph"> </div>
+
+First, we notice that any chain is able to compute its associated exponent:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="the_exponent_nat" class="idref" href="#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> (<a id="c:179" class="idref" href="#c:179"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a class="idref" href="additions.Addition_Chains.html#c:179"><span class="id" title="variable">c</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Monoid_instances.html#Natplus"><span class="id" title="instance">Natplus</span></a>) 1%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="the_exponent" class="idref" href="#the_exponent"><span class="id" title="definition">the_exponent</span></a> (<a id="c:180" class="idref" href="#c:180"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_execute"><span class="id" title="definition">chain_execute</span></a> <a class="idref" href="additions.Addition_Chains.html#c:180"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.add"><span class="id" title="definition">Pos.add</span></a>  1%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+Roughly, if cA is a computation on A and cB a computation on B,
+cA and cB are related through (computation_R A B R) if, during  their execution,
+the corresponding variables of type A  and B are always bound to related
+(w.r.t. <span class="inlinecode"><span class="id" title="var">R</span></span> ) values. 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="additions.Addition_Chains.html#computation_R"><span class="id" title="inductive">computation_R</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We say that a chain <span class="inlinecode"><span class="id" title="var">c</span></span> is <i>parametric</i> if
+   <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> and <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">B</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> are equivalent with respect to any  relation that
+   contains  the pair <span class="inlinecode">(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>)</span>.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="parametric" class="idref" href="#parametric"><span class="id" title="definition">parametric</span></a> (<a id="c:181" class="idref" href="#c:181"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:182" class="idref" href="#A:182"><span class="id" title="binder">A</span></a> <a id="B:183" class="idref" href="#B:183"><span class="id" title="binder">B</span></a> (<a id="R:184" class="idref" href="#R:184"><span class="id" title="binder">R</span></a>: <a class="idref" href="additions.Addition_Chains.html#A:182"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#B:183"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) (<a id="a:185" class="idref" href="#a:185"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#A:182"><span class="id" title="variable">A</span></a>) (<a id="b:186" class="idref" href="#b:186"><span class="id" title="binder">b</span></a>:<a class="idref" href="additions.Addition_Chains.html#B:183"><span class="id" title="variable">B</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#R:184"><span class="id" title="variable">R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:185"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#b:186"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#computation_R"><span class="id" title="inductive">computation_R</span></a>  <a class="idref" href="additions.Addition_Chains.html#A:182"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#B:183"><span class="id" title="variable">B</span></a> <a class="idref" href="additions.Addition_Chains.html#R:184"><span class="id" title="variable">R</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:181"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#A:182"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#a:185"><span class="id" title="variable">a</span></a>) (<a class="idref" href="additions.Addition_Chains.html#c:181"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#B:183"><span class="id" title="variable">B</span></a> <a class="idref" href="additions.Addition_Chains.html#b:186"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ The following statement cannot be proven in Coq (AFAIK) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="any_chain_parametric" class="idref" href="#any_chain_parametric"><span class="id" title="definition">any_chain_parametric</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="c:187" class="idref" href="#c:187"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>, <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:187"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">any_chain_parametric</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ Nevertheless, if we don't want to assume <span class="inlinecode"><span class="id" title="var">any_chain_parametric</span></span>, 
+ we can use the  following tactic for proving a given that a given chain <span class="inlinecode"><span class="id" title="var">c</span></span> 
+ is parametric. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">parametric_tac</span>  := <br/>
+&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> [ |- <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> ?<span class="id" title="var">c</span>] ⇒<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">red</span> ; <span class="id" title="tactic">intros</span>;<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">right</span>;[<span class="id" title="tactic">assumption</span> | <span class="id" title="tactic">assumption</span> | ]);  <span class="id" title="tactic">left</span>; <span class="id" title="tactic">assumption</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="P87" class="idref" href="#P87"><span class="id" title="definition">P87</span></a> : <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab349"></a><h2 class="section">computation of  compared to computation of </h2>
+
+
+<div class="paragraph"> </div>
+
+The following section is dedicated to prove that, if a chain  is parametric,
+it computes powers of the form , where  is obtained by applying the
+function <span class="inlinecode"><span class="id" title="var">the_exponent_nat</span></span> to .
+
+<div class="paragraph"> </div>
+
+Basically, we proceed by an induction on the hypothesis 
+<span class="inlinecode"><span class="id" title="var">equiv</span></span> <span class="inlinecode"><span class="id" title="var">gamma_A</span></span> <span class="inlinecode"><span class="id" title="var">gamma_nat</span></span> <span class="inlinecode"><span class="id" title="var">l</span></span> where <span class="inlinecode"><span class="id" title="var">gamma</span></span> is a computation on <span class="inlinecode"><span class="id" title="var">A</span></span>,
+gamma_nat a computation on <span class="inlinecode"><span class="id" title="var">nat</span></span> (with respect to the additive monoid on <span class="inlinecode"><span class="id" title="var">nat</span></span>),
+and <span class="inlinecode"><span class="id" title="var">l</span></span> is a list of pairs of the form .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Refinement_proof" class="idref" href="#Refinement_proof"><span class="id" title="section">Refinement_proof</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Refinement_proof.A" class="idref" href="#Refinement_proof.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Refinement_proof.E_op" class="idref" href="#Refinement_proof.E_op"><span class="id" title="variable">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Refinement_proof.E_one" class="idref" href="#Refinement_proof.E_one"><span class="id" title="variable">E_one</span></a> : <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Refinement_proof.E_eq" class="idref" href="#Refinement_proof.E_eq"><span class="id" title="variable">E_eq</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Context</span> (<a id="M:192" class="idref" href="#M:192"><span class="id" title="binder">M</span></a> : <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.E_op"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.E_one"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.E_eq"><span class="id" title="variable">E_eq</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Let us characterise the lists of pairs of the form , for a given 
+ and .
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="power_R" class="idref" href="#power_R"><span class="id" title="definition">power_R</span></a>  (<a id="a:193" class="idref" href="#a:193"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="x:194" class="idref" href="#x:194"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>)(<a id="n:195" class="idref" href="#n:195"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="additions.Addition_Chains.html#n:195"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="additions.Addition_Chains.html#x:194"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Addition_Chains.html#a:193"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Addition_Chains.html#n:195"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="power_R_Mult" class="idref" href="#power_R_Mult"><span class="id" title="lemma">power_R_Mult</span></a> : <span class="id" title="keyword">∀</span> <a id="a:196" class="idref" href="#a:196"><span class="id" title="binder">a</span></a> <a id="x:197" class="idref" href="#x:197"><span class="id" title="binder">x</span></a> <a id="y:198" class="idref" href="#y:198"><span class="id" title="binder">y</span></a> <a id="i:199" class="idref" href="#i:199"><span class="id" title="binder">i</span></a> <a id="j:200" class="idref" href="#j:200"><span class="id" title="binder">j</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:196"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#x:197"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#i:199"><span class="id" title="variable">i</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:196"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#y:198"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Addition_Chains.html#j:200"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:196"><span class="id" title="variable">a</span></a> (<a class="idref" href="additions.Addition_Chains.html#x:197"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#y:198"><span class="id" title="variable">y</span></a>) (<a class="idref" href="additions.Addition_Chains.html#i:199"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="additions.Addition_Chains.html#j:200"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="power_R_1" class="idref" href="#power_R_1"><span class="id" title="lemma">power_R_1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:201" class="idref" href="#a:201"><span class="id" title="binder">a</span></a>, <a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:201"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#a:201"><span class="id" title="variable">a</span></a> 1%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="power_R_is_a_refinement" class="idref" href="#power_R_is_a_refinement"><span class="id" title="lemma">power_R_is_a_refinement</span></a> (<a id="a:202" class="idref" href="#a:202"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>(<a id="gamma:203" class="idref" href="#gamma:203"><span class="id" title="binder">gamma</span></a> : @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="gamma_nat:204" class="idref" href="#gamma_nat:204"><span class="id" title="binder">gamma_nat</span></a> : @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_R"><span class="id" title="inductive">computation_R</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>  (<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:202"><span class="id" title="variable">a</span></a>) <a class="idref" href="additions.Addition_Chains.html#gamma:203"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Addition_Chains.html#gamma_nat:204"><span class="id" title="variable">gamma_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:202"><span class="id" title="variable">a</span></a> (<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> <a class="idref" href="additions.Addition_Chains.html#gamma:203"><span class="id" title="variable">gamma</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> (<span class="id" title="var">M</span>:= <a class="idref" href="additions.Monoid_instances.html#Natplus"><span class="id" title="instance">Natplus</span></a>) <a class="idref" href="additions.Addition_Chains.html#gamma_nat:204"><span class="id" title="variable">gamma_nat</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="param_correctness_aux" class="idref" href="#param_correctness_aux"><span class="id" title="lemma">param_correctness_aux</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="c:205" class="idref" href="#c:205"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>)(<a id="a:206" class="idref" href="#a:206"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_R"><span class="id" title="inductive">computation_R</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> (<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a> <a class="idref" href="additions.Addition_Chains.html#a:206"><span class="id" title="variable">a</span></a> ) (<a class="idref" href="additions.Addition_Chains.html#c:205"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#a:206"><span class="id" title="variable">a</span></a>) (<a class="idref" href="additions.Addition_Chains.html#c:205"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> 1%<span class="id" title="var">nat</span>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#power_R"><span class="id" title="definition">power_R</span></a>  <a class="idref" href="additions.Addition_Chains.html#a:206"><span class="id" title="variable">a</span></a> (<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:205"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#a:206"><span class="id" title="variable">a</span></a>)) (<a class="idref" href="additions.Addition_Chains.html#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#c:205"><span class="id" title="variable">c</span></a>) .<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Addition_Chains.html#Refinement_proof"><span class="id" title="section">Refinement_proof</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab350"></a><h2 class="section">Correctness Theorem</h2>
+
+
+<div class="paragraph"> </div>
+
+From our study of the <span class="inlinecode"><span class="id" title="var">computation</span>\<span class="id" title="var">_R</span></span> predicate, we infer that any parametric chain 
+is correct with respect to the number <span class="inlinecode"><span class="id" title="var">the_exponent_nat</span></span> <span class="inlinecode"><span class="id" title="var">c</span></span>
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exponent_nat_neq_0" class="idref" href="#exponent_nat_neq_0"><span class="id" title="lemma">exponent_nat_neq_0</span></a> : <span class="id" title="keyword">∀</span> <a id="c:207" class="idref" href="#c:207"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>,  <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:207"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#c:207"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exponent_pos2nat" class="idref" href="#exponent_pos2nat"><span class="id" title="lemma">exponent_pos2nat</span></a> : <span class="id" title="keyword">∀</span> <a id="c:208" class="idref" href="#c:208"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:208"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#c:208"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> (<a class="idref" href="additions.Addition_Chains.html#the_exponent"><span class="id" title="definition">the_exponent</span></a> <a class="idref" href="additions.Addition_Chains.html#c:208"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exponent_pos_of_nat" class="idref" href="#exponent_pos_of_nat"><span class="id" title="lemma">exponent_pos_of_nat</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="c:213" class="idref" href="#c:213"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>,  <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:213"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#the_exponent"><span class="id" title="definition">the_exponent</span></a> <a class="idref" href="additions.Addition_Chains.html#c:213"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> (<a class="idref" href="additions.Addition_Chains.html#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#c:213"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="param_correctness_nat" class="idref" href="#param_correctness_nat"><span class="id" title="lemma">param_correctness_nat</span></a> (<a id="c:214" class="idref" href="#c:214"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:214"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_correct_nat"><span class="id" title="definition">chain_correct_nat</span></a> (<a class="idref" href="additions.Addition_Chains.html#the_exponent_nat"><span class="id" title="definition">the_exponent_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#c:214"><span class="id" title="variable">c</span></a>) <a class="idref" href="additions.Addition_Chains.html#c:214"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="param_correctness" class="idref" href="#param_correctness"><span class="id" title="lemma">param_correctness</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="p:215" class="idref" href="#p:215"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="c:216" class="idref" href="#c:216"><span class="id" title="binder">c</span></a>:<a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#p:215"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Addition_Chains.html#the_exponent"><span class="id" title="definition">the_exponent</span></a> <a class="idref" href="additions.Addition_Chains.html#c:216"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:216"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Addition_Chains.html#p:215"><span class="id" title="variable">p</span></a>  <a class="idref" href="additions.Addition_Chains.html#c:216"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab351"></a><h3 class="section">Remark</h3>
+
+
+<div class="paragraph"> </div>
+
+If we admit that any chain is parametric, we infer the correctness of every chain.
+
+<div class="paragraph"> </div>
+
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="param_correctness_for_free" class="idref" href="#param_correctness_for_free"><span class="id" title="lemma">param_correctness_for_free</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#any_chain_parametric"><span class="id" title="definition">any_chain_parametric</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:217" class="idref" href="#p:217"><span class="id" title="binder">p</span></a> (<a id="c:218" class="idref" href="#c:218"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) ,<br/>
+<a class="idref" href="additions.Addition_Chains.html#p:217"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Addition_Chains.html#the_exponent"><span class="id" title="definition">the_exponent</span></a> <a class="idref" href="additions.Addition_Chains.html#c:218"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Addition_Chains.html#p:217"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Addition_Chains.html#c:218"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab352"></a><h3 class="section">Back to 87</h3>
+
+
+<div class="paragraph"> </div>
+
+We present two new-proofs of consistency of <span class="inlinecode"><span class="id" title="var">C87</span></span>.
+The first one uses the <span class="inlinecode"><span class="id" title="var">parametric_tac</span></span> tactic, the other one 
+using the "free refinement" method 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">param_chain_correct</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;[|- <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> ?<span class="id" title="var">p</span> ?<span class="id" title="var">c</span> ] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">H</span> := <span class="id" title="tactic">fresh</span> "H" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Addition_Chains.html#the_exponent"><span class="id" title="definition">the_exponent</span></a> <span class="id" title="var">c</span>) <span class="id" title="tactic">by</span> <span class="id" title="tactic">reflexivity</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="additions.Addition_Chains.html#param_correctness"><span class="id" title="lemma">param_correctness</span></a>;[<span class="id" title="tactic">trivial</span> | <span class="id" title="var">parametric_tac</span>]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="C87_ok'" class="idref" href="#C87_ok'"><span class="id" title="lemma">C87_ok'</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 87  <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L87''" class="idref" href="#L87''"><span class="id" title="lemma">L87''</span></a> : <a class="idref" href="additions.Addition_Chains.html#any_chain_parametric"><span class="id" title="definition">any_chain_parametric</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 87 <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab353"></a><h2 class="section">Correct by construction chains</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_generator" class="idref" href="#chain_generator"><span class="id" title="definition">chain_generator</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="correct_generator" class="idref" href="#correct_generator"><span class="id" title="definition">correct_generator</span></a> (<a id="gen:219" class="idref" href="#gen:219"><span class="id" title="binder">gen</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="p:220" class="idref" href="#p:220"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Addition_Chains.html#p:220"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Addition_Chains.html#gen:219"><span class="id" title="variable">gen</span></a> <a class="idref" href="additions.Addition_Chains.html#p:220"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Computing  using a chain generator 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cpower_pos" class="idref" href="#cpower_pos"><span class="id" title="definition">cpower_pos</span></a> (<a id="g:221" class="idref" href="#g:221"><span class="id" title="binder">g</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_generator"><span class="id" title="definition">chain_generator</span></a>)  <a id="p:222" class="idref" href="#p:222"><span class="id" title="binder">p</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="M:227" class="idref" href="#M:227"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:223" class="idref" href="#A:223"><span class="id" title="binder">A</span></a> <a id="E_op:224" class="idref" href="#E_op:224"><span class="id" title="binder">E_op</span></a> <a id="E_one:225" class="idref" href="#E_one:225"><span class="id" title="binder">E_one</span></a> <a id="E_eq:226" class="idref" href="#E_eq:226"><span class="id" title="binder">E_eq</span></a>} <a id="a:228" class="idref" href="#a:228"><span class="id" title="binder">a</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> (<a class="idref" href="additions.Addition_Chains.html#g:221"><span class="id" title="variable">g</span></a> <a class="idref" href="additions.Addition_Chains.html#p:222"><span class="id" title="variable">p</span></a>) (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Addition_Chains.html#M:227"><span class="id" title="variable">M</span></a>) <a class="idref" href="additions.Addition_Chains.html#a:228"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cpower" class="idref" href="#cpower"><span class="id" title="definition">cpower</span></a> (<a id="g:229" class="idref" href="#g:229"><span class="id" title="binder">g</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_generator"><span class="id" title="definition">chain_generator</span></a>)  <a id="n:230" class="idref" href="#n:230"><span class="id" title="binder">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="M:235" class="idref" href="#M:235"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:231" class="idref" href="#A:231"><span class="id" title="binder">A</span></a> <a id="E_op:232" class="idref" href="#E_op:232"><span class="id" title="binder">E_op</span></a> <a id="E_one:233" class="idref" href="#E_one:233"><span class="id" title="binder">E_one</span></a> <a id="E_eq:234" class="idref" href="#E_eq:234"><span class="id" title="binder">E_eq</span></a>} <a id="a:236" class="idref" href="#a:236"><span class="id" title="binder">a</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#n:230"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0%<span class="id" title="var">N</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#E_one:233"><span class="id" title="variable">E_one</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Addition_Chains.html#cpower_pos"><span class="id" title="definition">cpower_pos</span></a>  <a class="idref" href="additions.Addition_Chains.html#g:229"><span class="id" title="variable">g</span></a> <span class="id" title="var">p</span> <a class="idref" href="additions.Addition_Chains.html#a:236"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab354"></a><h3 class="section">Definition</h3>
+
+A chain generator is <i>optimal</i> if it returns chains whose length are less or 
+equal than the chains returned by any correct generator.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="optimal_generator" class="idref" href="#optimal_generator"><span class="id" title="definition">optimal_generator</span></a> (<a id="g:238" class="idref" href="#g:238"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> ) :=<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="p:239" class="idref" href="#p:239"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, <a class="idref" href="additions.Addition_Chains.html#optimal"><span class="id" title="definition">optimal</span></a> <a class="idref" href="additions.Addition_Chains.html#p:239"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Addition_Chains.html#g:238"><span class="id" title="variable">g</span></a> <a class="idref" href="additions.Addition_Chains.html#p:239"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab355"></a><h2 class="section">Some chain generators</h2>
+
+
+<div class="paragraph"> </div>
+
+We reinterpret the naïve and binary exponentiation algorithms in the framework 
+of addition chains.
+
+<div class="paragraph"> </div>
+
+Instead of directly computing  for some base  and exponent ,
+we build chains that describe the computations associated with both methods.
+Not surprisingly, this chain generation will be described in terms of recursive
+functions, once the underlying monoid is fixed.
+
+<div class="paragraph"> </div>
+
+<a id="lab356"></a><h2 class="section">Chains associated to the well-known binary exponentiation algorithm</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab357"></a><h3 class="section">computation of </h3>
+
+
+<div class="paragraph"> </div>
+
+As for the "classical" binary exponentiation algorithm,
+we define an auxiliary computation generator for computing the
+product of an accumulator  with an arbitrary power of some value 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="axp_scheme" class="idref" href="#axp_scheme"><span class="id" title="definition">axp_scheme</span></a>  {<a id="A:240" class="idref" href="#A:240"><span class="id" title="binder">A</span></a>} <a id="p:241" class="idref" href="#p:241"><span class="id" title="binder">p</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:240"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Addition_Chains.html#A:240"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#A:240"><span class="id" title="variable">A</span></a>   :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#p:241"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒  (<span class="id" title="keyword">fun</span> <a id="a:244" class="idref" href="#a:244"><span class="id" title="binder">a</span></a> <a id="x:245" class="idref" href="#x:245"><span class="id" title="binder">x</span></a> ⇒ <a id="y:246" class="idref" href="#y:246"><span class="id" title="binder">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#a:244"><span class="id" title="variable">a</span></a>  <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:245"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#y:246"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ (<span class="id" title="keyword">fun</span> <a id="a:247" class="idref" href="#a:247"><span class="id" title="binder">a</span></a> <a id="x:248" class="idref" href="#x:248"><span class="id" title="binder">x</span></a> ⇒ <a id="x2:249" class="idref" href="#x2:249"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:248"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a>  <a class="idref" href="additions.Addition_Chains.html#x:248"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Addition_Chains.html#axp_scheme:242"><span class="id" title="definition">axp_scheme</span></a> <span class="id" title="var">q</span> <a class="idref" href="additions.Addition_Chains.html#a:247"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:249"><span class="id" title="variable">x2</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ (<span class="id" title="keyword">fun</span> <a id="a:250" class="idref" href="#a:250"><span class="id" title="binder">a</span></a> <a id="x:251" class="idref" href="#x:251"><span class="id" title="binder">x</span></a> ⇒ <a id="ax:252" class="idref" href="#ax:252"><span class="id" title="binder">ax</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#a:250"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:251"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="x2:253" class="idref" href="#x2:253"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:251"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:251"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#axp_scheme:242"><span class="id" title="definition">axp_scheme</span></a> <span class="id" title="var">q</span> <a class="idref" href="additions.Addition_Chains.html#ax:252"><span class="id" title="variable">ax</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:253"><span class="id" title="variable">x2</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span>  <a id="bin_pow_scheme" class="idref" href="#bin_pow_scheme"><span class="id" title="definition">bin_pow_scheme</span></a> {<a id="A:254" class="idref" href="#A:254"><span class="id" title="binder">A</span></a>} (<a id="p:255" class="idref" href="#p:255"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)  : <a class="idref" href="additions.Addition_Chains.html#A:254"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#A:254"><span class="id" title="variable">A</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#p:255"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span>  |  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="x:258" class="idref" href="#x:258"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Addition_Chains.html#x:258"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span>  ⇒ <span class="id" title="keyword">fun</span> <a id="x:259" class="idref" href="#x:259"><span class="id" title="binder">x</span></a> ⇒ <a id="x2:260" class="idref" href="#x2:260"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:259"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:259"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Addition_Chains.html#axp_scheme"><span class="id" title="definition">axp_scheme</span></a> <span class="id" title="var">q</span> <a class="idref" href="additions.Addition_Chains.html#x:259"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#x2:260"><span class="id" title="variable">x2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <span class="id" title="keyword">fun</span> <a id="x:261" class="idref" href="#x:261"><span class="id" title="binder">x</span></a> ⇒ <a id="x2:262" class="idref" href="#x2:262"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Addition_Chains.html#x:261"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Addition_Chains.html#x:261"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Addition_Chains.html#bin_pow_scheme:256"><span class="id" title="definition">bin_pow_scheme</span></a> <span class="id" title="var">q</span> <a class="idref" href="additions.Addition_Chains.html#x2:262"><span class="id" title="variable">x2</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="binary_chain" class="idref" href="#binary_chain"><span class="id" title="definition">binary_chain</span></a> (<a id="p:263" class="idref" href="#p:263"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="A:264" class="idref" href="#A:264"><span class="id" title="binder">A</span></a> ⇒ <a class="idref" href="additions.Addition_Chains.html#bin_pow_scheme"><span class="id" title="definition">bin_pow_scheme</span></a> <a class="idref" href="additions.Addition_Chains.html#p:263"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab358"></a><h2 class="section">Correctness of the binary method</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="binary_power_proof" class="idref" href="#binary_power_proof"><span class="id" title="section">binary_power_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a id="binary_power_proof.A" class="idref" href="#binary_power_proof.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="binary_power_proof.E_op" class="idref" href="#binary_power_proof.E_op"><span class="id" title="variable">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Addition_Chains.html#A:265"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="binary_power_proof.E_one" class="idref" href="#binary_power_proof.E_one"><span class="id" title="variable">E_one</span></a> : <a class="idref" href="additions.Addition_Chains.html#A:265"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="binary_power_proof.E_eq" class="idref" href="#binary_power_proof.E_eq"><span class="id" title="variable">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Addition_Chains.html#A:265"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="M:269" class="idref" href="#M:269"><span class="id" title="binder">M</span></a> : <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="additions.Addition_Chains.html#binary_power_proof.E_op"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Addition_Chains.html#binary_power_proof.E_one"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Addition_Chains.html#binary_power_proof.E_eq"><span class="id" title="variable">E_eq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Existing Instance</span> <span class="id" title="var">Eop_proper</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ As for linear chains, we first establish the correctness of the auxiliary
+  function <span class="inlinecode"><span class="id" title="var">axp_scheme</span></span>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="axp_correct" class="idref" href="#axp_correct"><span class="id" title="lemma">axp_correct</span></a> : <span class="id" title="keyword">∀</span> <a id="p:270" class="idref" href="#p:270"><span class="id" title="binder">p</span></a> <a id="a:271" class="idref" href="#a:271"><span class="id" title="binder">a</span></a> <a id="x:272" class="idref" href="#x:272"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> (<a class="idref" href="additions.Addition_Chains.html#axp_scheme"><span class="id" title="definition">axp_scheme</span></a> <a class="idref" href="additions.Addition_Chains.html#p:270"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Addition_Chains.html#a:271"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#x:272"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#a:271"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Addition_Chains.html#x:272"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#p:270"><span class="id" title="variable">p</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="binary_correct" class="idref" href="#binary_correct"><span class="id" title="lemma">binary_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:273" class="idref" href="#p:273"><span class="id" title="binder">p</span></a> <a id="x:274" class="idref" href="#x:274"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_eval"><span class="id" title="definition">computation_eval</span></a> (<a class="idref" href="additions.Addition_Chains.html#bin_pow_scheme"><span class="id" title="definition">bin_pow_scheme</span></a> <a class="idref" href="additions.Addition_Chains.html#p:273"><span class="id" title="variable">p</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="additions.Addition_Chains.html#binary_power_proof.A"><span class="id" title="variable">A</span></a>) <a class="idref" href="additions.Addition_Chains.html#x:274"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#x:274"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Addition_Chains.html#p:273"><span class="id" title="variable">p</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span>  <a class="idref" href="additions.Addition_Chains.html#binary_power_proof"><span class="id" title="section">binary_power_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="binary_generator_correct" class="idref" href="#binary_generator_correct"><span class="id" title="lemma">binary_generator_correct</span></a> : <a class="idref" href="additions.Addition_Chains.html#correct_generator"><span class="id" title="definition">correct_generator</span></a> <a class="idref" href="additions.Addition_Chains.html#binary_chain"><span class="id" title="definition">binary_chain</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab359"></a><h2 class="section">The binary method is not optimal</h2>
+
+
+<div class="paragraph"> </div>
+
+    We use the function <span class="inlinecode"><span class="id" title="var">chain_length</span></span> and proofs of correctness 
+  for showing that <span class="inlinecode"><span class="id" title="var">binary_chain</span></span>, although correct, is not an optimal 
+  chain generator.
+
+<div class="paragraph"> </div>
+
+   Our proof  is structured as a proof by contradiction, under the 
+   hypothesis that <span class="inlinecode"><span class="id" title="var">binary_chain</span></span> is optimal.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="non_optimality_proof" class="idref" href="#non_optimality_proof"><span class="id" title="section">non_optimality_proof</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="non_optimality_proof.binary_opt" class="idref" href="#non_optimality_proof.binary_opt"><span class="id" title="variable">binary_opt</span></a> : <a class="idref" href="additions.Addition_Chains.html#optimal_generator"><span class="id" title="definition">optimal_generator</span></a> <a class="idref" href="additions.Addition_Chains.html#binary_chain"><span class="id" title="definition">binary_chain</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="binary_generator_not_optimal" class="idref" href="#binary_generator_not_optimal"><span class="id" title="lemma">binary_generator_not_optimal</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.  
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Addition_Chains.html#non_optimality_proof"><span class="id" title="section">non_optimality_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="S1" class="idref" href="#S1"><span class="id" title="section">S1</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="clog2" class="idref" href="#clog2"><span class="id" title="definition">clog2</span></a> (<a id="p:276" class="idref" href="#p:276"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#p:276"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ 1%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ 2%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="additions.Addition_Chains.html#clog2:277"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:276"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="additions.Addition_Chains.html#clog2:277"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:276"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="exp2" class="idref" href="#exp2"><span class="id" title="definition">exp2</span></a> (<a id="n:279" class="idref" href="#n:279"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Addition_Chains.html#n:279"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#exp2:280"><span class="id" title="definition">exp2</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="exp2_Plus" class="idref" href="#exp2_Plus"><span class="id" title="lemma">exp2_Plus</span></a> : <span class="id" title="keyword">∀</span> <a id="n:282" class="idref" href="#n:282"><span class="id" title="binder">n</span></a> <a id="p:283" class="idref" href="#p:283"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Addition_Chains.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="additions.Addition_Chains.html#n:282"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Addition_Chains.html#p:283"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="additions.Addition_Chains.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="additions.Addition_Chains.html#n:282"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:283"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="axp_scheme_length1" class="idref" href="#axp_scheme_length1"><span class="id" title="lemma">axp_scheme_length1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:284" class="idref" href="#p:284"><span class="id" title="binder">p</span></a>, (<a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> (<a class="idref" href="additions.Addition_Chains.html#axp_scheme"><span class="id" title="definition">axp_scheme</span></a>  <a class="idref" href="additions.Addition_Chains.html#p:284"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>  ) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#clog2"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:284"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bin_pow_scheme_length1" class="idref" href="#bin_pow_scheme_length1"><span class="id" title="lemma">bin_pow_scheme_length1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:285" class="idref" href="#p:285"><span class="id" title="binder">p</span></a>, (<a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> (<a class="idref" href="additions.Addition_Chains.html#bin_pow_scheme"><span class="id" title="definition">bin_pow_scheme</span></a>  <a class="idref" href="additions.Addition_Chains.html#p:285"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>  ) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#clog2"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:285"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="binary_chain_length" class="idref" href="#binary_chain_length"><span class="id" title="lemma">binary_chain_length</span></a> : <span class="id" title="keyword">∀</span> <a id="p:286" class="idref" href="#p:286"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Addition_Chains.html#chain_length"><span class="id" title="definition">chain_length</span></a> (<a class="idref" href="additions.Addition_Chains.html#binary_chain"><span class="id" title="definition">binary_chain</span></a> <a class="idref" href="additions.Addition_Chains.html#p:286"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#clog2"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:286"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="optimal_upper_bound" class="idref" href="#optimal_upper_bound"><span class="id" title="lemma">optimal_upper_bound</span></a>: <span class="id" title="keyword">∀</span> <a id="p:287" class="idref" href="#p:287"><span class="id" title="binder">p</span></a> <a id="c:288" class="idref" href="#c:288"><span class="id" title="binder">c</span></a>, <a class="idref" href="additions.Addition_Chains.html#optimal"><span class="id" title="definition">optimal</span></a> <a class="idref" href="additions.Addition_Chains.html#p:287"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Addition_Chains.html#c:288"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Addition_Chains.html#chain_length"><span class="id" title="definition">chain_length</span></a> <a class="idref" href="additions.Addition_Chains.html#c:288"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Addition_Chains.html#clog2"><span class="id" title="definition">clog2</span></a> <a class="idref" href="additions.Addition_Chains.html#p:287"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Addition_Chains.html#S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab360"></a><h1 class="section">How to prove chain correctness in general ?</h1>
+
+
+<div class="paragraph"> </div>
+
+ In previous sections, we showed proofs of correctness of two chain generators.
+ By definition, every chain returned by these generators is correct w.r.t. 
+ the given exponent.
+
+<div class="paragraph"> </div>
+
+ The problem with <span class="inlinecode"><span class="id" title="var">C87</span></span> is quite different. This chain 
+ has been given by hand, and we had to prove its correctness. 
+
+<div class="paragraph"> </div>
+
+ Although the proof script for lemma <span class="inlinecode"><span class="id" title="var">L87</span></span> is quite short, the resulting 
+ proof  is quite clumsy, consisting in a long chain of equivalences using
+ associativity of the multiplication.
+
+<div class="paragraph"> </div>
+
+ The reader can easily be convinced of this fact, just by the command 
+<span class="inlinecode"><span class="id" title="keyword">Print</span></span> <span class="inlinecode"><span class="id" title="var">L87</span>.</span>
+
+<div class="paragraph"> </div>
+
+In a similar way, it may happen that a correct chain generator is hard
+to certify in . In this case, one may chose to prove the 
+correctness of various chains returned by the generator.
+
+<div class="paragraph"> </div>
+
+In the rest of this section, we investigate some ways of proving 
+the correctness of given chains.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="S2" class="idref" href="#S2"><span class="id" title="section">S2</span></a>.<br/>
+</div>
+
+<div class="doc">
+Difficult and unfinished (conjecture)
+
+<div class="paragraph"> </div>
+
+ We would like to prove that for  any parametric chain c for p,
+c's length is greater or equal than  log2(p).
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variables</span> <a id="S2.A" class="idref" href="#S2.A"><span class="id" title="variable">A</span></a> <a id="S2.B" class="idref" href="#S2.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+<span class="id" title="keyword">Variables</span> (<a id="S2.a" class="idref" href="#S2.a"><span class="id" title="variable">a</span></a>:<a class="idref" href="additions.Addition_Chains.html#S2.A"><span class="id" title="variable">A</span></a>)(<a id="S2.b" class="idref" href="#S2.b"><span class="id" title="variable">b</span></a>:<a class="idref" href="additions.Addition_Chains.html#S2.B"><span class="id" title="variable">B</span></a>).<br/>
+<span class="id" title="keyword">Let</span> <a id="S2.R_true" class="idref" href="#S2.R_true"><span class="id" title="variable">R_true</span></a> := <span class="id" title="keyword">fun</span> (<a id="x:293" class="idref" href="#x:293"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Addition_Chains.html#S2.A"><span class="id" title="variable">A</span></a>)(<a id="y:294" class="idref" href="#y:294"><span class="id" title="binder">y</span></a>:<a class="idref" href="additions.Addition_Chains.html#S2.B"><span class="id" title="variable">B</span></a>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="L" class="idref" href="#L"><span class="id" title="lemma">L</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>(<a id="gammaA:295" class="idref" href="#gammaA:295"><span class="id" title="binder">gammaA</span></a> : @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.A"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="gammaB:296" class="idref" href="#gammaB:296"><span class="id" title="binder">gammaB</span></a> : @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.B"><span class="id" title="variable">B</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_R"><span class="id" title="inductive">computation_R</span></a>  <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="additions.Addition_Chains.html#S2.R_true"><span class="id" title="variable">R_true</span></a> <a class="idref" href="additions.Addition_Chains.html#gammaA:295"><span class="id" title="variable">gammaA</span></a> <a class="idref" href="additions.Addition_Chains.html#gammaB:296"><span class="id" title="variable">gammaB</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.a"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Addition_Chains.html#gammaA:295"><span class="id" title="variable">gammaA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.B"><span class="id" title="variable">B</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.b"><span class="id" title="variable">b</span></a> <a class="idref" href="additions.Addition_Chains.html#gammaB:296"><span class="id" title="variable">gammaB</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <span class="id" title="keyword">∀</span> <a id="c:297" class="idref" href="#c:297"><span class="id" title="binder">c</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>, <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Addition_Chains.html#c:297"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.a"><span class="id" title="variable">a</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:297"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Addition_Chains.html#computation_length"><span class="id" title="definition">computation_length</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.b"><span class="id" title="variable">b</span></a> (<a class="idref" href="additions.Addition_Chains.html#c:297"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.B"><span class="id" title="variable">B</span></a> <a class="idref" href="additions.Addition_Chains.html#S2.b"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Addition_Chains.html#S2"><span class="id" title="section">S2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Extraction</span> <span class="id" title="var">Language</span> <span class="id" title="keyword">Scheme</span>.<br/>
+<span class="id" title="keyword">Recursive Extraction</span> <span class="id" title="var">chain_apply</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.BinaryStrat.html b/theories/html/additions.BinaryStrat.html
new file mode 100644
index 00000000..96137d96
--- /dev/null
+++ b/theories/html/additions.BinaryStrat.html
@@ -0,0 +1,69 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.BinaryStrat</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.BinaryStrat</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Binary strategy, according to Bergeron, Brlek et al. 
+
+<div class="paragraph"> </div>
+
+  Let  be a positive number. We associate to  the half of .
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a> <a class="idref" href="additions.Compatibility.html#"><span class="id" title="library">Compatibility</span></a> <a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="additions.Strategies.html#"><span class="id" title="library">Strategies</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="half" class="idref" href="#half"><span class="id" title="definition">half</span></a> (<a id="p:1" class="idref" href="#p:1"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.BinaryStrat.html#p:1"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|    <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒  <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="two" class="idref" href="#two"><span class="id" title="definition">two</span></a> (<a id="p:3" class="idref" href="#p:3"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) := 2%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Binary_strat" class="idref" href="#Binary_strat"><span class="id" title="instance">Binary_strat</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.BinaryStrat.html#half"><span class="id" title="definition">half</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Two_strat" class="idref" href="#Two_strat"><span class="id" title="instance">Two_strat</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.BinaryStrat.html#two"><span class="id" title="definition">two</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Compatibility.html b/theories/html/additions.Compatibility.html
new file mode 100644
index 00000000..6eb50f03
--- /dev/null
+++ b/theories/html/additions.Compatibility.html
@@ -0,0 +1,295 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Compatibility</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Compatibility</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab361"></a><h1 class="section">Compatibility with StdLib exponentiation functions</h1>
+
+
+<div class="paragraph"> </div>
+
+Function for computing powers already  exist in 's standard library. We provide
+equivalence theorems with functions defined in our module <span class="inlinecode"><span class="id" title="var">Pow</span></span>.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab362"></a><h2 class="section">really logarithmic versions of N.pow, Pos.pow and Z.pow</h2>
+
+
+<div class="paragraph"> </div>
+
+We propose three functions that are extensionnaly equivalent to functions
+of the standard library.
+These functions are defined in the  module;  we just rename them for readability's sake.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N_pow" class="idref" href="#N_pow"><span class="id" title="definition">N_pow</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) :=<a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">Pow.N_bpow</span></a> <a class="idref" href="additions.Compatibility.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Compatibility.html#n:2"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Pos_pow" class="idref" href="#Pos_pow"><span class="id" title="definition">Pos_pow</span></a> (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) := <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pow.Pos_bpow</span></a> <a class="idref" href="additions.Compatibility.html#a:3"><span class="id" title="variable">a</span></a>  <a class="idref" href="additions.Compatibility.html#n:4"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Z_pow" class="idref" href="#Z_pow"><span class="id" title="definition">Z_pow</span></a> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Compatibility.html#y:6"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <br/>
+| 0%<span class="id" title="var">Z</span> ⇒ 1%<span class="id" title="var">Z</span><br/>
+| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pos"><span class="id" title="abbreviation">Z.pos</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">Pow.N_bpow</span></a> <a class="idref" href="additions.Compatibility.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a>  <span class="id" title="var">p</span>)<br/>
+| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.neg"><span class="id" title="abbreviation">Z.neg</span></a> <span class="id" title="var">_</span> ⇒ 0%<span class="id" title="var">Z</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab363"></a><h2 class="section">Equivalence between StdLib definitions and ours</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Equivalence" class="idref" href="#Equivalence"><span class="id" title="section">Equivalence</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> (<a id="Equivalence.A" class="idref" href="#Equivalence.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Equivalence.op" class="idref" href="#Equivalence.op"><span class="id" title="variable">op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Compatibility.html#A:8"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Equivalence.one" class="idref" href="#Equivalence.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.Compatibility.html#A:8"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="M:11" class="idref" href="#M:11"><span class="id" title="binder">M</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a>  <a class="idref" href="additions.Compatibility.html#Equivalence.op"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a>).<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="keyword">fun</span> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a> <a id="y:13" class="idref" href="#y:13"><span class="id" title="binder">y</span></a>:<a class="idref" href="additions.Compatibility.html#Equivalence.A"><span class="id" title="variable">A</span></a> ⇒  <a class="idref" href="additions.Compatibility.html#x:12"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Compatibility.html#y:13"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_rw</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (@<a class="idref" href="additions.Monoid_def.html#one_left"><span class="id" title="method">one_left</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.op"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.M"><span class="id" title="variable">M</span></a>) || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (@<a class="idref" href="additions.Monoid_def.html#one_right"><span class="id" title="method">one_right</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.op"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.M"><span class="id" title="variable">M</span></a>)|| <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (@<a class="idref" href="additions.Monoid_def.html#op_assoc"><span class="id" title="method">op_assoc</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.A"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.op"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.M"><span class="id" title="variable">M</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_simpl</span> := <span class="id" title="tactic">repeat</span> <span class="id" title="var">monoid_rw</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">power_simpl</span> := <span class="id" title="tactic">repeat</span> (<span class="id" title="var">monoid_rw</span> || <span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Equivalence.pos_iter_M" class="idref" href="#Equivalence.pos_iter_M"><span class="id" title="variable">pos_iter_M</span></a> <a id="x:14" class="idref" href="#x:14"><span class="id" title="binder">x</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.iter"><span class="id" title="definition">Pos.iter</span></a> (<a class="idref" href="additions.Compatibility.html#Equivalence.op"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Compatibility.html#x:14"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  During an execution of the binary exponentiation algorithm
+  through the function <span class="inlinecode"><span class="id" title="var">binary_power_mult</span></span>,
+  the  "accumulator" <span class="inlinecode"><span class="id" title="var">acc</span></span> is always a power of . Thus, even if the 
+  considered monoid is not abelian, the accumulator commutes with any 
+  other power of .
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Equivalence.is_power_of" class="idref" href="#Equivalence.is_power_of"><span class="id" title="variable">is_power_of</span></a> (<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a> <a id="acc:16" class="idref" href="#acc:16"><span class="id" title="binder">acc</span></a>:<a class="idref" href="additions.Compatibility.html#Equivalence.A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:17" class="idref" href="#i:17"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Compatibility.html#acc:16"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Compatibility.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#i:17"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Pos_iter_op_ok_0" class="idref" href="#Pos_iter_op_ok_0"><span class="id" title="lemma">Pos_iter_op_ok_0</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:18" class="idref" href="#p:18"><span class="id" title="binder">p</span></a> <a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> <a id="acc:20" class="idref" href="#acc:20"><span class="id" title="binder">acc</span></a>, <a class="idref" href="additions.Compatibility.html#Equivalence.is_power_of"><span class="id" title="variable">is_power_of</span></a> <a class="idref" href="additions.Compatibility.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#acc:20"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Compatibility.html#Equivalence.pos_iter_M"><span class="id" title="variable">pos_iter_M</span></a> <a class="idref" href="additions.Compatibility.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#acc:20"><span class="id" title="variable">acc</span></a>  <a class="idref" href="additions.Compatibility.html#p:18"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a> <a class="idref" href="additions.Compatibility.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#acc:20"><span class="id" title="variable">acc</span></a> <a class="idref" href="additions.Compatibility.html#p:18"><span class="id" title="variable">p</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Pos_iter_op_ok" class="idref" href="#Pos_iter_op_ok"><span class="id" title="lemma">Pos_iter_op_ok</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:21" class="idref" href="#p:21"><span class="id" title="binder">p</span></a> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Compatibility.html#Equivalence.pos_iter_M"><span class="id" title="variable">pos_iter_M</span></a> <a class="idref" href="additions.Compatibility.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a>  <a class="idref" href="additions.Compatibility.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>   <a class="idref" href="additions.Compatibility.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Compatibility.html#p:21"><span class="id" title="variable">p</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Pos_iter_ok" class="idref" href="#Pos_iter_ok"><span class="id" title="lemma">Pos_iter_ok</span></a>: <span class="id" title="keyword">∀</span> <a id="p:23" class="idref" href="#p:23"><span class="id" title="binder">p</span></a> <a id="x:24" class="idref" href="#x:24"><span class="id" title="binder">x</span></a>,  <a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">N_bpow</span></a> <a class="idref" href="additions.Compatibility.html#x:24"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="additions.Compatibility.html#p:23"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.pos_iter_M"><span class="id" title="variable">pos_iter_M</span></a> <a class="idref" href="additions.Compatibility.html#x:24"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#Equivalence.one"><span class="id" title="variable">one</span></a>   <a class="idref" href="additions.Compatibility.html#p:23"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Compatibility.html#Equivalence"><span class="id" title="section">Equivalence</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_pow_power" class="idref" href="#Pos_pow_power"><span class="id" title="lemma">Pos_pow_power</span></a> : <span class="id" title="keyword">∀</span> <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> <a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Compatibility.html#a:26"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#72c1e614b36b07dc51a712bc261ee813"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:25"><span class="id" title="variable">n</span></a> )%<span class="id" title="var">positive</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Compatibility.html#a:26"><span class="id" title="variable">a</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Compatibility.html#n:25"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Npos_power_compat" class="idref" href="#Npos_power_compat"><span class="id" title="lemma">Npos_power_compat</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> (<a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Compatibility.html#a:28"><span class="id" title="variable">a</span></a>  <a class="idref" href="additions.Compatibility.html#n:27"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="additions.Compatibility.html#a:28"><span class="id" title="variable">a</span></a>)  <a class="idref" href="additions.Compatibility.html#n:27"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_pow_power" class="idref" href="#N_pow_power"><span class="id" title="lemma">N_pow_power</span></a> : <span class="id" title="keyword">∀</span> <a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a> <a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> , (<a class="idref" href="additions.Compatibility.html#a:30"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#7878de73cb557151570f9fd311801cb8"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:29"><span class="id" title="variable">n</span></a> )%<span class="id" title="var">N</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Compatibility.html#a:30"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="additions.Compatibility.html#n:29"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_pow_compat" class="idref" href="#N_pow_compat"><span class="id" title="lemma">N_pow_compat</span></a> : <span class="id" title="keyword">∀</span> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> (<a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),   (<a class="idref" href="additions.Compatibility.html#a:32"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#7878de73cb557151570f9fd311801cb8"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:31"><span class="id" title="variable">n</span></a> )%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Compatibility.html#N_pow"><span class="id" title="definition">N_pow</span></a> <a class="idref" href="additions.Compatibility.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Compatibility.html#n:31"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_pow_compat" class="idref" href="#Pos_pow_compat"><span class="id" title="lemma">Pos_pow_compat</span></a> : <span class="id" title="keyword">∀</span> <a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a> (<a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),  (<a class="idref" href="additions.Compatibility.html#a:34"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#72c1e614b36b07dc51a712bc261ee813"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:33"><span class="id" title="variable">n</span></a> )%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Compatibility.html#Pos_pow"><span class="id" title="definition">Pos_pow</span></a>  <a class="idref" href="additions.Compatibility.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Compatibility.html#n:33"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_power_ok" class="idref" href="#nat_power_ok"><span class="id" title="lemma">nat_power_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a> <a id="b:36" class="idref" href="#b:36"><span class="id" title="binder">b</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="additions.Compatibility.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#b:36"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="additions.Compatibility.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#b:36"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">M</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Pos2Nat_morph" class="idref" href="#Pos2Nat_morph"><span class="id" title="lemma">Pos2Nat_morph</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:37" class="idref" href="#n:37"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> (<a class="idref" href="additions.Compatibility.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:37"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">M</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Compatibility.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#n:37"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z_pow_compat_pos" class="idref" href="#Z_pow_compat_pos"><span class="id" title="lemma">Z_pow_compat_pos</span></a> : <span class="id" title="keyword">∀</span> (<a id="p:39" class="idref" href="#p:39"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pow_pos"><span class="id" title="definition">Z.pow_pos</span></a> <a class="idref" href="additions.Compatibility.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#p:39"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">N_bpow</span></a> <a class="idref" href="additions.Compatibility.html#x:40"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a>  <a class="idref" href="additions.Compatibility.html#p:39"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z_pow_compat" class="idref" href="#Z_pow_compat"><span class="id" title="lemma">Z_pow_compat</span></a> : <span class="id" title="keyword">∀</span> <a id="x:41" class="idref" href="#x:41"><span class="id" title="binder">x</span></a> <a id="y:42" class="idref" href="#y:42"><span class="id" title="binder">y</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="additions.Compatibility.html#Z_pow"><span class="id" title="definition">Z_pow</span></a> <a class="idref" href="additions.Compatibility.html#x:41"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Compatibility.html#y:42"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="additions.Compatibility.html#x:41"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#14ef112b66e341f773bd1e9d05816f43"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Compatibility.html#y:42"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">Z</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab364"></a><h2 class="section">Tests</h2>
+
+
+<div class="paragraph"> </div>
+
+Let us chose a big exponent, and a computation that does not create 
+big numbers. So, the following computation time will be proportional to
+ the number of recursive calls, hence to the number of multiplications
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+ Time consuming !
+<br/>
+<span class="inlinecode"><span class="id" title="keyword">Time</span> <span class="id" title="keyword">Compute</span> (1 ^ 56666667)%<span class="id" title="var">Z</span>.
+<div class="paragraph"> </div>
+
+</span>
+</div>
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><span class="id" title="var">Finished</span> <span class="id" title="var">transaction</span> <span class="id" title="tactic">in</span> 3.604 <span class="id" title="var">secs</span> (3.587<span class="id" title="var">u</span>,0.007<span class="id" title="var">s</span>) 
+<div class="paragraph"> </div>
+
+</span>
+</div>
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><span class="id" title="var">Finished</span> <span class="id" title="var">transaction</span> <span class="id" title="tactic">in</span> 0. <span class="id" title="var">secs</span> (0.<span class="id" title="var">u</span>,0.<span class="id" title="var">s</span>) (<span class="id" title="var">successful</span>)
+<div class="paragraph"> </div>
+
+</span>
+<div class="paragraph"> </div>
+
+<a id="lab365"></a><h3 class="section">Adapting lemmas from Standard Library</h3>
+
+
+<div class="paragraph"> </div>
+
+Standard library already contains many lemmas about 
+<span class="inlinecode"><span class="id" title="var">N.pow</span></span>, <span class="inlinecode"><span class="id" title="var">Pos.pow</span></span> and <span class="inlinecode"><span class="id" title="var">Z.pow</span></span>. By using our extensional equivalence properties, one can 
+  easily prove the same properties for our implementation of the same functions.
+
+<div class="paragraph"> </div>
+
+We just give a simple example of this adaptation.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a id="Adaptation" class="idref" href="#Adaptation"><span class="id" title="section">Adaptation</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_size_gt" class="idref" href="#N_size_gt"><span class="id" title="lemma">N_size_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>, (<a class="idref" href="additions.Compatibility.html#n:43"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Compatibility.html#N_pow"><span class="id" title="definition">N_pow</span></a> 2 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.size"><span class="id" title="definition">N.size</span></a> <a class="idref" href="additions.Compatibility.html#n:43"><span class="id" title="variable">n</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Compatibility.html#Adaptation"><span class="id" title="section">Adaptation</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Comments on Stdlib's exponentiation 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pow"><span class="id" title="definition">Z.pow</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pow_pos"><span class="id" title="definition">Z.pow_pos</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span>   <span class="id" title="keyword">fun</span>  (<a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.iter"><span class="id" title="definition">Pos.iter</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Compatibility.html#x:44"><span class="id" title="variable">x</span></a> 12%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Demo.html b/theories/html/additions.Demo.html
new file mode 100644
index 00000000..6f174284
--- /dev/null
+++ b/theories/html/additions.Demo.html
@@ -0,0 +1,47 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Demo</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Demo</h1>
+
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Alt" class="idref" href="#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.double" class="idref" href="#Alt.double"><span class="id" title="definition">double</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Demo.html#n:1"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Demo.html#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Even.html#"><span class="id" title="library">Even</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="alt_double_ok" class="idref" href="#alt_double_ok"><span class="id" title="lemma">alt_double_ok</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">Nat.double</span></a> <a class="idref" href="additions.Demo.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Demo.html#double"><span class="id" title="definition">Alt.double</span></a> <a class="idref" href="additions.Demo.html#n:2"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Demo_power.html b/theories/html/additions.Demo_power.html
new file mode 100644
index 00000000..8e45eae2
--- /dev/null
+++ b/theories/html/additions.Demo_power.html
@@ -0,0 +1,47 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Demo_power</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Demo_power</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#"><span class="id" title="module">Uint63</span></a>.<br/>
+<span class="id" title="keyword">Search</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">int</span>).<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_Z"><span class="id" title="definition">of_Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_Z"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_Z"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_Z"><span class="id" title="definition">int</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Dichotomy.html b/theories/html/additions.Dichotomy.html
new file mode 100644
index 00000000..2eebf686
--- /dev/null
+++ b/theories/html/additions.Dichotomy.html
@@ -0,0 +1,154 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Dichotomy</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Dichotomy</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Dichotomic strategy, according to Bergeron, Brlek et al. 
+
+<div class="paragraph"> </div>
+
+  Let  be a positive number. We associate to  the integer 
+   .
+  For instance, if , the result is 10.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="additions.Strategies.html#"><span class="id" title="library">Strategies</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a> <a class="idref" href="additions.Compatibility.html#"><span class="id" title="library">Compatibility</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Computes 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="var">dicho_aux</span> (<a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Dichotomy.html#p:1"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;| 1%<span class="id" title="var">positive</span>   ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><br/>
+&nbsp;&nbsp;&nbsp;| 2%<span class="id" title="var">positive</span> | 3%<span class="id" title="var">positive</span>  ⇒ 2 <br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span>) | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span>) | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span>) | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux:2"><span class="id" title="definition">dicho_aux</span></a> <span class="id" title="var">q</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dicho" class="idref" href="#dicho"><span class="id" title="definition">dicho</span></a>  (<a id="p:31" class="idref" href="#p:31"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.div"><span class="id" title="definition">N.div</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="additions.Dichotomy.html#p:31"><span class="id" title="variable">p</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:31"><span class="id" title="variable">p</span></a>))).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+rewriting lemmas 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_xOxO" class="idref" href="#dicho_aux_xOxO"><span class="id" title="lemma">dicho_aux_xOxO</span></a> :  <span class="id" title="keyword">∀</span> <a id="p:32" class="idref" href="#p:32"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.Dichotomy.html#p:32"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:32"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_xOxI" class="idref" href="#dicho_aux_xOxI"><span class="id" title="lemma">dicho_aux_xOxI</span></a> :  <span class="id" title="keyword">∀</span> <a id="p:33" class="idref" href="#p:33"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="additions.Dichotomy.html#p:33"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:33"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_xIxO" class="idref" href="#dicho_aux_xIxO"><span class="id" title="lemma">dicho_aux_xIxO</span></a> :  <span class="id" title="keyword">∀</span> <a id="p:34" class="idref" href="#p:34"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.Dichotomy.html#p:34"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:34"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_xIxI" class="idref" href="#dicho_aux_xIxI"><span class="id" title="lemma">dicho_aux_xIxI</span></a> :  <span class="id" title="keyword">∀</span> <a id="p:35" class="idref" href="#p:35"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="additions.Dichotomy.html#p:35"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:35"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_le_xOXO" class="idref" href="#dicho_aux_le_xOXO"><span class="id" title="lemma">dicho_aux_le_xOXO</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> ,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:36"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:36"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="additions.Dichotomy.html#p:36"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0~0</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:36"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0~0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_le_xOXI" class="idref" href="#dicho_aux_le_xOXI"><span class="id" title="lemma">dicho_aux_le_xOXI</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:37" class="idref" href="#p:37"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:37"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:37"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="additions.Dichotomy.html#p:37"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:37"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_le_xIXO" class="idref" href="#dicho_aux_le_xIXO"><span class="id" title="lemma">dicho_aux_le_xIXO</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:38" class="idref" href="#p:38"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:38"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:38"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="additions.Dichotomy.html#p:38"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:38"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#35713fcadfdd66af247cbf95acf859fa"><span class="id" title="notation">~0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_le_xIXI" class="idref" href="#dicho_aux_le_xIXI"><span class="id" title="lemma">dicho_aux_le_xIXI</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:39" class="idref" href="#p:39"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:39"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:39"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> (<a class="idref" href="additions.Dichotomy.html#p:39"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1~1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.Dichotomy.html#p:39"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPosDef.html#119ad21c8e0c4947cd973f1f72e5feea"><span class="id" title="notation">~1~1</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">dicho_aux_le_xOXO</span> <span class="id" title="var">dicho_aux_le_xOXI</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">dicho_aux_le_xIXO</span> <span class="id" title="var">dicho_aux_le_xIXI</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_lt" class="idref" href="#dicho_aux_lt"><span class="id" title="lemma">dicho_aux_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="p:40" class="idref" href="#p:40"><span class="id" title="binder">p</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#p:40"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:40"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a>  <a class="idref" href="additions.Dichotomy.html#p:40"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_aux_gt" class="idref" href="#dicho_aux_gt"><span class="id" title="lemma">dicho_aux_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="p:41" class="idref" href="#p:41"><span class="id" title="binder">p</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#p:41"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#dicho_aux"><span class="id" title="definition">dicho_aux</span></a> <a class="idref" href="additions.Dichotomy.html#p:41"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_lt" class="idref" href="#dicho_lt"><span class="id" title="lemma">dicho_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="p:42" class="idref" href="#p:42"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#p:42"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a>  <a class="idref" href="additions.Dichotomy.html#p:42"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#p:42"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dicho_gt" class="idref" href="#dicho_gt"><span class="id" title="lemma">dicho_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="p:43" class="idref" href="#p:43"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#p:43"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a>  <a class="idref" href="additions.Dichotomy.html#p:43"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Dicho_strat" class="idref" href="#Dicho_strat"><span class="id" title="instance">Dicho_strat</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Euclidean_Chains</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Euclidean_Chains</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab366"></a><h1 class="section">Euclidean chains</h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab367"></a><h2 class="section">Introduction</h2>
+
+
+<div class="paragraph"> </div>
+
+  In this module, we study a way to build efficiently efficient chains.
+  Our approach is recursive (compositional?). Chains associated with big exponents are built by composition of smaller chains. Thus, the construction of a 
+small computation may be parameterized by the context in which it will be 
+  used. In other terms, we shall use Continuation Passing Style 
+
+<div class="paragraph"> </div>
+
+  Euclidean chains are introduced by 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Inverse_Image</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Inclusion</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.PArith.html#"><span class="id" title="library">PArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Lexicographic_Product.html#"><span class="id" title="library">Lexicographic_Product</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Addition_Chains.html#"><span class="id" title="library">Addition_Chains</span></a>    <a class="idref" href="additions.Compatibility.html#"><span class="id" title="library">Compatibility</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a> <a class="idref" href="additions.Wf_transparent.html#"><span class="id" title="library">Wf_transparent</span></a>   <a class="idref" href="additions.Dichotomy.html#"><span class="id" title="library">Dichotomy</span></a> <a class="idref" href="additions.BinaryStrat.html#"><span class="id" title="library">BinaryStrat</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="module">Morphisms</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="module">Monoid_def</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">add_op_proper</span> <span class="id" title="var">M</span> <span class="id" title="var">H</span> := <br/>
+&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">h</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="var">H</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (@<a class="idref" href="additions.Monoid_def.html#Eop_proper"><span class="id" title="method">Eop_proper</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span> <span class="id" title="var">h</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab368"></a><h1 class="section">CPS chain construction</h1>
+
+
+<div class="paragraph"> </div>
+
+ Type of chain continuations 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fkont" class="idref" href="#Fkont"><span class="id" title="definition">Fkont</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) := <a class="idref" href="additions.Euclidean_Chains.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:1"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fchain" class="idref" href="#Fchain"><span class="id" title="definition">Fchain</span></a> := <span class="id" title="keyword">∀</span> <a id="A:2" class="idref" href="#A:2"><span class="id" title="binder">A</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Fkont"><span class="id" title="definition">Fkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:2"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:2"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:2"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">F3</span></span> <span class="inlinecode"><span class="id" title="var">k</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> computes , then executes the computation associated
+   with <span class="inlinecode"><span class="id" title="var">k</span></span> <span class="inlinecode"><span class="id" title="var">z</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F3" class="idref" href="#F3"><span class="id" title="definition">F3</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> := <br/>
+&nbsp;<span class="id" title="keyword">fun</span>  <a id="A:3" class="idref" href="#A:3"><span class="id" title="binder">A</span></a> <a id="k:4" class="idref" href="#k:4"><span class="id" title="binder">k</span></a>  (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:3"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;<a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="z:7" class="idref" href="#z:7"><span class="id" title="binder">z</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#k:4"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#z:7"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F1" class="idref" href="#F1"><span class="id" title="definition">F1</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> := <br/>
+&nbsp;<span class="id" title="keyword">fun</span> <a id="A:8" class="idref" href="#A:8"><span class="id" title="binder">A</span></a> <a id="k:9" class="idref" href="#k:9"><span class="id" title="binder">k</span></a> (<a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:8"><span class="id" title="variable">A</span></a>) ⇒ <a class="idref" href="additions.Euclidean_Chains.html#k:9"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:10"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F2" class="idref" href="#F2"><span class="id" title="definition">F2</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> := <br/>
+<span class="id" title="keyword">fun</span>  <a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a> <a id="k:12" class="idref" href="#k:12"><span class="id" title="binder">k</span></a>  (<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:11"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;<a id="y:14" class="idref" href="#y:14"><span class="id" title="binder">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#k:12"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#y:14"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+An Fchain <span class="inlinecode"><span class="id" title="var">f</span></span> can be considered as a function that takes as 
+    argument another chain <span class="inlinecode"><span class="id" title="var">c</span></span> for continueing the computation.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fapply" class="idref" href="#Fapply"><span class="id" title="definition">Fapply</span></a> (<a id="f:15" class="idref" href="#f:15"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) (<a id="c:16" class="idref" href="#c:16"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>) : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a>  :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> (<a id="A:17" class="idref" href="#A:17"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="x:18" class="idref" href="#x:18"><span class="id" title="binder">x</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#A:17"><span class="id" title="variable">A</span></a>)  ⇒  <a class="idref" href="additions.Euclidean_Chains.html#f:15"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:17"><span class="id" title="variable">A</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#c:16"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:17"><span class="id" title="variable">A</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:18"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fcompose" class="idref" href="#Fcompose"><span class="id" title="definition">Fcompose</span></a> (<a id="f1:19" class="idref" href="#f1:19"><span class="id" title="binder">f1</span></a> <a id="f2:20" class="idref" href="#f2:20"><span class="id" title="binder">f2</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>  :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span>   <a id="A:21" class="idref" href="#A:21"><span class="id" title="binder">A</span></a> <a id="k:22" class="idref" href="#k:22"><span class="id" title="binder">k</span></a> <a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> ⇒  <a class="idref" href="additions.Euclidean_Chains.html#f1:19"><span class="id" title="variable">f1</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#A:21"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="y:24" class="idref" href="#y:24"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#f2:20"><span class="id" title="variable">f2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:21"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:22"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:24"><span class="id" title="variable">y</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:23"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Any Fchain can be transformed into a plain chain 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F2C" class="idref" href="#F2C"><span class="id" title="definition">F2C</span></a> (<a id="f:25" class="idref" href="#f:25"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> (<a id="A:26" class="idref" href="#A:26"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) ⇒ <a class="idref" href="additions.Euclidean_Chains.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:26"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> .<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Composition of Fchains 
+
+<div class="paragraph"> </div>
+
+Fchains are used for building correct exponentiation schemes by composition 
+of correct components. So, we have to define composition of Fchains.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="F9" class="idref" href="#F9"><span class="id" title="definition">F9</span></a> := <a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Fchains associated with powers of 2 
+<div class="paragraph"> </div>
+
+ computes  then send this value to  
+<div class="paragraph"> </div>
+
+ The neutral element for Fcompose 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F1_neutral_l" class="idref" href="#F1_neutral_l"><span class="id" title="lemma">F1_neutral_l</span></a> : <span class="id" title="keyword">∀</span> <a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f:27"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F1_neutral_r" class="idref" href="#F1_neutral_r"><span class="id" title="lemma">F1_neutral_r</span></a> : <span class="id" title="keyword">∀</span> <a id="f:28" class="idref" href="#f:28"><span class="id" title="binder">f</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f:28"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f:28"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fixpoint</span>  <a id="Fexp2_of_nat" class="idref" href="#Fexp2_of_nat"><span class="id" title="definition">Fexp2_of_nat</span></a> (<a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#n:29"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F2"><span class="id" title="definition">F2</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2_of_nat:30"><span class="id" title="definition">Fexp2_of_nat</span></a> <span class="id" title="var">p</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fexp2" class="idref" href="#Fexp2"><span class="id" title="definition">Fexp2</span></a> (<a id="p:32" class="idref" href="#p:32"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fexp2_of_nat"><span class="id" title="definition">Fexp2_of_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:32"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="F9_correct" class="idref" href="#F9_correct"><span class="id" title="lemma">F9_correct</span></a> :<a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 9 (<a class="idref" href="additions.Euclidean_Chains.html#F2C"><span class="id" title="definition">F2C</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F9"><span class="id" title="definition">F9</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A first attempt to define Fchain correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad" class="idref" href="#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Bad.Fchain_correct" class="idref" href="#Bad.Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="fc:34" class="idref" href="#fc:34"><span class="id" title="binder">fc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a> `(<a id="M:39" class="idref" href="#M:39"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:35"><span class="id" title="variable">A</span></a> <a id="op:36" class="idref" href="#op:36"><span class="id" title="binder">op</span></a> <a id="E_one:37" class="idref" href="#E_one:37"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:38" class="idref" href="#E_equiv:38"><span class="id" title="binder">E_equiv</span></a>) <a id="k:40" class="idref" href="#k:40"><span class="id" title="binder">k</span></a> (<a id="a:41" class="idref" href="#a:41"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:35"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:36"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#fc:34"><span class="id" title="variable">fc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:35"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:40"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#a:41"><span class="id" title="variable">a</span></a>)<a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:36"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#k:40"><span class="id" title="variable">k</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#a:41"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:33"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Bad.F3_correct" class="idref" href="#Bad.F3_correct"><span class="id" title="lemma">F3_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Bad.Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 3 <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a>.  <span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Equivalence on computations 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="computation_equiv" class="idref" href="#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> {<a id="A:42" class="idref" href="#A:42"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="op:43" class="idref" href="#op:43"><span class="id" title="binder">op</span></a>: <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:42"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="equiv:44" class="idref" href="#equiv:44"><span class="id" title="binder">equiv</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:42"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="c:45" class="idref" href="#c:45"><span class="id" title="binder">c</span></a> <a id="c':46" class="idref" href="#c':46"><span class="id" title="binder">c'</span></a>: @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:42"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:43"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#c:45"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:43"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#c':46"><span class="id" title="variable">c'</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Comp_equiv" class="idref" href="#Comp_equiv"><span class="id" title="instance">Comp_equiv</span></a> {<a id="A:47" class="idref" href="#A:47"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="op:48" class="idref" href="#op:48"><span class="id" title="binder">op</span></a>: <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:47"><span class="id" title="variable">A</span></a>) (<a id="equiv:49" class="idref" href="#equiv:49"><span class="id" title="binder">equiv</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:47"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> (@<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:47"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">equiv</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="comp_equiv_equivalence" class="idref" href="#comp_equiv_equivalence"><span class="id" title="instance">comp_equiv_equivalence</span></a> {<a id="A:50" class="idref" href="#A:50"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="op:51" class="idref" href="#op:51"><span class="id" title="binder">op</span></a>: <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:50"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="equiv:52" class="idref" href="#equiv:52"><span class="id" title="binder">equiv</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:50"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#equiv:52"><span class="id" title="variable">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:51"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#equiv:52"><span class="id" title="variable">equiv</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Fkonts that respect E_equiv 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Fkont_proper" class="idref" href="#Fkont_proper"><span class="id" title="record">Fkont_proper</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="M:57" class="idref" href="#M:57"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:53" class="idref" href="#A:53"><span class="id" title="binder">A</span></a> <a id="op:54" class="idref" href="#op:54"><span class="id" title="binder">op</span></a> <a id="E_one:55" class="idref" href="#E_one:55"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:56" class="idref" href="#E_equiv:56"><span class="id" title="binder">E_equiv</span></a>) (<a id="k:58" class="idref" href="#k:58"><span class="id" title="binder">k</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#Fkont"><span class="id" title="definition">Fkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:53"><span class="id" title="variable">A</span></a> )  :=<br/>
+&nbsp;&nbsp;<a id="Fkont_proper_prf" class="idref" href="#Fkont_proper_prf"><span class="id" title="projection">Fkont_proper_prf</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:54"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:56"><span class="id" title="variable">E_equiv</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#k:58"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Return_proper" class="idref" href="#Return_proper"><span class="id" title="instance">Return_proper</span></a> `(<a id="M:65" class="idref" href="#M:65"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:61" class="idref" href="#A:61"><span class="id" title="binder">A</span></a> <a id="op:62" class="idref" href="#op:62"><span class="id" title="binder">op</span></a> <a id="E_one:63" class="idref" href="#E_one:63"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:64" class="idref" href="#E_equiv:64"><span class="id" title="binder">E_equiv</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fkont_proper"><span class="id" title="class">Fkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:65"><span class="id" title="variable">M</span></a> (@<a class="idref" href="additions.Addition_Chains.html#Return"><span class="id" title="constructor">Return</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:61"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Fchain correctness (for exponent of type <span class="inlinecode"><span class="id" title="var">nat</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fchain_correct_nat" class="idref" href="#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> (<a id="n:66" class="idref" href="#n:66"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:67" class="idref" href="#f:67"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a> `(<a id="M:72" class="idref" href="#M:72"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:68"><span class="id" title="variable">A</span></a> <a id="op:69" class="idref" href="#op:69"><span class="id" title="binder">op</span></a> <a id="E_one:70" class="idref" href="#E_one:70"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:71" class="idref" href="#E_equiv:71"><span class="id" title="binder">E_equiv</span></a>) <a id="k:73" class="idref" href="#k:73"><span class="id" title="binder">k</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hk:74" class="idref" href="#Hk:74"><span class="id" title="binder">Hk</span></a> :<a class="idref" href="additions.Euclidean_Chains.html#Fkont_proper"><span class="id" title="class">Fkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:72"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:73"><span class="id" title="variable">k</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:75" class="idref" href="#a:75"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:68"><span class="id" title="variable">A</span></a>) ,<br/>
+&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:69"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#f:67"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:68"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:73"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#a:75"><span class="id" title="variable">a</span></a>) <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:69"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#k:73"><span class="id" title="variable">k</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:66"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fchain_correct" class="idref" href="#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a id="p:76" class="idref" href="#p:76"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="f:77" class="idref" href="#f:77"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) :=<br/>
+&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:76"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#f:77"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F1_correct" class="idref" href="#F1_correct"><span class="id" title="lemma">F1_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 1 <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F3_correct" class="idref" href="#F3_correct"><span class="id" title="lemma">F3_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 3 <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F2_correct" class="idref" href="#F2_correct"><span class="id" title="lemma">F2_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 2 <a class="idref" href="additions.Euclidean_Chains.html#F2"><span class="id" title="definition">F2</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+F2C preserves correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F2C_correct" class="idref" href="#F2C_correct"><span class="id" title="lemma">F2C_correct</span></a> (<a id="p:78" class="idref" href="#p:78"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="fc:79" class="idref" href="#fc:79"><span class="id" title="binder">fc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:78"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc:79"><span class="id" title="variable">fc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:78"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#F2C"><span class="id" title="definition">F2C</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc:79"><span class="id" title="variable">fc</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad2" class="idref" href="#Bad2"><span class="id" title="module">Bad2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Bad2.Fcompose_correct" class="idref" href="#Bad2.Fcompose_correct"><span class="id" title="lemma">Fcompose_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:80" class="idref" href="#f1:80"><span class="id" title="binder">f1</span></a> <a id="f2:81" class="idref" href="#f2:81"><span class="id" title="binder">f2</span></a> <a id="n1:82" class="idref" href="#n1:82"><span class="id" title="binder">n1</span></a> <a id="n2:83" class="idref" href="#n2:83"><span class="id" title="binder">n2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n1:82"><span class="id" title="variable">n1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f1:80"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:83"><span class="id" title="variable">n2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f2:81"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#n1:82"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:83"><span class="id" title="variable">n2</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f1:80"><span class="id" title="variable">f1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f2:81"><span class="id" title="variable">f2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Bad2"><span class="id" title="module">Bad2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Fisrt attempt to define Fchain_proper 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad3" class="idref" href="#Bad3"><span class="id" title="module">Bad3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Bad3.Fchain_proper" class="idref" href="#Bad3.Fchain_proper"><span class="id" title="record">Fchain_proper</span></a> (<a id="fc:86" class="idref" href="#fc:86"><span class="id" title="binder">fc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) := <a id="Bad3.Fchain_proper_bad_prf" class="idref" href="#Bad3.Fchain_proper_bad_prf"><span class="id" title="projection">Fchain_proper_bad_prf</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span>  `(<a id="M:92" class="idref" href="#M:92"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a> <a id="op:89" class="idref" href="#op:89"><span class="id" title="binder">op</span></a> <a id="E_one:90" class="idref" href="#E_one:90"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:91" class="idref" href="#E_equiv:91"><span class="id" title="binder">E_equiv</span></a>) <a id="k:93" class="idref" href="#k:93"><span class="id" title="binder">k</span></a>  ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fkont_proper"><span class="id" title="class">Fkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:92"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:93"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:94" class="idref" href="#x:94"><span class="id" title="binder">x</span></a> <a id="y:95" class="idref" href="#y:95"><span class="id" title="binder">y</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#x:94"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:95"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <span class="id" title="var">_</span> <a class="idref" href="additions.Euclidean_Chains.html#op:89"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:91"><span class="id" title="variable">E_equiv</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#fc:86"><span class="id" title="variable">fc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:93"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:94"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#fc:86"><span class="id" title="variable">fc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:93"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:95"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Bad3.Fcompose_proper" class="idref" href="#Bad3.Fcompose_proper"><span class="id" title="instance">Fcompose_proper</span></a> (<a id="f1:97" class="idref" href="#f1:97"><span class="id" title="binder">f1</span></a> <a id="f2:98" class="idref" href="#f2:98"><span class="id" title="binder">f2</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Bad3.Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f1:97"><span class="id" title="variable">f1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Bad3.Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f2:98"><span class="id" title="variable">f2</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Bad3.Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f1:97"><span class="id" title="variable">f1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f2:98"><span class="id" title="variable">f2</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Bad3"><span class="id" title="module">Bad3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Correct definition 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Fkont_equiv" class="idref" href="#Fkont_equiv"><span class="id" title="definition">Fkont_equiv</span></a>  `(<a id="M:103" class="idref" href="#M:103"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:99" class="idref" href="#A:99"><span class="id" title="binder">A</span></a> <a id="op:100" class="idref" href="#op:100"><span class="id" title="binder">op</span></a> <a id="E_one:101" class="idref" href="#E_one:101"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:102" class="idref" href="#E_equiv:102"><span class="id" title="binder">E_equiv</span></a>)<br/>
+&nbsp;(<a id="k:104" class="idref" href="#k:104"><span class="id" title="binder">k</span></a> <a id="k':105" class="idref" href="#k':105"><span class="id" title="binder">k'</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#Fkont"><span class="id" title="definition">Fkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:99"><span class="id" title="variable">A</span></a> )  := <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="x:106" class="idref" href="#x:106"><span class="id" title="binder">x</span></a> <a id="y:107" class="idref" href="#y:107"><span class="id" title="binder">y</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:99"><span class="id" title="variable">A</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#x:106"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:107"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:100"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:102"><span class="id" title="variable">E_equiv</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#k:104"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:106"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#k':105"><span class="id" title="variable">k'</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:107"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Fchain_proper" class="idref" href="#Fchain_proper"><span class="id" title="record">Fchain_proper</span></a> (<a id="fc:108" class="idref" href="#fc:108"><span class="id" title="binder">fc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) := <a id="Fchain_proper_prf" class="idref" href="#Fchain_proper_prf"><span class="id" title="projection">Fchain_proper_prf</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span>  `(<a id="M:114" class="idref" href="#M:114"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:110" class="idref" href="#A:110"><span class="id" title="binder">A</span></a> <a id="op:111" class="idref" href="#op:111"><span class="id" title="binder">op</span></a> <a id="E_one:112" class="idref" href="#E_one:112"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:113" class="idref" href="#E_equiv:113"><span class="id" title="binder">E_equiv</span></a>) <a id="k:115" class="idref" href="#k:115"><span class="id" title="binder">k</span></a> <a id="k':116" class="idref" href="#k':116"><span class="id" title="binder">k'</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fkont_proper"><span class="id" title="class">Fkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:114"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:115"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fkont_proper"><span class="id" title="class">Fkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:114"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':116"><span class="id" title="variable">k'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fkont_equiv"><span class="id" title="definition">Fkont_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:114"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:115"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':116"><span class="id" title="variable">k'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:117" class="idref" href="#x:117"><span class="id" title="binder">x</span></a> <a id="y:118" class="idref" href="#y:118"><span class="id" title="binder">y</span></a>,  <a class="idref" href="additions.Euclidean_Chains.html#x:117"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:118"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <span class="id" title="var">_</span> <a class="idref" href="additions.Euclidean_Chains.html#op:111"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:113"><span class="id" title="variable">E_equiv</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#fc:108"><span class="id" title="variable">fc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:110"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:115"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:117"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#fc:108"><span class="id" title="variable">fc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:110"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':116"><span class="id" title="variable">k'</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:118"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="F1_proper" class="idref" href="#F1_proper"><span class="id" title="instance">F1_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="F3_proper" class="idref" href="#F3_proper"><span class="id" title="instance">F3_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="F2_proper" class="idref" href="#F2_proper"><span class="id" title="instance">F2_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F2"><span class="id" title="definition">F2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Fcompose respects correctness and properness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Fcompose_correct_nat" class="idref" href="#Fcompose_correct_nat"><span class="id" title="lemma">Fcompose_correct_nat</span></a> : <span class="id" title="keyword">∀</span> <a id="fc1:120" class="idref" href="#fc1:120"><span class="id" title="binder">fc1</span></a> <a id="fc2:121" class="idref" href="#fc2:121"><span class="id" title="binder">fc2</span></a> <a id="n1:122" class="idref" href="#n1:122"><span class="id" title="binder">n1</span></a> <a id="n2:123" class="idref" href="#n2:123"><span class="id" title="binder">n2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n1:122"><span class="id" title="variable">n1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:120"><span class="id" title="variable">fc1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:123"><span class="id" title="variable">n2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:121"><span class="id" title="variable">fc2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:121"><span class="id" title="variable">fc2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#n1:122"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:123"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:120"><span class="id" title="variable">fc1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:121"><span class="id" title="variable">fc2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Fcompose_correct" class="idref" href="#Fcompose_correct"><span class="id" title="lemma">Fcompose_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="fc1:129" class="idref" href="#fc1:129"><span class="id" title="binder">fc1</span></a> <a id="fc2:130" class="idref" href="#fc2:130"><span class="id" title="binder">fc2</span></a> <a id="n1:131" class="idref" href="#n1:131"><span class="id" title="binder">n1</span></a> <a id="n2:132" class="idref" href="#n2:132"><span class="id" title="binder">n2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n1:131"><span class="id" title="variable">n1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:129"><span class="id" title="variable">fc1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:132"><span class="id" title="variable">n2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:130"><span class="id" title="variable">fc2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:130"><span class="id" title="variable">fc2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#n1:131"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n2:132"><span class="id" title="variable">n2</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:129"><span class="id" title="variable">fc1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:130"><span class="id" title="variable">fc2</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Fcompose_proper" class="idref" href="#Fcompose_proper"><span class="id" title="instance">Fcompose_proper</span></a> (<a id="fc1:133" class="idref" href="#fc1:133"><span class="id" title="binder">fc1</span></a> <a id="fc2:134" class="idref" href="#fc2:134"><span class="id" title="binder">fc2</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:133"><span class="id" title="variable">fc1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:134"><span class="id" title="variable">fc2</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc1:133"><span class="id" title="variable">fc1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fc2:134"><span class="id" title="variable">fc2</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Fexp2_nat_proper" class="idref" href="#Fexp2_nat_proper"><span class="id" title="instance">Fexp2_nat_proper</span></a> (<a id="n:135" class="idref" href="#n:135"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2_of_nat"><span class="id" title="definition">Fexp2_of_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:135"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Fexp2_nat_correct" class="idref" href="#Fexp2_nat_correct"><span class="id" title="lemma">Fexp2_nat_correct</span></a> (<a id="n:136" class="idref" href="#n:136"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> (2  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:136"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#Fexp2_of_nat"><span class="id" title="definition">Fexp2_of_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:136"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Fexp2_correct" class="idref" href="#Fexp2_correct"><span class="id" title="lemma">Fexp2_correct</span></a> (<a id="p:137" class="idref" href="#p:137"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#72c1e614b36b07dc51a712bc261ee813"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:137"><span class="id" title="variable">p</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:137"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="Fexp2_proper" class="idref" href="#Fexp2_proper"><span class="id" title="instance">Fexp2_proper</span></a> (<a id="p:138" class="idref" href="#p:138"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:138"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab369"></a><h2 class="section">Remark</h2>
+
+
+<div class="paragraph"> </div>
+
+We are now  able to build chains for any exponent of the form 
+, using Fcompose and previous lemmas.
+
+<div class="paragraph"> </div>
+
+Let us look at a simple example 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">F1_correct</span> <span class="id" title="var">F1_proper</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">F3_correct</span> <span class="id" title="var">F3_proper</span> <span class="id" title="var">Fcompose_correct</span> <span class="id" title="var">Fcompose_proper</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">Fexp2_correct</span> <span class="id" title="var">Fexp2_proper</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="F144" class="idref" href="#F144"><span class="id" title="definition">F144</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="f:139" class="idref" href="#f:139"><span class="id" title="binder">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 144 <a class="idref" href="additions.Euclidean_Chains.html#f:139"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#f:139"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.  
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Bad solution 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad4" class="idref" href="#Bad4"><span class="id" title="module">Bad4</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Bad4.Fplus" class="idref" href="#Bad4.Fplus"><span class="id" title="definition">Fplus</span></a> (<a id="f1:140" class="idref" href="#f1:140"><span class="id" title="binder">f1</span></a> <a id="f2:141" class="idref" href="#f2:141"><span class="id" title="binder">f2</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="A:142" class="idref" href="#A:142"><span class="id" title="binder">A</span></a> <a id="k:143" class="idref" href="#k:143"><span class="id" title="binder">k</span></a> <a id="x:144" class="idref" href="#x:144"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#f1:140"><span class="id" title="variable">f1</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:142"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="y:145" class="idref" href="#y:145"><span class="id" title="binder">y</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#f2:141"><span class="id" title="variable">f2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:142"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="z:146" class="idref" href="#z:146"><span class="id" title="binder">z</span></a> ⇒ <a id="t:147" class="idref" href="#t:147"><span class="id" title="binder">t</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#z:146"><span class="id" title="variable">z</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:145"><span class="id" title="variable">y</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:143"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#t:147"><span class="id" title="variable">t</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:144"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:144"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Bad4.F23" class="idref" href="#Bad4.F23"><span class="id" title="definition">F23</span></a> := <a class="idref" href="additions.Euclidean_Chains.html#Bad4.Fplus"><span class="id" title="definition">Fplus</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Bad4.Fplus"><span class="id" title="definition">Fplus</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> 4) (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> 2)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Bad4.F23_ok" class="idref" href="#Bad4.F23_ok"><span class="id" title="lemma">F23_ok</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 23 (<a class="idref" href="additions.Euclidean_Chains.html#F2C"><span class="id" title="definition">F2C</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Bad4.F23"><span class="id" title="definition">F23</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Bad4"><span class="id" title="module">Bad4</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Continuations with two arguments 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kkont" class="idref" href="#Kkont"><span class="id" title="definition">Kkont</span></a> <a id="A:148" class="idref" href="#A:148"><span class="id" title="binder">A</span></a>:=  <a class="idref" href="additions.Euclidean_Chains.html#A:148"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:148"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:148"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+CPS chain builders for  two exponents  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kchain" class="idref" href="#Kchain"><span class="id" title="definition">Kchain</span></a> :=  <span class="id" title="keyword">∀</span> <a id="A:149" class="idref" href="#A:149"><span class="id" title="binder">A</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:149"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:149"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> @<a class="idref" href="additions.Addition_Chains.html#computation"><span class="id" title="inductive">computation</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:149"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Kchain for  and  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="k3_1" class="idref" href="#k3_1"><span class="id" title="definition">k3_1</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a> := <span class="id" title="keyword">fun</span> <a id="A:150" class="idref" href="#A:150"><span class="id" title="binder">A</span></a> (<a id="k:151" class="idref" href="#k:151"><span class="id" title="binder">k</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:150"><span class="id" title="variable">A</span></a>) (<a id="x:152" class="idref" href="#x:152"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:150"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;<a id="x2:153" class="idref" href="#x2:153"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:152"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:152"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x3:154" class="idref" href="#x3:154"><span class="id" title="binder">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x2:153"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:152"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#k:151"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x3:154"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:152"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Kchain for  and  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="k7_3" class="idref" href="#k7_3"><span class="id" title="definition">k7_3</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a> := <span class="id" title="keyword">fun</span> <a id="A:155" class="idref" href="#A:155"><span class="id" title="binder">A</span></a> (<a id="k:156" class="idref" href="#k:156"><span class="id" title="binder">k</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:155"><span class="id" title="variable">A</span></a>)   (<a id="x:157" class="idref" href="#x:157"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#A:155"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;<a id="x2:158" class="idref" href="#x2:158"><span class="id" title="binder">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:157"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x3:159" class="idref" href="#x3:159"><span class="id" title="binder">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x2:158"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x6:160" class="idref" href="#x6:160"><span class="id" title="binder">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x3:159"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x3:159"><span class="id" title="variable">x3</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a id="x7:161" class="idref" href="#x7:161"><span class="id" title="binder">x7</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x6:160"><span class="id" title="variable">x6</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#k:156"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#x7:161"><span class="id" title="variable">x7</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x3:159"><span class="id" title="variable">x3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The Definition of correct chains and proper chains and 
+  continuations are adapted to Kchains 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kkont_proper" class="idref" href="#Kkont_proper"><span class="id" title="definition">Kkont_proper</span></a> `(<a id="M:166" class="idref" href="#M:166"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:162" class="idref" href="#A:162"><span class="id" title="binder">A</span></a> <a id="op:163" class="idref" href="#op:163"><span class="id" title="binder">op</span></a> <a id="E_one:164" class="idref" href="#E_one:164"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:165" class="idref" href="#E_equiv:165"><span class="id" title="binder">E_equiv</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="k:167" class="idref" href="#k:167"><span class="id" title="binder">k</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:162"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:163"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:165"><span class="id" title="variable">E_equiv</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#k:167"><span class="id" title="variable">k</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kkont_equiv" class="idref" href="#Kkont_equiv"><span class="id" title="definition">Kkont_equiv</span></a>  `(<a id="M:172" class="idref" href="#M:172"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:168" class="idref" href="#A:168"><span class="id" title="binder">A</span></a> <a id="op:169" class="idref" href="#op:169"><span class="id" title="binder">op</span></a> <a id="E_one:170" class="idref" href="#E_one:170"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:171" class="idref" href="#E_equiv:171"><span class="id" title="binder">E_equiv</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="k:173" class="idref" href="#k:173"><span class="id" title="binder">k</span></a> <a id="k':174" class="idref" href="#k':174"><span class="id" title="binder">k'</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:168"><span class="id" title="variable">A</span></a> )  := <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="x:175" class="idref" href="#x:175"><span class="id" title="binder">x</span></a> <a id="y:176" class="idref" href="#y:176"><span class="id" title="binder">y</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:168"><span class="id" title="variable">A</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#x:175"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:176"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="z:177" class="idref" href="#z:177"><span class="id" title="binder">z</span></a> <a id="t:178" class="idref" href="#t:178"><span class="id" title="binder">t</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#z:177"><span class="id" title="variable">z</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Euclidean_Chains.html#t:178"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:169"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:171"><span class="id" title="variable">E_equiv</span></a>   (<a class="idref" href="additions.Euclidean_Chains.html#k:173"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#x:175"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Euclidean_Chains.html#z:177"><span class="id" title="variable">z</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#k':174"><span class="id" title="variable">k'</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:176"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Euclidean_Chains.html#t:178"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A Kchain is correct with respect to two exponents  and  
+  if it computes  and  for every  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kchain_correct_nat" class="idref" href="#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a> (<a id="n:179" class="idref" href="#n:179"><span class="id" title="binder">n</span></a> <a id="p:180" class="idref" href="#p:180"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="kc:181" class="idref" href="#kc:181"><span class="id" title="binder">kc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="A:182" class="idref" href="#A:182"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="op:183" class="idref" href="#op:183"><span class="id" title="binder">op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a>) (<a id="E_one:184" class="idref" href="#E_one:184"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a>) (<a id="E_equiv:185" class="idref" href="#E_equiv:185"><span class="id" title="binder">E_equiv</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M:186" class="idref" href="#M:186"><span class="id" title="binder">M</span></a> : <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:183"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_one:184"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:185"><span class="id" title="variable">E_equiv</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="k:187" class="idref" href="#k:187"><span class="id" title="binder">k</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kkont_proper"><span class="id" title="definition">Kkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:186"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:187"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="a:188" class="idref" href="#a:188"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a>) ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:183"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#kc:181"><span class="id" title="variable">kc</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#A:182"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:187"><span class="id" title="variable">k</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#a:188"><span class="id" title="variable">a</span></a>) <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Addition_Chains.html#computation_execute"><span class="id" title="definition">computation_execute</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:183"><span class="id" title="variable">op</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#k:187"><span class="id" title="variable">k</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#a:188"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:179"><span class="id" title="variable">n</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#a:188"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:180"><span class="id" title="variable">p</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Kchain_correct" class="idref" href="#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> (<a id="n:189" class="idref" href="#n:189"><span class="id" title="binder">n</span></a> <a id="p:190" class="idref" href="#p:190"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="kc:191" class="idref" href="#kc:191"><span class="id" title="binder">kc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:189"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:190"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#kc:191"><span class="id" title="variable">kc</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Kchain_proper" class="idref" href="#Kchain_proper"><span class="id" title="record">Kchain_proper</span></a> (<a id="kc:192" class="idref" href="#kc:192"><span class="id" title="binder">kc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) :=<br/>
+<a id="Kchain_proper_prf" class="idref" href="#Kchain_proper_prf"><span class="id" title="projection">Kchain_proper_prf</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span> `(<a id="M:198" class="idref" href="#M:198"><span class="id" title="binder">M</span></a> : @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:194" class="idref" href="#A:194"><span class="id" title="binder">A</span></a> <a id="op:195" class="idref" href="#op:195"><span class="id" title="binder">op</span></a> <a id="E_one:196" class="idref" href="#E_one:196"><span class="id" title="binder">E_one</span></a> <a id="E_equiv:197" class="idref" href="#E_equiv:197"><span class="id" title="binder">E_equiv</span></a>) <a id="k:199" class="idref" href="#k:199"><span class="id" title="binder">k</span></a> <a id="k':200" class="idref" href="#k':200"><span class="id" title="binder">k'</span></a> <a id="x:201" class="idref" href="#x:201"><span class="id" title="binder">x</span></a> <a id="y:202" class="idref" href="#y:202"><span class="id" title="binder">y</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kkont_proper"><span class="id" title="definition">Kkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:198"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:199"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kkont_proper"><span class="id" title="definition">Kkont_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:198"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':200"><span class="id" title="variable">k'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kkont_equiv"><span class="id" title="definition">Kkont_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#M:198"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:199"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':200"><span class="id" title="variable">k'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#E_equiv:197"><span class="id" title="variable">E_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:201"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:202"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#computation_equiv"><span class="id" title="definition">computation_equiv</span></a> <a class="idref" href="additions.Euclidean_Chains.html#op:195"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Euclidean_Chains.html#E_equiv:197"><span class="id" title="variable">E_equiv</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#kc:192"><span class="id" title="variable">kc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:194"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:199"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:201"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#kc:192"><span class="id" title="variable">kc</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:194"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k':200"><span class="id" title="variable">k'</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:202"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="k7_3_proper" class="idref" href="#k7_3_proper"><span class="id" title="instance">k7_3_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k7_3"><span class="id" title="definition">k7_3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="k7_3_correct" class="idref" href="#k7_3_correct"><span class="id" title="lemma">k7_3_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> 7 3 <a class="idref" href="additions.Euclidean_Chains.html#k7_3"><span class="id" title="definition">k7_3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+conversion between several definitions of correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Kchain_correct_conv" class="idref" href="#Kchain_correct_conv"><span class="id" title="lemma">Kchain_correct_conv</span></a> (<a id="kc:204" class="idref" href="#kc:204"><span class="id" title="binder">kc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) (<a id="n:205" class="idref" href="#n:205"><span class="id" title="binder">n</span></a> <a id="p:206" class="idref" href="#p:206"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;0%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:205"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 0%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:206"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:205"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:206"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kc:204"><span class="id" title="variable">kc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:205"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:206"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#kc:204"><span class="id" title="variable">kc</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab370"></a><h2 class="section">More chain combinators</h2>
+
+
+<div class="paragraph"> </div>
+
+  Since we are working with two types of functional chains, we have to define
+  several ways of composing them. Each of these operators is certified to
+ preserve correctnes and properness 
+<div class="paragraph"> </div>
+
+ Conversion of Kchains into Fchains 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="K2F" class="idref" href="#K2F"><span class="id" title="definition">K2F</span></a> (<a id="knp:207" class="idref" href="#knp:207"><span class="id" title="binder">knp</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="A:208" class="idref" href="#A:208"><span class="id" title="binder">A</span></a> (<a id="k:209" class="idref" href="#k:209"><span class="id" title="binder">k</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#Fkont"><span class="id" title="definition">Fkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:208"><span class="id" title="variable">A</span></a>) ⇒ <a class="idref" href="additions.Euclidean_Chains.html#knp:207"><span class="id" title="variable">knp</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:208"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span>  <a id="y:210" class="idref" href="#y:210"><span class="id" title="binder">y</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#k:209"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:210"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="K2F_correct_nat" class="idref" href="#K2F_correct_nat"><span class="id" title="lemma">K2F_correct_nat</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="knp:211" class="idref" href="#knp:211"><span class="id" title="binder">knp</span></a> <a id="n:212" class="idref" href="#n:212"><span class="id" title="binder">n</span></a> <a id="p:213" class="idref" href="#p:213"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#n:212"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:213"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#knp:211"><span class="id" title="variable">knp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:212"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#K2F"><span class="id" title="definition">K2F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#knp:211"><span class="id" title="variable">knp</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="K2F_correct" class="idref" href="#K2F_correct"><span class="id" title="lemma">K2F_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="kc:218" class="idref" href="#kc:218"><span class="id" title="binder">kc</span></a> <a id="n:219" class="idref" href="#n:219"><span class="id" title="binder">n</span></a> <a id="p:220" class="idref" href="#p:220"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:219"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:220"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kc:218"><span class="id" title="variable">kc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:219"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#K2F"><span class="id" title="definition">K2F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kc:218"><span class="id" title="variable">kc</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="K2F_proper" class="idref" href="#K2F_proper"><span class="id" title="instance">K2F_proper</span></a> (<a id="kc:221" class="idref" href="#kc:221"><span class="id" title="binder">kc</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>)(<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kc:221"><span class="id" title="variable">kc</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#K2F"><span class="id" title="definition">K2F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kc:221"><span class="id" title="variable">kc</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Using <span class="inlinecode"><span class="id" title="var">kbr</span></span> for  computing  and , then using <span class="inlinecode"><span class="id" title="var">Cq</span></span> for
+  computing , then sending  and  to the continuation
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="KFK" class="idref" href="#KFK"><span class="id" title="definition">KFK</span></a> (<a id="kbr:222" class="idref" href="#kbr:222"><span class="id" title="binder">kbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) (<a id="fq:223" class="idref" href="#fq:223"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>  :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="A:224" class="idref" href="#A:224"><span class="id" title="binder">A</span></a> <a id="k:225" class="idref" href="#k:225"><span class="id" title="binder">k</span></a> <a id="a:226" class="idref" href="#a:226"><span class="id" title="binder">a</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#kbr:222"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:224"><span class="id" title="variable">A</span></a>  (<span class="id" title="keyword">fun</span> <a id="xb:227" class="idref" href="#xb:227"><span class="id" title="binder">xb</span></a> <a id="xr:228" class="idref" href="#xr:228"><span class="id" title="binder">xr</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#fq:223"><span class="id" title="variable">fq</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:224"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="y:229" class="idref" href="#y:229"><span class="id" title="binder">y</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="z:230" class="idref" href="#z:230"><span class="id" title="binder">z</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">&lt;---</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:229"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">times</span></a> <a class="idref" href="additions.Euclidean_Chains.html#xr:228"><span class="id" title="variable">xr</span></a><a class="idref" href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556"><span class="id" title="notation">;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k:225"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#z:230"><span class="id" title="variable">z</span></a> <a class="idref" href="additions.Euclidean_Chains.html#xb:227"><span class="id" title="variable">xb</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#xb:227"><span class="id" title="variable">xb</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#a:226"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="KFF" class="idref" href="#KFF"><span class="id" title="definition">KFF</span></a> (<a id="kbr:231" class="idref" href="#kbr:231"><span class="id" title="binder">kbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) (<a id="fq:232" class="idref" href="#fq:232"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#K2F"><span class="id" title="definition">K2F</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:231"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:232"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="FFK" class="idref" href="#FFK"><span class="id" title="definition">FFK</span></a> (<a id="fp:233" class="idref" href="#fp:233"><span class="id" title="binder">fp</span></a> <a id="fq:234" class="idref" href="#fq:234"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="A:235" class="idref" href="#A:235"><span class="id" title="binder">A</span></a> <a id="k:236" class="idref" href="#k:236"><span class="id" title="binder">k</span></a> <a id="a:237" class="idref" href="#a:237"><span class="id" title="binder">a</span></a> ⇒  <a class="idref" href="additions.Euclidean_Chains.html#fp:233"><span class="id" title="variable">fp</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:235"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="xb:238" class="idref" href="#xb:238"><span class="id" title="binder">xb</span></a>  ⇒ <a class="idref" href="additions.Euclidean_Chains.html#fq:234"><span class="id" title="variable">fq</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:235"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="y:239" class="idref" href="#y:239"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#k:236"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:239"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Euclidean_Chains.html#xb:238"><span class="id" title="variable">xb</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#xb:238"><span class="id" title="variable">xb</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#a:237"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="FK" class="idref" href="#FK"><span class="id" title="definition">FK</span></a> (<a id="f:240" class="idref" href="#f:240"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="A:241" class="idref" href="#A:241"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="k:242" class="idref" href="#k:242"><span class="id" title="binder">k</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:241"><span class="id" title="variable">A</span></a>) (<a id="a:243" class="idref" href="#a:243"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#A:241"><span class="id" title="variable">A</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#f:240"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:241"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="y:244" class="idref" href="#y:244"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#k:242"><span class="id" title="variable">k</span></a> <a class="idref" href="additions.Euclidean_Chains.html#y:244"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Euclidean_Chains.html#a:243"><span class="id" title="variable">a</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#a:243"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="k17_7" class="idref" href="#k17_7"><span class="id" title="definition">k17_7</span></a> := <a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k7_3"><span class="id" title="definition">k7_3</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> 1).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+In the following section, we prove that the constructions KFK and KFF
+   respect properness and correctness 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="KFK_proof" class="idref" href="#KFK_proof"><span class="id" title="section">KFK_proof</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> <a id="KFK_proof.b" class="idref" href="#KFK_proof.b"><span class="id" title="variable">b</span></a> <a id="KFK_proof.q" class="idref" href="#KFK_proof.q"><span class="id" title="variable">q</span></a> <a id="KFK_proof.r" class="idref" href="#KFK_proof.r"><span class="id" title="variable">r</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="KFK_proof.kbr" class="idref" href="#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="KFK_proof.fq" class="idref" href="#KFK_proof.fq"><span class="id" title="variable">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="KFK_proof.Hbr" class="idref" href="#KFK_proof.Hbr"><span class="id" title="variable">Hbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.b"><span class="id" title="variable">b</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.r"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="KFK_proof.Hq" class="idref" href="#KFK_proof.Hq"><span class="id" title="variable">Hq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.q"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="KFK_proof.Hbr_prop" class="idref" href="#KFK_proof.Hbr_prop"><span class="id" title="variable">Hbr_prop</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="KFK_proof.Hq_prop" class="idref" href="#KFK_proof.Hq_prop"><span class="id" title="variable">Hq_prop</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="KFK_correct_nat" class="idref" href="#KFK_correct_nat"><span class="id" title="lemma">KFK_correct_nat</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.r"><span class="id" title="variable">r</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.b"><span class="id" title="variable">b</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>).<br/>
+</div>
+
+<div class="doc">
+simplifying the hypotheses 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="KFF_correct_nat" class="idref" href="#KFF_correct_nat"><span class="id" title="lemma">KFF_correct_nat</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.r"><span class="id" title="variable">r</span></a>)%<span class="id" title="var">nat</span> (<a class="idref" href="additions.Euclidean_Chains.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KFK_proper" class="idref" href="#KFK_proper"><span class="id" title="lemma">KFK_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="KFF_proper" class="idref" href="#KFF_proper"><span class="id" title="instance">KFF_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.kbr"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof.fq"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#KFK_proof"><span class="id" title="section">KFK_proof</span></a>.<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KFK_correct" class="idref" href="#KFK_correct"><span class="id" title="lemma">KFK_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="b:290" class="idref" href="#b:290"><span class="id" title="binder">b</span></a> <a id="q:291" class="idref" href="#q:291"><span class="id" title="binder">q</span></a> <a id="r:292" class="idref" href="#r:292"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="kbr:293" class="idref" href="#kbr:293"><span class="id" title="binder">kbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) (<a id="fq:294" class="idref" href="#fq:294"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#b:290"><span class="id" title="variable">b</span></a> <a class="idref" href="additions.Euclidean_Chains.html#r:292"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:293"><span class="id" title="variable">kbr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:291"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:294"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:293"><span class="id" title="variable">kbr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:294"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#b:290"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:291"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#r:292"><span class="id" title="variable">r</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#b:290"><span class="id" title="variable">b</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:293"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:294"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span>  <a class="idref" href="additions.Euclidean_Chains.html#KFK_proper"><span class="id" title="lemma">KFK_proper</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span>  <a class="idref" href="additions.Euclidean_Chains.html#KFK_proper"><span class="id" title="lemma">KFK_proper</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KFF_correct" class="idref" href="#KFF_correct"><span class="id" title="lemma">KFF_correct</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="b:295" class="idref" href="#b:295"><span class="id" title="binder">b</span></a> <a id="q:296" class="idref" href="#q:296"><span class="id" title="binder">q</span></a> <a id="r:297" class="idref" href="#r:297"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="kbr:298" class="idref" href="#kbr:298"><span class="id" title="binder">kbr</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a>) (<a id="fq:299" class="idref" href="#fq:299"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#b:295"><span class="id" title="variable">b</span></a> <a class="idref" href="additions.Euclidean_Chains.html#r:297"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:298"><span class="id" title="variable">kbr</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:296"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:299"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:298"><span class="id" title="variable">kbr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:299"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#b:295"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:296"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#r:297"><span class="id" title="variable">r</span></a>) (<a class="idref" href="additions.Euclidean_Chains.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.Euclidean_Chains.html#kbr:298"><span class="id" title="variable">kbr</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:299"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FFK_correct_nat" class="idref" href="#FFK_correct_nat"><span class="id" title="lemma">FFK_correct_nat</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="p:300" class="idref" href="#p:300"><span class="id" title="binder">p</span></a> <a id="q:301" class="idref" href="#q:301"><span class="id" title="binder">q</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="fp:302" class="idref" href="#fp:302"><span class="id" title="binder">fp</span></a> <a id="fq:303" class="idref" href="#fq:303"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:300"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:302"><span class="id" title="variable">fp</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct_nat"><span class="id" title="definition">Fchain_correct_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:301"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:303"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:302"><span class="id" title="variable">fp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:303"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct_nat"><span class="id" title="definition">Kchain_correct_nat</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#p:300"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:301"><span class="id" title="variable">q</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#p:300"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#FFK"><span class="id" title="definition">FFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:302"><span class="id" title="variable">fp</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:303"><span class="id" title="variable">fq</span></a>).<br/>
+</div>
+
+<div class="doc">
+simplifying the hypotheses 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FFK_correct" class="idref" href="#FFK_correct"><span class="id" title="lemma">FFK_correct</span></a>  (<a id="p:316" class="idref" href="#p:316"><span class="id" title="binder">p</span></a> <a id="q:317" class="idref" href="#q:317"><span class="id" title="binder">q</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="fp:318" class="idref" href="#fp:318"><span class="id" title="binder">fp</span></a> <a id="fq:319" class="idref" href="#fq:319"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:316"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:318"><span class="id" title="variable">fp</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:317"><span class="id" title="variable">q</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:319"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:318"><span class="id" title="variable">fp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:319"><span class="id" title="variable">fq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#p:316"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:317"><span class="id" title="variable">q</span></a> ) <a class="idref" href="additions.Euclidean_Chains.html#p:316"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#FFK"><span class="id" title="definition">FFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:318"><span class="id" title="variable">fp</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:319"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="FFK_proper" class="idref" href="#FFK_proper"><span class="id" title="instance">FFK_proper</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fp:320" class="idref" href="#fp:320"><span class="id" title="binder">fp</span></a> <a id="fq:321" class="idref" href="#fq:321"><span class="id" title="binder">fq</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> :   <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:320"><span class="id" title="variable">fp</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">_</span> :  <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:321"><span class="id" title="variable">fq</span></a>)<br/>
+&nbsp;&nbsp;:  <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#FFK"><span class="id" title="definition">FFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fp:320"><span class="id" title="variable">fp</span></a> <a class="idref" href="additions.Euclidean_Chains.html#fq:321"><span class="id" title="variable">fq</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FK_correct" class="idref" href="#FK_correct"><span class="id" title="lemma">FK_correct</span></a> : <span class="id" title="keyword">∀</span> (<a id="p:322" class="idref" href="#p:322"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="Fp:323" class="idref" href="#Fp:323"><span class="id" title="binder">Fp</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#p:322"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fp:323"><span class="id" title="variable">Fp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fp:323"><span class="id" title="variable">Fp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#p:322"><span class="id" title="variable">p</span></a> 1 (<a class="idref" href="additions.Euclidean_Chains.html#FK"><span class="id" title="definition">FK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fp:323"><span class="id" title="variable">Fp</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="FK_proper" class="idref" href="#FK_proper"><span class="id" title="instance">FK_proper</span></a>  (<a id="Fp:328" class="idref" href="#Fp:328"><span class="id" title="binder">Fp</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a>) (<span class="id" title="var">_</span> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fp:328"><span class="id" title="variable">Fp</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#FK"><span class="id" title="definition">FK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Fp:328"><span class="id" title="variable">Fp</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">KFF_correct</span> <span class="id" title="var">KFF_proper</span> <span class="id" title="var">KFK_correct</span> <span class="id" title="var">KFK_proper</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="k3_1_correct" class="idref" href="#k3_1_correct"><span class="id" title="lemma">k3_1_correct</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a> 3 1 <a class="idref" href="additions.Euclidean_Chains.html#k3_1"><span class="id" title="definition">k3_1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="k3_1_proper" class="idref" href="#k3_1_proper"><span class="id" title="lemma">k3_1_proper</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k3_1"><span class="id" title="definition">k3_1</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">k3_1_correct</span> <span class="id" title="var">k3_1_proper</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+an example of correct chain construction  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F87" class="idref" href="#F87"><span class="id" title="definition">F87</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="k7_3:329" class="idref" href="#k7_3:329"><span class="id" title="binder">k7_3</span></a> :=  <a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k3_1"><span class="id" title="definition">k3_1</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> 1) <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="k10_7:330" class="idref" href="#k10_7:330"><span class="id" title="binder">k10_7</span></a> := <a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k7_3:329"><span class="id" title="variable">k7_3</span></a> <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#KFF"><span class="id" title="definition">KFF</span></a> <a class="idref" href="additions.Euclidean_Chains.html#k10_7:330"><span class="id" title="variable">k10_7</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> 3).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="OK87" class="idref" href="#OK87"><span class="id" title="lemma">OK87</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> 87 <a class="idref" href="additions.Euclidean_Chains.html#F87"><span class="id" title="definition">F87</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">compute_chain</span> <span class="id" title="var">ch</span> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">X</span> := <span class="id" title="tactic">fresh</span> "x" <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">Y</span> := <span class="id" title="tactic">fresh</span> "y" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">X</span> := <span class="id" title="keyword">constr</span>:(<span class="id" title="var">ch</span>) <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">Y</span> := (<span class="id" title="tactic">eval</span> <span class="id" title="keyword">vm_compute</span> <span class="id" title="tactic">in</span>  <span class="id" title="var">X</span>) <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="tactic">exact</span> <span class="id" title="var">Y</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="C87'" class="idref" href="#C87'"><span class="id" title="definition">C87'</span></a> := <span class="id" title="keyword">ltac</span>:( <span class="id" title="var">compute_chain</span> <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a> ).<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="additions.Euclidean_Chains.html#C87'"><span class="id" title="definition">C87'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="PF87" class="idref" href="#PF87"><span class="id" title="lemma">PF87</span></a>:  <a class="idref" href="additions.Addition_Chains.html#parametric"><span class="id" title="definition">parametric</span></a> <a class="idref" href="additions.Euclidean_Chains.html#C87'"><span class="id" title="definition">C87'</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab371"></a><h3 class="section">Automatic generation of correct euclidean chains</h3>
+
+
+<div class="paragraph"> </div>
+
+We want to define a function that builds a correct chain
+for any positive exponent, using the previously defined
+and certified composition operators : Fcompose, KFK, etc.
+
+<div class="paragraph"> </div>
+
+Obviously, we have to define total mutually recursive functions:
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> A function that builds an Fchain for any positive exponent p
+
+</li>
+<li> A function that builds a Kchain for any pair of exponents
+   (n,p) where 
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+ In Coq, various ways of building functions are available:
+<ul class="doclist">
+<li> Structural <span class="inlinecode"><span class="id" title="var">mutual</span></span> recursion with <span class="inlinecode"><span class="id" title="keyword">Fixpoint</span></span>
+
+</li>
+<li> Using <span class="inlinecode"><span class="id" title="var">Program</span></span> <span class="inlinecode"><span class="id" title="keyword">Fixpoint</span></span>  
+
+</li>
+<li> Using <span class="inlinecode"><span class="id" title="keyword">Function</span></span>
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+ For simplicity's sake, we chosed to avoid dependent elimination
+ and used <span class="inlinecode"><span class="id" title="keyword">Function</span></span> with a decreasing measure.
+ For this purpose, we define a single data-type for associated with
+ the generation of F- and K-chains.
+
+<div class="paragraph"> </div>
+
+For specifying the computation of a Kchain for  and 
+where , we use the pair of positive numbers ,
+thus avoiding to propagate the constraint  in 
+our definitions.
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="signature" class="idref" href="#signature"><span class="id" title="inductive">signature</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
+| <a id="gen_F" class="idref" href="#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a id="n:333" class="idref" href="#n:333"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) </div>
+
+<div class="doc">
+Fchain for the exponent n 
+</div>
+<div class="code">
+| <a id="gen_K" class="idref" href="#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a id="p:334" class="idref" href="#p:334"><span class="id" title="binder">p</span></a> <a id="d:335" class="idref" href="#d:335"><span class="id" title="binder">d</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) </div>
+
+<div class="doc">
+Kchain for the exponents p+d  and p </div>
+<div class="code">
+.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Unifying  statements about chain generation 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="signature_exponent" class="idref" href="#signature_exponent"><span class="id" title="definition">signature_exponent</span></a> (<a id="s:336" class="idref" href="#s:336"><span class="id" title="binder">s</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:336"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <br/>
+| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">n</span> ⇒ <span class="id" title="var">n</span> <br/>
+| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">d</span>  ⇒  <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="kont_type" class="idref" href="#kont_type"><span class="id" title="definition">kont_type</span></a> (<a id="s:338" class="idref" href="#s:338"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>)(<a id="A:339" class="idref" href="#A:339"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:338"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <br/>
+| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">_</span>  ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fkont"><span class="id" title="definition">Fkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:339"><span class="id" title="variable">A</span></a> <br/>
+| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>   ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Kkont"><span class="id" title="definition">Kkont</span></a> <a class="idref" href="additions.Euclidean_Chains.html#A:339"><span class="id" title="variable">A</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_type" class="idref" href="#chain_type"><span class="id" title="definition">chain_type</span></a> (<a id="s:341" class="idref" href="#s:341"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:341"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <br/>
+| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">_</span>   ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fchain"><span class="id" title="definition">Fchain</span></a><br/>
+|  <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>  ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Kchain"><span class="id" title="definition">Kchain</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="correctness_statement" class="idref" href="#correctness_statement"><span class="id" title="definition">correctness_statement</span></a> (<a id="s:343" class="idref" href="#s:343"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <br/>
+<a class="idref" href="additions.Euclidean_Chains.html#chain_type"><span class="id" title="definition">chain_type</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:343"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:343"><span class="id" title="variable">s</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">p</span> ⇒ <span class="id" title="keyword">fun</span> <a id="ch:345" class="idref" href="#ch:345"><span class="id" title="binder">ch</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fchain_correct"><span class="id" title="definition">Fchain_correct</span></a> <span class="id" title="var">p</span> <a class="idref" href="additions.Euclidean_Chains.html#ch:345"><span class="id" title="variable">ch</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">d</span>   ⇒ <span class="id" title="keyword">fun</span> <a id="ch:346" class="idref" href="#ch:346"><span class="id" title="binder">ch</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Kchain_correct"><span class="id" title="definition">Kchain_correct</span></a>  (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>) <span class="id" title="var">p</span> <a class="idref" href="additions.Euclidean_Chains.html#ch:346"><span class="id" title="variable">ch</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="proper_statement" class="idref" href="#proper_statement"><span class="id" title="definition">proper_statement</span></a> (<a id="s:347" class="idref" href="#s:347"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <br/>
+<a class="idref" href="additions.Euclidean_Chains.html#chain_type"><span class="id" title="definition">chain_type</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:347"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:347"><span class="id" title="variable">s</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">_</span> ⇒ <span class="id" title="keyword">fun</span> <a id="ch:349" class="idref" href="#ch:349"><span class="id" title="binder">ch</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fchain_proper"><span class="id" title="class">Fchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#ch:349"><span class="id" title="variable">ch</span></a> <br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>   ⇒ <span class="id" title="keyword">fun</span> <a id="ch:350" class="idref" href="#ch:350"><span class="id" title="binder">ch</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Kchain_proper"><span class="id" title="class">Kchain_proper</span></a> <a class="idref" href="additions.Euclidean_Chains.html#ch:350"><span class="id" title="variable">ch</span></a> <br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab372"></a><h2 class="section">Full correctness</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="OK" class="idref" href="#OK"><span class="id" title="definition">OK</span></a> (<a id="s:351" class="idref" href="#s:351"><span class="id" title="binder">s</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) <br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">fun</span> <a id="c:352" class="idref" href="#c:352"><span class="id" title="binder">c</span></a>: <a class="idref" href="additions.Euclidean_Chains.html#chain_type"><span class="id" title="definition">chain_type</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:351"><span class="id" title="variable">s</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#correctness_statement"><span class="id" title="definition">correctness_statement</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:351"><span class="id" title="variable">s</span></a> <a class="idref" href="additions.Euclidean_Chains.html#c:352"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#proper_statement"><span class="id" title="definition">proper_statement</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:351"><span class="id" title="variable">s</span></a> <a class="idref" href="additions.Euclidean_Chains.html#c:352"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">pos_gt_3</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Gamma" class="idref" href="#Gamma"><span class="id" title="section">Gamma</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Gamma.gamma" class="idref" href="#Gamma.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="Hgamma:354" class="idref" href="#Hgamma:354"><span class="id" title="binder">Hgamma</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="signature_measure" class="idref" href="#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a id="s:355" class="idref" href="#s:355"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:355"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">n</span> ⇒ 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <span class="id" title="var">n</span> <br/>
+&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a>  <span class="id" title="var">p</span> <span class="id" title="var">d</span> ⇒ 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a>1<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO1" class="idref" href="#PO1"><span class="id" title="lemma">PO1</span></a> :<span class="id" title="keyword">∀</span> (<a id="s:357" class="idref" href="#s:357"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="i:358" class="idref" href="#i:358"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:357"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:359" class="idref" href="#anonymous:359"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:359"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous0:360" class="idref" href="#anonymous0:360"><span class="id" title="binder">anonymous0</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous0:360"><span class="id" title="variable">anonymous0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:361" class="idref" href="#q:361"><span class="id" title="binder">q</span></a> <a id="r:362" class="idref" href="#r:362"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:362"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:361"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:361"><span class="id" title="variable">q</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:358"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO2" class="idref" href="#PO2"><span class="id" title="lemma">PO2</span></a> :  <span class="id" title="keyword">∀</span> (<a id="s:363" class="idref" href="#s:363"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="i:364" class="idref" href="#i:364"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:363"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:365" class="idref" href="#anonymous:365"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:365"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous0:366" class="idref" href="#anonymous0:366"><span class="id" title="binder">anonymous0</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous0:366"><span class="id" title="variable">anonymous0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:367" class="idref" href="#q:367"><span class="id" title="binder">q</span></a> <a id="r:368" class="idref" href="#r:368"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:368"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:367"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:364"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO3" class="idref" href="#PO3"><span class="id" title="lemma">PO3</span></a> :  <span class="id" title="keyword">∀</span> (<a id="s:369" class="idref" href="#s:369"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="i:370" class="idref" href="#i:370"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:369"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:371" class="idref" href="#anonymous:371"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:371"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous0:372" class="idref" href="#anonymous0:372"><span class="id" title="binder">anonymous0</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous0:372"><span class="id" title="variable">anonymous0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="q:373" class="idref" href="#q:373"><span class="id" title="binder">q</span></a> <a id="r:374" class="idref" href="#r:374"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="p:375" class="idref" href="#p:375"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:374"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:375"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:373"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:375"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:373"><span class="id" title="variable">q</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:370"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO4" class="idref" href="#PO4"><span class="id" title="lemma">PO4</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:376" class="idref" href="#s:376"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="i:377" class="idref" href="#i:377"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:376"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:378" class="idref" href="#anonymous:378"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:378"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous0:379" class="idref" href="#anonymous0:379"><span class="id" title="binder">anonymous0</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous0:379"><span class="id" title="variable">anonymous0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="q:380" class="idref" href="#q:380"><span class="id" title="binder">q</span></a> <a id="r:381" class="idref" href="#r:381"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="p:382" class="idref" href="#p:382"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:381"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:382"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:380"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:382"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:382"><span class="id" title="variable">p</span></a>)) (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:382"><span class="id" title="variable">p</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#i:377"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO6" class="idref" href="#PO6"><span class="id" title="lemma">PO6</span></a>: <span class="id" title="keyword">∀</span> (<a id="s:383" class="idref" href="#s:383"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="p:384" class="idref" href="#p:384"><span class="id" title="binder">p</span></a> <a id="d:385" class="idref" href="#d:385"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:383"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:385"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:386" class="idref" href="#anonymous:386"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:386"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:387" class="idref" href="#q:387"><span class="id" title="binder">q</span></a> <a id="r:388" class="idref" href="#r:388"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:388"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:385"><span class="id" title="variable">d</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:387"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:387"><span class="id" title="variable">q</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:384"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:385"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="PO8" class="idref" href="#PO8"><span class="id" title="lemma">PO8</span></a> :<span class="id" title="keyword">∀</span> (<a id="s:389" class="idref" href="#s:389"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="p:390" class="idref" href="#p:390"><span class="id" title="binder">p</span></a> <a id="d:391" class="idref" href="#d:391"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:389"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:391"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:392" class="idref" href="#anonymous:392"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:392"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="q:393" class="idref" href="#q:393"><span class="id" title="binder">q</span></a> <a id="r:394" class="idref" href="#r:394"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="p0:395" class="idref" href="#p0:395"><span class="id" title="binder">p0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:394"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:395"><span class="id" title="variable">p0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:391"><span class="id" title="variable">d</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:393"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:395"><span class="id" title="variable">p0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#q:393"><span class="id" title="variable">q</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:390"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:391"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PO9" class="idref" href="#PO9"><span class="id" title="lemma">PO9</span></a> :<span class="id" title="keyword">∀</span> (<a id="s:396" class="idref" href="#s:396"><span class="id" title="binder">s</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) (<a id="p:397" class="idref" href="#p:397"><span class="id" title="binder">p</span></a> <a id="d:398" class="idref" href="#d:398"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#s:396"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:398"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="anonymous:399" class="idref" href="#anonymous:399"><span class="id" title="binder">anonymous</span></a> : <a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="additions.Euclidean_Chains.html#anonymous:399"><span class="id" title="variable">anonymous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="q:400" class="idref" href="#q:400"><span class="id" title="binder">q</span></a> <a id="r:401" class="idref" href="#r:401"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="p0:402" class="idref" href="#p0:402"><span class="id" title="binder">p0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#r:401"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:402"><span class="id" title="variable">p0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:398"><span class="id" title="variable">d</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Euclidean_Chains.html#q:400"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:402"><span class="id" title="variable">p0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:402"><span class="id" title="variable">p0</span></a>)) (<a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p0:402"><span class="id" title="variable">p0</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:397"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Euclidean_Chains.html#d:398"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">nat</span>.<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="var">chain_gen</span>  (<a id="s:412" class="idref" href="#s:412"><span class="id" title="binder">s</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>) {<span class="id" title="keyword">measure</span> <a class="idref" href="additions.Euclidean_Chains.html#signature_measure"><span class="id" title="definition">signature_measure</span></a>}<br/>
+:  <a class="idref" href="additions.Euclidean_Chains.html#chain_type"><span class="id" title="definition">chain_type</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:403"><span class="id" title="variable">s</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Euclidean_Chains.html#s:403"><span class="id" title="variable">s</span></a>  <span class="id" title="keyword">return</span> <a class="idref" href="additions.Euclidean_Chains.html#chain_type"><span class="id" title="definition">chain_type</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:405"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">i</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">i</span> 1 <span class="id" title="keyword">then</span> <a class="idref" href="additions.Euclidean_Chains.html#F1"><span class="id" title="definition">F1</span></a> <span class="id" title="keyword">else</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">i</span> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="additions.Euclidean_Chains.html#F3"><span class="id" title="definition">F3</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#Fexp2"><span class="id" title="definition">Fexp2</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <span class="id" title="var">i</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#Fcompose"><span class="id" title="definition">Fcompose</span></a>  (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  ⇒ <a class="idref" href="additions.Euclidean_Chains.html#KFF"><span class="id" title="definition">KFF</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#Gamma.gamma"><span class="id" title="variable">gamma</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">d</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.More_on_positive.html#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> <span class="id" title="var">p</span> 1 <span class="id" title="keyword">then</span> <a class="idref" href="additions.Euclidean_Chains.html#FK"><span class="id" title="definition">FK</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> (<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <span class="id" title="var">d</span>)  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="additions.Euclidean_Chains.html#FFK"><span class="id" title="definition">FFK</span></a>   (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <span class="id" title="var">p</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  ⇒ <a class="idref" href="additions.Euclidean_Chains.html#KFK"><span class="id" title="definition">KFK</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_K"><span class="id" title="constructor">gen_K</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">_r</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Euclidean_Chains.html#chain_gen:404"><span class="id" title="definition">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <span class="id" title="var">q</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="make_chain" class="idref" href="#make_chain"><span class="id" title="definition">make_chain</span></a> (<a id="n:470" class="idref" href="#n:470"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Addition_Chains.html#chain"><span class="id" title="definition">chain</span></a> :=<br/>
+&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#F2C"><span class="id" title="definition">F2C</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:470"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="chain_gen_OK" class="idref" href="#chain_gen_OK"><span class="id" title="lemma">chain_gen_OK</span></a> : <span class="id" title="keyword">∀</span> <a id="s:471" class="idref" href="#s:471"><span class="id" title="binder">s</span></a>:<a class="idref" href="additions.Euclidean_Chains.html#signature"><span class="id" title="inductive">signature</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Euclidean_Chains.html#OK"><span class="id" title="definition">OK</span></a> <a class="idref" href="additions.Euclidean_Chains.html#s:471"><span class="id" title="variable">s</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#s:471"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="make_chain_correct" class="idref" href="#make_chain_correct"><span class="id" title="lemma">make_chain_correct</span></a> : <span class="id" title="keyword">∀</span> <a id="p:472" class="idref" href="#p:472"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:472"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Euclidean_Chains.html#p:472"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Gamma"><span class="id" title="section">Gamma</span></a>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <span class="id" title="var">gamma</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> .<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#"><span class="id" title="library">Int63.Uint63</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="additions.Addition_Chains.html#cpower"><span class="id" title="definition">cpower</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a>) 10.<br/>
+<span class="id" title="keyword">Module</span>  <a id="Examples" class="idref" href="#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+<span class="id" title="keyword">Search</span> (<span class="id" title="var">int</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>).<br/>
+<span class="id" title="keyword">Search</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">int</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.fast_int63_power" class="idref" href="#Examples.fast_int63_power"><span class="id" title="definition">fast_int63_power</span></a> (<a id="x:473" class="idref" href="#x:473"><span class="id" title="binder">x</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)(<a id="n:474" class="idref" href="#n:474"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#to_Z"><span class="id" title="definition">to_Z</span></a> (<a class="idref" href="additions.Addition_Chains.html#cpower"><span class="id" title="definition">cpower</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#n:474"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_pos"><span class="id" title="definition">of_pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:473"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.slow_int31_power" class="idref" href="#Examples.slow_int31_power"><span class="id" title="definition">slow_int31_power</span></a> (<a id="x:475" class="idref" href="#x:475"><span class="id" title="binder">x</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)(<a id="n:476" class="idref" href="#n:476"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#to_Z"><span class="id" title="definition">to_Z</span></a> (<a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_pos"><span class="id" title="definition">of_pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:475"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="additions.Euclidean_Chains.html#n:476"><span class="id" title="variable">n</span></a>) ).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.binary_int31_power" class="idref" href="#Examples.binary_int31_power"><span class="id" title="definition">binary_int31_power</span></a> (<a id="x:477" class="idref" href="#x:477"><span class="id" title="binder">x</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)(<a id="n:478" class="idref" href="#n:478"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#to_Z"><span class="id" title="definition">to_Z</span></a> (<a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">N_bpow</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#of_pos"><span class="id" title="definition">of_pos</span></a> <a class="idref" href="additions.Euclidean_Chains.html#x:477"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#n:478"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+long computations ... 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.big_chain" class="idref" href="#Examples.big_chain"><span class="id" title="definition">big_chain</span></a> := <span class="id" title="keyword">ltac</span>:(<span class="id" title="var">compute_chain</span>  (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> 45319)).<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="additions.Euclidean_Chains.html#Examples.big_chain"><span class="id" title="definition">big_chain</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Euclidean_Chains.html#Examples.big_chain"><span class="id" title="definition">big_chain</span></a> <span class="id" title="var">_</span>%<span class="id" title="var">type_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Examples.RM" class="idref" href="#Examples.RM"><span class="id" title="lemma">RM</span></a> : (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> 56789)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.M" class="idref" href="#Examples.M"><span class="id" title="definition">M</span></a> := <a class="idref" href="additions.Monoid_instances.html#Nmod_Monoid"><span class="id" title="instance">Nmod_Monoid</span></a>  56789%<span class="id" title="var">N</span> <a class="idref" href="additions.Euclidean_Chains.html#Examples.RM"><span class="id" title="lemma">RM</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.exp56789" class="idref" href="#Examples.exp56789"><span class="id" title="definition">exp56789</span></a> <a id="x:479" class="idref" href="#x:479"><span class="id" title="binder">x</span></a> := <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a class="idref" href="additions.Euclidean_Chains.html#Examples.big_chain"><span class="id" title="definition">big_chain</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Euclidean_Chains.html#Examples.M"><span class="id" title="definition">M</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:479"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">cbv</span> <span class="id" title="keyword">iota</span>  <span class="id" title="keyword">match</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">big_chain</span> <span class="id" title="var">chain_apply</span> <span class="id" title="var">computation_eval</span>  ]  <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="tactic">in</span>  <span class="id" title="keyword">fun</span> <a id="x:480" class="idref" href="#x:480"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a>  <a class="idref" href="additions.Euclidean_Chains.html#Examples.big_chain"><span class="id" title="definition">big_chain</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Euclidean_Chains.html#Examples.M"><span class="id" title="definition">M</span></a>) <a class="idref" href="additions.Euclidean_Chains.html#x:480"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Examples.C87'" class="idref" href="#Examples.C87'"><span class="id" title="definition">C87'</span></a> := <span class="id" title="keyword">ltac</span>:( <span class="id" title="var">compute_chain</span> <a class="idref" href="additions.Addition_Chains.html#C87"><span class="id" title="definition">C87</span></a> ).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Examples.Unnamed_thm" class="idref" href="#Examples.Unnamed_thm"><span class="id" title="definition">parametric</span></a> <a id="Examples.Unnamed_thm" class="idref" href="#Examples.Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Examples.Unnamed_thm" class="idref" href="#Examples.Unnamed_thm"><span class="id" title="definition">make_chain</span></a> <a id="Examples.Unnamed_thm" class="idref" href="#Examples.Unnamed_thm"><span class="id" title="definition">dicho</span></a> <a id="Examples.Unnamed_thm" class="idref" href="#Examples.Unnamed_thm"><span class="id" title="definition">45319)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Examples.big_correct" class="idref" href="#Examples.big_correct"><span class="id" title="lemma">big_correct</span></a> :<a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 45319 (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> 45319).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Examples.big_correct'" class="idref" href="#Examples.big_correct'"><span class="id" title="lemma">big_correct'</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 453 (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> 453).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Examples.big_correct'''" class="idref" href="#Examples.big_correct'''"><span class="id" title="lemma">big_correct'''</span></a> : <a class="idref" href="additions.Addition_Chains.html#chain_correct"><span class="id" title="definition">chain_correct</span></a> 453 (<a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> 453).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Euclidean_Chains.html#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.extraction.Extraction.html#"><span class="id" title="library">Extraction</span></a>.<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">exp56789</span>.<br/>
+<span class="id" title="keyword">Extraction</span> <span class="id" title="var">Language</span> <span class="id" title="var">OCaml</span>.<br/>
+<span class="id" title="keyword">Extraction</span> "bigmod" <span class="id" title="var">Examples.exp56789</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Recursive Extraction</span> <span class="id" title="var">cpower</span>.<br/>
+<span class="id" title="keyword">Recursive Extraction</span> <span class="id" title="var">make_chain</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Fib2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Fib2</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.Ring.html#"><span class="id" title="library">Ring</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_instances.html#"><span class="id" title="library">Monoid_instances</span></a> <a class="idref" href="additions.Euclidean_Chains.html#"><span class="id" title="library">Euclidean_Chains</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Strategies.html#"><span class="id" title="library">Strategies</span></a> <a class="idref" href="additions.Dichotomy.html#"><span class="id" title="library">Dichotomy</span></a> <a class="idref" href="additions.BinaryStrat.html#"><span class="id" title="library">BinaryStrat</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.Addition_Chains.html#"><span class="id" title="module">Addition_Chains</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">N_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fib" class="idref" href="#fib"><span class="id" title="definition">fib</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Fib2.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0%<span class="id" title="var">nat</span> | 1%<span class="id" title="var">nat</span> ⇒ 1<br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ((<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">q</span>)) ⇒ <a class="idref" href="additions.Fib2.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Fib2.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fib_ind" class="idref" href="#fib_ind"><span class="id" title="lemma">fib_ind</span></a> (<a id="P:4" class="idref" href="#P:4"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a> 0%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a> 1%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a>, <a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a> <a class="idref" href="additions.Fib2.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:5"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:5"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>, <a class="idref" href="additions.Fib2.html#P:4"><span class="id" title="variable">P</span></a> <a class="idref" href="additions.Fib2.html#n:6"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fib_SSn" class="idref" href="#fib_SSn"><span class="id" title="lemma">fib_SSn</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) , <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:7"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.Fib2.html#n:7"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:7"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Yves' encoding 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mul2" class="idref" href="#mul2"><span class="id" title="definition">mul2</span></a> (<a id="p:8" class="idref" href="#p:8"><span class="id" title="binder">p</span></a> <a id="q:9" class="idref" href="#q:9"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Fib2.html#p:8"><span class="id" title="variable">p</span></a>, <a class="idref" href="additions.Fib2.html#q:9"><span class="id" title="variable">q</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">d</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a><span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a><span class="id" title="var">d</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a><span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a><span class="id" title="var">d</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="neutral_l" class="idref" href="#neutral_l"><span class="id" title="lemma">neutral_l</span></a> <a id="p:12" class="idref" href="#p:12"><span class="id" title="binder">p</span></a> : <a class="idref" href="additions.Fib2.html#mul2"><span class="id" title="definition">mul2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Fib2.html#p:12"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Fib2.html#p:12"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="neutral_r" class="idref" href="#neutral_r"><span class="id" title="lemma">neutral_r</span></a> <a id="p:13" class="idref" href="#p:13"><span class="id" title="binder">p</span></a> : <a class="idref" href="additions.Fib2.html#mul2"><span class="id" title="definition">mul2</span></a> <a class="idref" href="additions.Fib2.html#p:13"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Fib2.html#p:13"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Mul2" class="idref" href="#Mul2"><span class="id" title="instance">Mul2</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a>  <a class="idref" href="additions.Fib2.html#mul2"><span class="id" title="definition">mul2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="next_fib" class="idref" href="#next_fib"><span class="id" title="lemma">next_fib</span></a> (<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.Fib2.html#mul2"><span class="id" title="definition">mul2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:14"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.Fib2.html#n:14"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:14"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:14"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fib_mul2" class="idref" href="#fib_mul2"><span class="id" title="definition">fib_mul2</span></a> <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> := <span class="id" title="keyword">let</span> (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a>,<a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a>) := <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Fib2.html#Mul2"><span class="id" title="instance">Mul2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Fib2.html#n:15"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> (<a class="idref" href="additions.Fib2.html#a:16"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><a class="idref" href="additions.Fib2.html#b:17"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fib_mul2_OK_0" class="idref" href="#fib_mul2_OK_0"><span class="id" title="lemma">fib_mul2_OK_0</span></a> (<a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Fib2.html#Mul2"><span class="id" title="instance">Mul2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:18"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Fib2.html#n:18"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.Fib2.html#n:18"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fib_mul2_OK" class="idref" href="#fib_mul2_OK"><span class="id" title="lemma">fib_mul2_OK</span></a> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> : <a class="idref" href="additions.Fib2.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.Fib2.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Fib2.html#fib_mul2"><span class="id" title="definition">fib_mul2</span></a> <a class="idref" href="additions.Fib2.html#n:19"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fib_pos" class="idref" href="#fib_pos"><span class="id" title="definition">fib_pos</span></a> <a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a>,<a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a>) := <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> (<span class="id" title="var">M</span>:= <a class="idref" href="additions.Fib2.html#Mul2"><span class="id" title="instance">Mul2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Fib2.html#n:20"><span class="id" title="variable">n</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;(<a class="idref" href="additions.Fib2.html#a:21"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><a class="idref" href="additions.Fib2.html#b:22"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">chain_apply</span>.<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fib_eucl" class="idref" href="#fib_eucl"><span class="id" title="definition">fib_eucl</span></a> <a id="gamma:23" class="idref" href="#gamma:23"><span class="id" title="binder">gamma</span></a> `{<a id="Hgamma:24" class="idref" href="#Hgamma:24"><span class="id" title="binder">Hgamma</span></a>: <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.Fib2.html#gamma:23"><span class="id" title="variable">gamma</span></a>} <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="c:26" class="idref" href="#c:26"><span class="id" title="binder">c</span></a> := <a class="idref" href="additions.Euclidean_Chains.html#make_chain"><span class="id" title="definition">make_chain</span></a> <a class="idref" href="additions.Fib2.html#gamma:23"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="additions.Fib2.html#n:25"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <a id="r:27" class="idref" href="#r:27"><span class="id" title="binder">r</span></a> := <a class="idref" href="additions.Addition_Chains.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a class="idref" href="additions.Fib2.html#c:26"><span class="id" title="variable">c</span></a> (<span class="id" title="var">M</span>:=<a class="idref" href="additions.Fib2.html#Mul2"><span class="id" title="instance">Mul2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="additions.Fib2.html#r:27"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="additions.Fib2.html#r:27"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.AM.html#"><span class="id" title="library">AM</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fib_with_chain" class="idref" href="#fib_with_chain"><span class="id" title="definition">fib_with_chain</span></a> <a id="c:28" class="idref" href="#c:28"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.AM.html#chain_apply"><span class="id" title="definition">chain_apply</span></a> <a class="idref" href="additions.Fib2.html#c:28"><span class="id" title="variable">c</span></a>  <a class="idref" href="additions.Fib2.html#Mul2"><span class="id" title="instance">Mul2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">b</span>) | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="c153" class="idref" href="#c153"><span class="id" title="definition">c153</span></a> := <a class="idref" href="additions.AM.html#chain_gen"><span class="id" title="lemma">chain_gen</span></a> <a class="idref" href="additions.Dichotomy.html#dicho"><span class="id" title="definition">dicho</span></a> (<a class="idref" href="additions.Euclidean_Chains.html#gen_F"><span class="id" title="constructor">gen_F</span></a> 153%<span class="id" title="var">positive</span>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.FirstSteps.html b/theories/html/additions.FirstSteps.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/additions.FirstSteps.html
@@ -0,0 +1,270 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.FirstSteps</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.FirstSteps</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Polymorphic versions of exponentiation functions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Strings.String.html#"><span class="id" title="library">String</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Polymorphic exponentiation functions 
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Definitions" class="idref" href="#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Definitions.A" class="idref" href="#Definitions.A"><span class="id" title="variable">A</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Definitions.mult" class="idref" href="#Definitions.mult"><span class="id" title="variable">mult</span></a> : <a class="idref" href="additions.FirstSteps.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Definitions.one" class="idref" href="#Definitions.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.FirstSteps.html#A:1"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Infix</span> "*" := <a class="idref" href="additions.FirstSteps.html#Definitions.mult"><span class="id" title="variable">mult</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> "1" := <a class="idref" href="additions.FirstSteps.html#Definitions.one"><span class="id" title="variable">one</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Naive (linear) implementation 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="power" class="idref" href="#power"><span class="id" title="definition">power</span></a> (<a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.FirstSteps.html#Definitions.A"><span class="id" title="variable">A</span></a>)(<a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.FirstSteps.html#Definitions.A"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.FirstSteps.html#n:5"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| 0%<span class="id" title="var">nat</span> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒   <a class="idref" href="additions.FirstSteps.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#072addf1f952576a9d67e98c63e20939"><span class="id" title="notation">^</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="072addf1f952576a9d67e98c63e20939" class="idref" href="#072addf1f952576a9d67e98c63e20939"><span class="id" title="notation">&quot;</span></a>x ^ n" := (<a class="idref" href="additions.FirstSteps.html#power:6"><span class="id" title="definition">power</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Logarithmic implementation (with exponents in <span class="inlinecode"><span class="id" title="var">N</span></span>)  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="binary_power_mult" class="idref" href="#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a> (<a id="x:8" class="idref" href="#x:8"><span class="id" title="binder">x</span></a> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.FirstSteps.html#Definitions.A"><span class="id" title="variable">A</span></a>)(<a id="p:10" class="idref" href="#p:10"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.FirstSteps.html#Definitions.A"><span class="id" title="variable">A</span></a> <br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.FirstSteps.html#p:10"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒  <a class="idref" href="additions.FirstSteps.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.FirstSteps.html#binary_power_mult:11"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.FirstSteps.html#a:9"><span class="id" title="variable">a</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒  <a class="idref" href="additions.FirstSteps.html#binary_power_mult:11"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.FirstSteps.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:8"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Pos_bpow" class="idref" href="#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a>   (<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.FirstSteps.html#Definitions.A"><span class="id" title="variable">A</span></a>)(<a id="p:14" class="idref" href="#p:14"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.FirstSteps.html#p:14"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.FirstSteps.html#Pos_bpow:15"><span class="id" title="definition">Pos_bpow</span></a>  (<a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.FirstSteps.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>   (<a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.FirstSteps.html#x:13"><span class="id" title="variable">x</span></a> <span class="id" title="var">q</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N_bpow" class="idref" href="#N_bpow"><span class="id" title="definition">N_bpow</span></a>  <a id="x:17" class="idref" href="#x:17"><span class="id" title="binder">x</span></a> (<a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.FirstSteps.html#n:18"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| 0%<span class="id" title="var">N</span> ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.FirstSteps.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.FirstSteps.html#x:17"><span class="id" title="variable">x</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.FirstSteps.html#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.FirstSteps.html#N_bpow"><span class="id" title="definition">N_bpow</span></a>  {<span class="id" title="var">A</span>}.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.FirstSteps.html#power"><span class="id" title="definition">power</span></a>  {<span class="id" title="var">A</span>}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab373"></a><h2 class="section">Examples</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">string_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Exponentiation on 2x2 matrices 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="M2" class="idref" href="#M2"><span class="id" title="module">M2</span></a>.<br/>
+<span class="id" title="keyword">Section</span> <a id="M2.M2_Definitions" class="idref" href="#M2.M2_Definitions"><span class="id" title="section">M2_Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="M2.M2_Definitions.A" class="idref" href="#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2.M2_Definitions.zero" class="idref" href="#M2.M2_Definitions.zero"><span class="id" title="variable">zero</span></a> <a id="M2.M2_Definitions.one" class="idref" href="#M2.M2_Definitions.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.FirstSteps.html#A:20"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2.M2_Definitions.plus" class="idref" href="#M2.M2_Definitions.plus"><span class="id" title="variable">plus</span></a> <a id="M2.M2_Definitions.mult" class="idref" href="#M2.M2_Definitions.mult"><span class="id" title="variable">mult</span></a>  : <a class="idref" href="additions.FirstSteps.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:20"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="M2.M2_Definitions.rt" class="idref" href="#M2.M2_Definitions.rt"><span class="id" title="variable">rt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.Ring_theory.html#semi_ring_theory"><span class="id" title="record">semi_ring_theory</span></a>  <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.zero"><span class="id" title="variable">zero</span></a> <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.plus"><span class="id" title="variable">plus</span></a> <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.mult"><span class="id" title="variable">mult</span></a>   (@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Add</span> <span class="id" title="var">Ring</span> <span class="id" title="var">Aring</span> : <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.rt"><span class="id" title="variable">rt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2.M2_Definitions.:::'0'" class="idref" href="#M2.M2_Definitions.:::'0'"><span class="id" title="notation">&quot;</span></a>0" := <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.zero"><span class="id" title="variable">zero</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2.M2_Definitions.:::'1'" class="idref" href="#M2.M2_Definitions.:::'1'"><span class="id" title="notation">&quot;</span></a>1" := <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.one"><span class="id" title="variable">one</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="792efe48a5e5591a3ac8904aa574c15c" class="idref" href="#792efe48a5e5591a3ac8904aa574c15c"><span class="id" title="notation">&quot;</span></a>x + y" := (<a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.plus"><span class="id" title="variable">plus</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="e19429236df0008b7680aa787e1bf5a5" class="idref" href="#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">&quot;</span></a>x * y " := (<a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.mult"><span class="id" title="variable">mult</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Structure</span> <a id="M2.t" class="idref" href="#M2.t"><span class="id" title="record">t</span></a> : <span class="id" title="keyword">Type</span> := <a id="M2.mat" class="idref" href="#M2.mat"><span class="id" title="constructor">mat</span></a>{<a id="M2.c00" class="idref" href="#M2.c00"><span class="id" title="projection">c00</span></a> : <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>;  <a id="M2.c01" class="idref" href="#M2.c01"><span class="id" title="projection">c01</span></a> : <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="M2.c10" class="idref" href="#M2.c10"><span class="id" title="projection">c10</span></a> : <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>;  <a id="M2.c11" class="idref" href="#M2.c11"><span class="id" title="projection">c11</span></a> : <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions.A"><span class="id" title="variable">A</span></a>}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="M2.Id2" class="idref" href="#M2.Id2"><span class="id" title="definition">Id2</span></a> : <a class="idref" href="additions.FirstSteps.html#M2.t"><span class="id" title="record">t</span></a> := <a class="idref" href="additions.FirstSteps.html#M2.mat"><span class="id" title="constructor">mat</span></a> 1 0 0 1.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="M2.M2_mult" class="idref" href="#M2.M2_mult"><span class="id" title="definition">M2_mult</span></a> (<a id="M:31" class="idref" href="#M:31"><span class="id" title="binder">M</span></a> <a id="M':32" class="idref" href="#M':32"><span class="id" title="binder">M'</span></a>:<a class="idref" href="additions.FirstSteps.html#M2.t"><span class="id" title="record">t</span></a>) : <a class="idref" href="additions.FirstSteps.html#M2.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.FirstSteps.html#M2.mat"><span class="id" title="constructor">mat</span></a> (<a class="idref" href="additions.FirstSteps.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.FirstSteps.html#792efe48a5e5591a3ac8904aa574c15c"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.FirstSteps.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.FirstSteps.html#792efe48a5e5591a3ac8904aa574c15c"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.FirstSteps.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.FirstSteps.html#792efe48a5e5591a3ac8904aa574c15c"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.FirstSteps.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.FirstSteps.html#792efe48a5e5591a3ac8904aa574c15c"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.FirstSteps.html#M:31"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.FirstSteps.html#M':32"><span class="id" title="variable">M'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.FirstSteps.html#M2.M2_Definitions"><span class="id" title="section">M2_Definitions</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.FirstSteps.html#M2"><span class="id" title="module">M2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.FirstSteps.html#M2"><span class="id" title="module">M2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.FirstSteps.html#M2_mult"><span class="id" title="definition">M2_mult</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">plus</span> <span class="id" title="var">mult</span>  <span class="id" title="var">_</span>  <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.FirstSteps.html#mat"><span class="id" title="constructor">mat</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.FirstSteps.html#Id2"><span class="id" title="definition">Id2</span></a> {<span class="id" title="var">A</span>}  <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fibonacci" class="idref" href="#fibonacci"><span class="id" title="definition">fibonacci</span></a> (<a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) :=<br/>
+&nbsp;<a class="idref" href="additions.FirstSteps.html#c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>  (<a class="idref" href="additions.FirstSteps.html#N_bpow"><span class="id" title="definition">N_bpow</span></a>  (<a class="idref" href="additions.FirstSteps.html#M2_mult"><span class="id" title="definition">M2_mult</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#Nplus"><span class="id" title="abbreviation">Nplus</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#Nmult"><span class="id" title="abbreviation">Nmult</span></a>) (<a class="idref" href="additions.FirstSteps.html#Id2"><span class="id" title="definition">Id2</span></a>  0%<span class="id" title="var">N</span> 1%<span class="id" title="var">N</span>)(<a class="idref" href="additions.FirstSteps.html#mat"><span class="id" title="constructor">mat</span></a>  1 1 1 0)%<span class="id" title="var">N</span> <a class="idref" href="additions.FirstSteps.html#n:33"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="power_t" class="idref" href="#power_t"><span class="id" title="definition">power_t</span></a> := <span class="id" title="keyword">∀</span> (<a id="A:34" class="idref" href="#A:34"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="mult:35" class="idref" href="#mult:35"><span class="id" title="binder">mult</span></a> : <a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="one:36" class="idref" href="#one:36"><span class="id" title="binder">one</span></a>:<a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:37" class="idref" href="#x:37"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="n:38" class="idref" href="#n:38"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <a class="idref" href="additions.FirstSteps.html#A:34"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab374"></a><h1 class="section">A wrong definition of correctness</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad" class="idref" href="#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Bad.correct_expt_function" class="idref" href="#Bad.correct_expt_function"><span class="id" title="definition">correct_expt_function</span></a> (<a id="f:39" class="idref" href="#f:39"><span class="id" title="binder">f</span></a> : <a class="idref" href="additions.FirstSteps.html#power_t"><span class="id" title="definition">power_t</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:40" class="idref" href="#A:40"><span class="id" title="binder">A</span></a> (<a id="mult:41" class="idref" href="#mult:41"><span class="id" title="binder">mult</span></a> : <a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a>) (<a id="one:42" class="idref" href="#one:42"><span class="id" title="binder">one</span></a>:<a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:43" class="idref" href="#x:43"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a>) (<a id="n:44" class="idref" href="#n:44"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <a class="idref" href="additions.FirstSteps.html#power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.FirstSteps.html#mult:41"><span class="id" title="variable">mult</span></a> <a class="idref" href="additions.FirstSteps.html#one:42"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.FirstSteps.html#x:43"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="additions.FirstSteps.html#n:44"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.FirstSteps.html#f:39"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.FirstSteps.html#A:40"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.FirstSteps.html#mult:41"><span class="id" title="variable">mult</span></a> <a class="idref" href="additions.FirstSteps.html#one:42"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.FirstSteps.html#x:43"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.FirstSteps.html#n:44"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Bad.CounterExample" class="idref" href="#Bad.CounterExample"><span class="id" title="section">CounterExample</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Bad.CounterExample.mul" class="idref" href="#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> (<a id="n:45" class="idref" href="#n:45"><span class="id" title="binder">n</span></a> <a id="p:46" class="idref" href="#p:46"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="additions.FirstSteps.html#n:45"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.FirstSteps.html#p:46"><span class="id" title="variable">p</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Bad.CounterExample.one" class="idref" href="#Bad.CounterExample.one"><span class="id" title="variable">one</span></a> := 0.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+With our fake definition, <span class="inlinecode"><span class="id" title="var">N_bpow</span></span> is not correct! 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Bad.mul_not_associative" class="idref" href="#Bad.mul_not_associative"><span class="id" title="lemma">mul_not_associative</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a>  <a id="n:47" class="idref" href="#n:47"><span class="id" title="binder">n</span></a> <a id="p:48" class="idref" href="#p:48"><span class="id" title="binder">p</span></a> <a id="q:49" class="idref" href="#q:49"><span class="id" title="binder">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.FirstSteps.html#n:47"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.FirstSteps.html#p:48"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.FirstSteps.html#q:49"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> (<a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.FirstSteps.html#n:47"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.FirstSteps.html#p:48"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.FirstSteps.html#q:49"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Bad.one_not_neutral" class="idref" href="#Bad.one_not_neutral"><span class="id" title="lemma">one_not_neutral</span></a>  :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.mul"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.FirstSteps.html#Bad.CounterExample.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.FirstSteps.html#n:50"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.FirstSteps.html#n:50"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Bad.correct_exp_too_strong" class="idref" href="#Bad.correct_exp_too_strong"><span class="id" title="lemma">correct_exp_too_strong</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="additions.FirstSteps.html#Bad.correct_expt_function"><span class="id" title="definition">correct_expt_function</span></a> (@<a class="idref" href="additions.FirstSteps.html#N_bpow"><span class="id" title="definition">N_bpow</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="additions.FirstSteps.html#Bad.CounterExample"><span class="id" title="section">CounterExample</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.FirstSteps.html#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Fibonacci matrices 
+</div>
+<div class="code">
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,292 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Monoid_def</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Monoid_def</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="library">Morphisms</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Strings.String.html#"><span class="id" title="library">String</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab375"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="var">Monoid</span></span> type class (with Operational Type Classes)</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Mult_op" class="idref" href="#Mult_op"><span class="id" title="record">Mult_op</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) := <a id="mult_op" class="idref" href="#mult_op"><span class="id" title="projection">mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_def.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_def.html#A:1"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="additions.Monoid_def.html#mult_op"><span class="id" title="definition">mult_op</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="A:4" class="idref" href="#A:4"><span class="id" title="binder">A</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="op:5" class="idref" href="#op:5"><span class="id" title="binder">op</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">:</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">Mult_op</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">A</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">),</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">@</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">mult_op</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">A</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">op</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">op</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">M</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="e1cb395850f224a43e9550833db388b3" class="idref" href="#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="additions.Monoid_def.html#mult_op"><span class="id" title="definition">mult_op</span></a> : <span class="id" title="var">M_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Demo" class="idref" href="#Demo"><span class="id" title="module">Demo</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="Demo.nat_mult_op" class="idref" href="#Demo.nat_mult_op"><span class="id" title="instance">nat_mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#mul"><span class="id" title="definition">Nat.mul</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Set Printing All</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Check</span> 3 <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> 4.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Unset Printing All</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Monoid_def.html#Demo"><span class="id" title="module">Demo</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="string_op" class="idref" href="#string_op"><span class="id" title="instance">string_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Strings.String.html#append"><span class="id" title="definition">append</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">string_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_string" class="idref" href="#ex_string"><span class="id" title="definition">ex_string</span></a> : "ab" <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> "cde" <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> "abcde".<br/>
+ 
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="bool_and_binop" class="idref" href="#bool_and_binop"><span class="id" title="instance">bool_and_binop</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_bool" class="idref" href="#ex_bool"><span class="id" title="definition">ex_bool</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+within M_scope, a term of the form (x * y) is an abbreviation of
+(mult_op A op x y) where op : Mult_op A and x, y : A.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Monoid" class="idref" href="#Monoid"><span class="id" title="record">Monoid</span></a> {<a id="A:6" class="idref" href="#A:6"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="op:7" class="idref" href="#op:7"><span class="id" title="binder">op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:6"><span class="id" title="variable">A</span></a>)(<a id="one:8" class="idref" href="#one:8"><span class="id" title="binder">one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:6"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="op_assoc" class="idref" href="#op_assoc"><span class="id" title="projection">op_assoc</span></a> : <span class="id" title="keyword">∀</span> <a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a> <a id="y:11" class="idref" href="#y:11"><span class="id" title="binder">y</span></a> <a id="z:12" class="idref" href="#z:12"><span class="id" title="binder">z</span></a>, <a class="idref" href="additions.Monoid_def.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Monoid_def.html#y:11"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#z:12"><span class="id" title="variable">z</span></a><a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_def.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#y:11"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#z:12"><span class="id" title="variable">z</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="one_left" class="idref" href="#one_left"><span class="id" title="projection">one_left</span></a> : <span class="id" title="keyword">∀</span> <a id="x:14" class="idref" href="#x:14"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Monoid_def.html#one:8"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#x:14"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_def.html#x:14"><span class="id" title="variable">x</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="one_right" class="idref" href="#one_right"><span class="id" title="projection">one_right</span></a> : <span class="id" title="keyword">∀</span> <a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Monoid_def.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#one:8"><span class="id" title="variable">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_def.html#x:16"><span class="id" title="variable">x</span></a><br/>
+}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab376"></a><h3 class="section">Exercice</h3>
+
+
+<div class="paragraph"> </div>
+
+Define a class for semi-groups, and re-define monoids as semi-groups with a neutral element
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab377"></a><h3 class="section">Monoids and Equivalence Relations</h3>
+
+
+<div class="paragraph"> </div>
+
+In some situations, the previous definition may be too restrictive.
+For instance, consider the computation of   where
+ and  are positive integers, and .
+
+<div class="paragraph"> </div>
+
+Although it could possible to compute with values of the dependent 
+type  <span class="inlinecode">{<span class="id" title="var">n</span>:<span class="id" title="var">N</span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">m</span>}</span>, it looks simpler to compute with numbers of type
+<span class="inlinecode"><span class="id" title="var">N</span></span>, and consider the multiplication .
+
+<div class="paragraph"> </div>
+
+It is easy to prove that this operation is associative, using library
+ . Unfortunately, it is not possible to prove the 
+following proposition,
+required for building an instance of <span class="inlinecode"><span class="id" title="var">Monoid</span></span>:
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<span class="id" title="var">N</span>, 1 × <span class="id" title="var">x</span>  <span class="id" title="var">mod</span> <span class="id" title="var">m</span> = <span class="id" title="var">x</span>.
+<div class="paragraph"> </div>
+
+</span>
+<div class="paragraph"> </div>
+
+Thus, we define a more general class, parameterized by an equivalence
+relation <span class="inlinecode"><span class="id" title="var">Aeq</span></span> on type <span class="inlinecode"><span class="id" title="var">A</span></span>, compatible with the multiplication . 
+The laws of associativity and neutral element
+are not expressed as Leibniz equalities but as equivalence statements:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Equiv" class="idref" href="#Equiv"><span class="id" title="record">Equiv</span></a> <a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> := <a id="equiv" class="idref" href="#equiv"><span class="id" title="projection">equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="additions.Monoid_def.html#A:18"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="2596316c6f2ff161cb03e25fd8914390" class="idref" href="#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">&quot;</span></a>==" := <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70) : <span class="id" title="var">type_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="EMonoid" class="idref" href="#EMonoid"><span class="id" title="record">EMonoid</span></a> (<a id="A:21" class="idref" href="#A:21"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="E_op:22" class="idref" href="#E_op:22"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:21"><span class="id" title="variable">A</span></a>)(<a id="E_one:23" class="idref" href="#E_one:23"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:21"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="E_eq:24" class="idref" href="#E_eq:24"><span class="id" title="binder">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:21"><span class="id" title="variable">A</span></a>): <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Eq_equiv" class="idref" href="#Eq_equiv"><span class="id" title="projection">Eq_equiv</span></a> :&gt; <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Eop_proper" class="idref" href="#Eop_proper"><span class="id" title="projection">Eop_proper</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Monoid_def.html#E_op:22"><span class="id" title="variable">E_op</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Eop_assoc" class="idref" href="#Eop_assoc"><span class="id" title="projection">Eop_assoc</span></a> : <span class="id" title="keyword">∀</span> <a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a> <a id="y:29" class="idref" href="#y:29"><span class="id" title="binder">y</span></a> <a id="z:30" class="idref" href="#z:30"><span class="id" title="binder">z</span></a>, <a class="idref" href="additions.Monoid_def.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Monoid_def.html#y:29"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#z:30"><span class="id" title="variable">z</span></a><a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Monoid_def.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#y:29"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#z:30"><span class="id" title="variable">z</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Eone_left" class="idref" href="#Eone_left"><span class="id" title="projection">Eone_left</span></a> : <span class="id" title="keyword">∀</span> <a id="x:32" class="idref" href="#x:32"><span class="id" title="binder">x</span></a>,  <a class="idref" href="additions.Monoid_def.html#E_one:23"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#x:32"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Monoid_def.html#x:32"><span class="id" title="variable">x</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Eone_right" class="idref" href="#Eone_right"><span class="id" title="projection">Eone_right</span></a> : <span class="id" title="keyword">∀</span> <a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a>,  <a class="idref" href="additions.Monoid_def.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#E_one:23"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Monoid_def.html#x:34"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Equiv_Equiv" class="idref" href="#Equiv_Equiv"><span class="id" title="instance">Equiv_Equiv</span></a> (<a id="A:36" class="idref" href="#A:36"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="E_op:37" class="idref" href="#E_op:37"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:36"><span class="id" title="variable">A</span></a>)(<a id="E_one:38" class="idref" href="#E_one:38"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:36"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="E_eq:39" class="idref" href="#E_eq:39"><span class="id" title="binder">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:36"><span class="id" title="variable">A</span></a>)(<a id="M:40" class="idref" href="#M:40"><span class="id" title="binder">M</span></a> :<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Monoid_def.html#E_op:37"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Monoid_def.html#E_one:38"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:39"><span class="id" title="variable">E_eq</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:39"><span class="id" title="variable">E_eq</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Equiv_Refl" class="idref" href="#Equiv_Refl"><span class="id" title="instance">Equiv_Refl</span></a> (<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="E_op:42" class="idref" href="#E_op:42"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:41"><span class="id" title="variable">A</span></a>)(<a id="E_one:43" class="idref" href="#E_one:43"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:41"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="E_eq:44" class="idref" href="#E_eq:44"><span class="id" title="binder">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:41"><span class="id" title="variable">A</span></a>)(<a id="M:45" class="idref" href="#M:45"><span class="id" title="binder">M</span></a> :<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Monoid_def.html#E_op:42"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Monoid_def.html#E_one:43"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:44"><span class="id" title="variable">E_eq</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:44"><span class="id" title="variable">E_eq</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Equiv_Sym" class="idref" href="#Equiv_Sym"><span class="id" title="instance">Equiv_Sym</span></a> (<a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="E_op:47" class="idref" href="#E_op:47"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:46"><span class="id" title="variable">A</span></a>)(<a id="E_one:48" class="idref" href="#E_one:48"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:46"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="E_eq:49" class="idref" href="#E_eq:49"><span class="id" title="binder">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:46"><span class="id" title="variable">A</span></a>)(<a id="M:50" class="idref" href="#M:50"><span class="id" title="binder">M</span></a> :<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Monoid_def.html#E_op:47"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Monoid_def.html#E_one:48"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:49"><span class="id" title="variable">E_eq</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Symmetric"><span class="id" title="class">Symmetric</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:49"><span class="id" title="variable">E_eq</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Equiv_Trans" class="idref" href="#Equiv_Trans"><span class="id" title="instance">Equiv_Trans</span></a> (<a id="A:51" class="idref" href="#A:51"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="E_op:52" class="idref" href="#E_op:52"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_def.html#A:51"><span class="id" title="variable">A</span></a>)(<a id="E_one:53" class="idref" href="#E_one:53"><span class="id" title="binder">E_one</span></a> : <a class="idref" href="additions.Monoid_def.html#A:51"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="E_eq:54" class="idref" href="#E_eq:54"><span class="id" title="binder">E_eq</span></a>: <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:51"><span class="id" title="variable">A</span></a>)(<a id="M:55" class="idref" href="#M:55"><span class="id" title="binder">M</span></a> :<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Monoid_def.html#E_op:52"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Monoid_def.html#E_one:53"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:54"><span class="id" title="variable">E_eq</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="additions.Monoid_def.html#E_eq:54"><span class="id" title="variable">E_eq</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab378"></a><h3 class="section">Coercion from Monoid to EMonoid</h3>
+
+
+<div class="paragraph"> </div>
+
+Every instance of class  <span class="inlinecode"><span class="id" title="var">Monoid</span></span> can be transformed into an instance of
+<span class="inlinecode"><span class="id" title="var">EMonoid</span></span>, considering Leibniz' equality <span class="inlinecode"><span class="id" title="var">eq</span></span>.
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="eq_equiv" class="idref" href="#eq_equiv"><span class="id" title="instance">eq_equiv</span></a> {<a id="A:56" class="idref" href="#A:56"><span class="id" title="binder">A</span></a>} : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Monoid_def.html#A:56"><span class="id" title="variable">A</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Monoid_EMonoid" class="idref" href="#Monoid_EMonoid"><span class="id" title="instance">Monoid_EMonoid</span></a> `(<a id="M:60" class="idref" href="#M:60"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a id="A:57" class="idref" href="#A:57"><span class="id" title="binder">A</span></a> <a id="op:58" class="idref" href="#op:58"><span class="id" title="binder">op</span></a> <a id="one:59" class="idref" href="#one:59"><span class="id" title="binder">one</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="additions.Monoid_def.html#op:58"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Monoid_def.html#one:59"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Monoid_def.html#eq_equiv"><span class="id" title="instance">eq_equiv</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We can now register <span class="inlinecode"><span class="id" title="var">Monoid_EMonoid</span></span> as a <i>coercion</i>:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="additions.Monoid_def.html#Monoid_EMonoid"><span class="id" title="instance">Monoid_EMonoid</span></a> <a class="idref" href="additions.Monoid_def.html#Monoid_EMonoid"><span class="id" title="instance">:</span></a> <a class="idref" href="additions.Monoid_def.html#Monoid_EMonoid"><span class="id" title="instance">Monoid</span></a> <a class="idref" href="additions.Monoid_def.html#Monoid_EMonoid"><span class="id" title="instance">&gt;-&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#Monoid_EMonoid"><span class="id" title="instance">EMonoid</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab379"></a><h3 class="section">Commutative Monoids</h3>
+
+
+<div class="paragraph"> </div>
+
+The following type class definitions allow to take advantage of
+  the possible commutativity of the  operation 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Abelian_EMonoid" class="idref" href="#Abelian_EMonoid"><span class="id" title="record">Abelian_EMonoid</span></a> `(<a id="M:65" class="idref" href="#M:65"><span class="id" title="binder">M</span></a>:@<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:61" class="idref" href="#A:61"><span class="id" title="binder">A</span></a> <a id="op:62" class="idref" href="#op:62"><span class="id" title="binder">op</span></a> <a id="one:63" class="idref" href="#one:63"><span class="id" title="binder">one</span></a> <a id="Aeq:64" class="idref" href="#Aeq:64"><span class="id" title="binder">Aeq</span></a> ):= {<br/>
+&nbsp;&nbsp;<a id="Eop_comm" class="idref" href="#Eop_comm"><span class="id" title="projection">Eop_comm</span></a> : <span class="id" title="keyword">∀</span> <a id="x:67" class="idref" href="#x:67"><span class="id" title="binder">x</span></a> <a id="y:68" class="idref" href="#y:68"><span class="id" title="binder">y</span></a>, <a class="idref" href="additions.Monoid_def.html#op:62"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Monoid_def.html#x:67"><span class="id" title="variable">x</span></a>  <a class="idref" href="additions.Monoid_def.html#y:68"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Monoid_def.html#op:62"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Monoid_def.html#y:68"><span class="id" title="variable">y</span></a>  <a class="idref" href="additions.Monoid_def.html#x:67"><span class="id" title="variable">x</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Abelian_Monoid" class="idref" href="#Abelian_Monoid"><span class="id" title="record">Abelian_Monoid</span></a> `(<a id="M:73" class="idref" href="#M:73"><span class="id" title="binder">M</span></a>:<a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> ):= {<br/>
+&nbsp;&nbsp;<a id="op_comm" class="idref" href="#op_comm"><span class="id" title="projection">op_comm</span></a> : <span class="id" title="keyword">∀</span> <a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a> <a id="y:76" class="idref" href="#y:76"><span class="id" title="binder">y</span></a>, <a class="idref" href="additions.Monoid_def.html#op:71"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Monoid_def.html#x:75"><span class="id" title="variable">x</span></a>  <a class="idref" href="additions.Monoid_def.html#y:76"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_def.html#op:71"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.Monoid_def.html#y:76"><span class="id" title="variable">y</span></a>  <a class="idref" href="additions.Monoid_def.html#x:75"><span class="id" title="variable">x</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">add_op_proper</span> <span class="id" title="var">M</span> <span class="id" title="var">H</span> := <br/>
+&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">h</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="var">H</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (@<a class="idref" href="additions.Monoid_def.html#Eop_proper"><span class="id" title="method">Eop_proper</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span> <span class="id" title="var">h</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Monoid_instances</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Monoid_instances</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab380"></a><h2 class="section">Some useful instances of Monoid classes</h2>
+
+
+<div class="paragraph"> </div>
+
+File 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="library">Morphisms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.PArith.html#"><span class="id" title="library">PArith</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Ring63.html#"><span class="id" title="library">Ring63</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab381"></a><h3 class="section">Multiplicative monoid on <span class="inlinecode"><span class="id" title="var">Z</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Z_mult_op" class="idref" href="#Z_mult_op"><span class="id" title="instance">Z_mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.mul"><span class="id" title="definition">Z.mul</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="ZMult" class="idref" href="#ZMult"><span class="id" title="instance">ZMult</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a>  <a class="idref" href="additions.Monoid_instances.html#Z_mult_op"><span class="id" title="instance">Z_mult_op</span></a> 1.  
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="ZMult_Abelian" class="idref" href="#ZMult_Abelian"><span class="id" title="instance">ZMult_Abelian</span></a> : <a class="idref" href="additions.Monoid_def.html#Abelian_Monoid"><span class="id" title="class">Abelian_Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#ZMult"><span class="id" title="instance">ZMult</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab382"></a><h3 class="section">Multiplicative monoid on <span class="inlinecode"><span class="id" title="var">nat</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="nat_mult_op" class="idref" href="#nat_mult_op"><span class="id" title="instance">nat_mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> | 5 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.mul"><span class="id" title="definition">Nat.mul</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="Natmult" class="idref" href="#Natmult"><span class="id" title="instance">Natmult</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#nat_mult_op"><span class="id" title="instance">nat_mult_op</span></a>  1%<span class="id" title="var">nat</span> | 5.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab383"></a><h3 class="section">Additive monoid on <span class="inlinecode"><span class="id" title="var">nat</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+The following monoid is useful for proving the correctness of complex
+exponentiation algorithms. In effect, the -th "power" of  is
+equal to . See Sect.
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="nat_plus_op" class="idref" href="#nat_plus_op"><span class="id" title="instance">nat_plus_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> | 12 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.add"><span class="id" title="definition">Nat.add</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Natplus" class="idref" href="#Natplus"><span class="id" title="instance">Natplus</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#nat_plus_op"><span class="id" title="instance">nat_plus_op</span></a>  0%<span class="id" title="var">nat</span> | 12.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">N_scope</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="N_mult_op" class="idref" href="#N_mult_op"><span class="id" title="instance">N_mult_op</span></a>  : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> | 5 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.mul"><span class="id" title="definition">N.mul</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="NMult" class="idref" href="#NMult"><span class="id" title="instance">NMult</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#N_mult_op"><span class="id" title="instance">N_mult_op</span></a> 1 | 5.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="additions.Monoid_instances.html#NMult"><span class="id" title="instance">NMult</span></a> : <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.mul"><span class="id" title="definition">N.mul</span></a> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="N_plus_op" class="idref" href="#N_plus_op"><span class="id" title="instance">N_plus_op</span></a>  : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> | 12 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.add"><span class="id" title="definition">N.add</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="NPlus" class="idref" href="#NPlus"><span class="id" title="instance">NPlus</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#N_plus_op"><span class="id" title="instance">N_plus_op</span></a> 0 | 12.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Multiplicative Monoid on <span class="inlinecode"><span class="id" title="var">positive</span></span> 
+
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="P_mult_op" class="idref" href="#P_mult_op"><span class="id" title="instance">P_mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> | 5  := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.mul"><span class="id" title="definition">Pos.mul</span></a> .<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="PMult" class="idref" href="#PMult"><span class="id" title="instance">PMult</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#P_mult_op"><span class="id" title="instance">P_mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> | 5.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#"><span class="id" title="module">Uint63</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">int63_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab384"></a><h3 class="section">Multiplicative monoid on 63-bits integers</h3>
+
+
+<div class="paragraph"> </div>
+
+Cyclic numeric types are  good candidates for testing exponentiations
+with big exponents, since the size of data is bounded.
+
+<div class="paragraph"> </div>
+
+The type <span class="inlinecode"><span class="id" title="var">int63</span></span> is defined in Standard Library in Module
+<span class="inlinecode"><span class="id" title="var">Coq.Numbers.Cyclic.Int63.Uint63</span></span>.
+
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="int63_mult_op" class="idref" href="#int63_mult_op"><span class="id" title="instance">int63_mult_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <span class="id" title="var">int</span> := <span class="id" title="var">mul</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="Int63mult" class="idref" href="#Int63mult"><span class="id" title="instance">Int63mult</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a> <a class="idref" href="additions.Monoid_instances.html#int63_mult_op"><span class="id" title="instance">int63_mult_op</span></a>  1.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad" class="idref" href="#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Bad.int63_from_nat" class="idref" href="#Bad.int63_from_nat"><span class="id" title="definition">int63_from_nat</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<span class="id" title="var">int</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Monoid_instances.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#148675259b0fe2bf16829f061c3d9d0a"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Monoid_instances.html#int63_from_nat:2"><span class="id" title="definition">int63_from_nat</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Coercion</span> <a class="idref" href="additions.Monoid_instances.html#Bad.int63_from_nat"><span class="id" title="definition">int63_from_nat</span></a> <a class="idref" href="additions.Monoid_instances.html#Bad.int63_from_nat"><span class="id" title="definition">:</span></a> <a class="idref" href="additions.Monoid_instances.html#Bad.int63_from_nat"><span class="id" title="definition">nat</span></a> <a class="idref" href="additions.Monoid_instances.html#Bad.int63_from_nat"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="additions.Monoid_instances.html#Bad.int63_from_nat"><span class="id" title="definition">int</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Bad.fact" class="idref" href="#Bad.fact"><span class="id" title="definition">fact</span></a> (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="var">int</span> := <span class="id" title="keyword">match</span> <a class="idref" href="additions.Monoid_instances.html#n:4"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Monoid_instances.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.Cyclic.Int63.Uint63.html#720ef075ebbaeae9ce7a13e9baa894ff"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#fact:5"><span class="id" title="definition">fact</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Monoid_instances.html#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">int63_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab385"></a><h3 class="section">Monoid of 2x2 matrices</h3>
+
+
+<div class="paragraph"> </div>
+
+Let  be some type, provided with a ring structure. We define the multiplication 
+of -matrices, the coefficients of which have type .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M2_def" class="idref" href="#M2_def"><span class="id" title="section">M2_def</span></a>.<br/>
+<span class="id" title="keyword">Variables</span> (<a id="M2_def.A" class="idref" href="#M2_def.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2_def.zero" class="idref" href="#M2_def.zero"><span class="id" title="variable">zero</span></a> <a id="M2_def.one" class="idref" href="#M2_def.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.Monoid_instances.html#A:7"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2_def.plus" class="idref" href="#M2_def.plus"><span class="id" title="variable">plus</span></a> <a id="M2_def.mult" class="idref" href="#M2_def.mult"><span class="id" title="variable">mult</span></a>  : <a class="idref" href="additions.Monoid_instances.html#A:7"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_instances.html#A:7"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_instances.html#A:7"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2_def.:::'0'" class="idref" href="#M2_def.:::'0'"><span class="id" title="notation">&quot;</span></a>0" := <a class="idref" href="additions.Monoid_instances.html#M2_def.zero"><span class="id" title="variable">zero</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2_def.:::'1'" class="idref" href="#M2_def.:::'1'"><span class="id" title="notation">&quot;</span></a>1" := <a class="idref" href="additions.Monoid_instances.html#M2_def.one"><span class="id" title="variable">one</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="b0d4ffdfad44d7e0423365f5b6c1abdc" class="idref" href="#b0d4ffdfad44d7e0423365f5b6c1abdc"><span class="id" title="notation">&quot;</span></a>x + y" := (<a class="idref" href="additions.Monoid_instances.html#M2_def.plus"><span class="id" title="variable">plus</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="3d24eb4e8ded7098f168acd26997b293" class="idref" href="#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">&quot;</span></a>x * y " := (<a class="idref" href="additions.Monoid_instances.html#M2_def.mult"><span class="id" title="variable">mult</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="M2_def.rt" class="idref" href="#M2_def.rt"><span class="id" title="variable">rt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.Ring_theory.html#semi_ring_theory"><span class="id" title="record">semi_ring_theory</span></a>  <a class="idref" href="additions.Monoid_instances.html#M2_def.zero"><span class="id" title="variable">zero</span></a> <a class="idref" href="additions.Monoid_instances.html#M2_def.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Monoid_instances.html#M2_def.plus"><span class="id" title="variable">plus</span></a> <a class="idref" href="additions.Monoid_instances.html#M2_def.mult"><span class="id" title="variable">mult</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="additions.Monoid_instances.html#M2_def.A"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Add</span>  <span class="id" title="var">Ring</span> <span class="id" title="var">Aring</span> : <a class="idref" href="additions.Monoid_instances.html#M2_def.rt"><span class="id" title="variable">rt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Structure</span> <a id="M2" class="idref" href="#M2"><span class="id" title="record">M2</span></a> : <span class="id" title="keyword">Type</span> := {<a id="c00" class="idref" href="#c00"><span class="id" title="projection">c00</span></a> : <a class="idref" href="additions.Monoid_instances.html#M2_def.A"><span class="id" title="variable">A</span></a>;  <a id="c01" class="idref" href="#c01"><span class="id" title="projection">c01</span></a> : <a class="idref" href="additions.Monoid_instances.html#M2_def.A"><span class="id" title="variable">A</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="c10" class="idref" href="#c10"><span class="id" title="projection">c10</span></a> : <a class="idref" href="additions.Monoid_instances.html#M2_def.A"><span class="id" title="variable">A</span></a>;  <a id="c11" class="idref" href="#c11"><span class="id" title="projection">c11</span></a> : <a class="idref" href="additions.Monoid_instances.html#M2_def.A"><span class="id" title="variable">A</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Id2" class="idref" href="#Id2"><span class="id" title="definition">Id2</span></a> : <a class="idref" href="additions.Monoid_instances.html#M2"><span class="id" title="record">M2</span></a> := <a class="idref" href="additions.Monoid_instances.html#Build_M2"><span class="id" title="constructor">Build_M2</span></a> 1 0 0 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="M2_mult" class="idref" href="#M2_mult"><span class="id" title="definition">M2_mult</span></a> (<a id="m:18" class="idref" href="#m:18"><span class="id" title="binder">m</span></a> <a id="m':19" class="idref" href="#m':19"><span class="id" title="binder">m'</span></a>:<a class="idref" href="additions.Monoid_instances.html#M2"><span class="id" title="record">M2</span></a>) : <a class="idref" href="additions.Monoid_instances.html#M2"><span class="id" title="record">M2</span></a> :=<br/>
+&nbsp;<a class="idref" href="additions.Monoid_instances.html#Build_M2"><span class="id" title="constructor">Build_M2</span></a> (<a class="idref" href="additions.Monoid_instances.html#c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a> <a class="idref" href="additions.Monoid_instances.html#b0d4ffdfad44d7e0423365f5b6c1abdc"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Monoid_instances.html#c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Monoid_instances.html#c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a> <a class="idref" href="additions.Monoid_instances.html#b0d4ffdfad44d7e0423365f5b6c1abdc"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Monoid_instances.html#c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Monoid_instances.html#c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a> <a class="idref" href="additions.Monoid_instances.html#b0d4ffdfad44d7e0423365f5b6c1abdc"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Monoid_instances.html#c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Monoid_instances.html#c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a> <a class="idref" href="additions.Monoid_instances.html#b0d4ffdfad44d7e0423365f5b6c1abdc"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Monoid_instances.html#c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Monoid_instances.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Monoid_instances.html#m':19"><span class="id" title="variable">m'</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="M2_eq_intros" class="idref" href="#M2_eq_intros"><span class="id" title="lemma">M2_eq_intros</span></a> : <span class="id" title="keyword">∀</span> <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> <a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a> <a id="c:22" class="idref" href="#c:22"><span class="id" title="binder">c</span></a> <a id="d:23" class="idref" href="#d:23"><span class="id" title="binder">d</span></a> <a id="a':24" class="idref" href="#a':24"><span class="id" title="binder">a'</span></a> <a id="b':25" class="idref" href="#b':25"><span class="id" title="binder">b'</span></a> <a id="c':26" class="idref" href="#c':26"><span class="id" title="binder">c'</span></a> <a id="d':27" class="idref" href="#d':27"><span class="id" title="binder">d'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Monoid_instances.html#a:20"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="additions.Monoid_instances.html#a':24"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_instances.html#b:21"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="additions.Monoid_instances.html#b':25"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_instances.html#c:22"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="additions.Monoid_instances.html#c':26"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Monoid_instances.html#d:23"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="additions.Monoid_instances.html#d':27"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Monoid_instances.html#Build_M2"><span class="id" title="constructor">Build_M2</span></a> <a class="idref" href="additions.Monoid_instances.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_instances.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="additions.Monoid_instances.html#c:22"><span class="id" title="variable">c</span></a> <a class="idref" href="additions.Monoid_instances.html#d:23"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_instances.html#Build_M2"><span class="id" title="constructor">Build_M2</span></a> <a class="idref" href="additions.Monoid_instances.html#a':24"><span class="id" title="variable">a'</span></a> <a class="idref" href="additions.Monoid_instances.html#b':25"><span class="id" title="variable">b'</span></a> <a class="idref" href="additions.Monoid_instances.html#c':26"><span class="id" title="variable">c'</span></a> <a class="idref" href="additions.Monoid_instances.html#d':27"><span class="id" title="variable">d'</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="M2_op" class="idref" href="#M2_op"><span class="id" title="instance">M2_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Monoid_instances.html#M2"><span class="id" title="record">M2</span></a> := <a class="idref" href="additions.Monoid_instances.html#M2_mult"><span class="id" title="definition">M2_mult</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="M2_Monoid" class="idref" href="#M2_Monoid"><span class="id" title="instance">M2_Monoid</span></a> : <a class="idref" href="additions.Monoid_def.html#Monoid"><span class="id" title="class">Monoid</span></a>   <a class="idref" href="additions.Monoid_instances.html#M2_op"><span class="id" title="instance">M2_op</span></a> <a class="idref" href="additions.Monoid_instances.html#Id2"><span class="id" title="definition">Id2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Monoid_instances.html#M2_def"><span class="id" title="section">M2_def</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Monoid_instances.html#M2_Monoid"><span class="id" title="instance">M2_Monoid</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">zero</span> <span class="id" title="var">one</span> <span class="id" title="var">plus</span> <span class="id" title="var">mult</span>} <span class="id" title="var">rt</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="additions.Monoid_instances.html#Build_M2"><span class="id" title="constructor">Build_M2</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Matrices over N 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="M2N" class="idref" href="#M2N"><span class="id" title="definition">M2N</span></a> := <a class="idref" href="additions.Monoid_instances.html#M2_Monoid"><span class="id" title="instance">M2_Monoid</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.InitialRing.html#Nth"><span class="id" title="lemma">Nth</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab386"></a><h3 class="section">Integers modulo m</h3>
+
+
+<div class="paragraph"> </div>
+
+The following instance of <span class="inlinecode"><span class="id" title="var">EMonoid</span></span> describes the set of integers modulo
+, where  is some integer greater or equal than .
+For simplicity's sake, we represent such values using the type <span class="inlinecode"><span class="id" title="var">N</span></span>,
+and consider "equivalence modulo " instead of equality.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Nmodulo" class="idref" href="#Nmodulo"><span class="id" title="section">Nmodulo</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Nmodulo.m" class="idref" href="#Nmodulo.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Nmodulo.m_gt_1" class="idref" href="#Nmodulo.m_gt_1"><span class="id" title="variable">m_gt_1</span></a> : 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Monoid_instances.html#Nmodulo.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_neq_0" class="idref" href="#m_neq_0"><span class="id" title="lemma">m_neq_0</span></a> : <a class="idref" href="additions.Monoid_instances.html#Nmodulo.m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">m_neq_0</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="mult_mod" class="idref" href="#mult_mod"><span class="id" title="definition">mult_mod</span></a> (<a id="x:30" class="idref" href="#x:30"><span class="id" title="binder">x</span></a> <a id="y:31" class="idref" href="#y:31"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Monoid_instances.html#x:30"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#y:31"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">mod</span></a> <a class="idref" href="additions.Monoid_instances.html#Nmodulo.m"><span class="id" title="variable">m</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="mod_eq" class="idref" href="#mod_eq"><span class="id" title="definition">mod_eq</span></a> (<a id="x:32" class="idref" href="#x:32"><span class="id" title="binder">x</span></a> <a id="y:33" class="idref" href="#y:33"><span class="id" title="binder">y</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <a class="idref" href="additions.Monoid_instances.html#x:32"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">mod</span></a> <a class="idref" href="additions.Monoid_instances.html#Nmodulo.m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_instances.html#y:33"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">mod</span></a> <a class="idref" href="additions.Monoid_instances.html#Nmodulo.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="mod_equiv" class="idref" href="#mod_equiv"><span class="id" title="instance">mod_equiv</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> := <a class="idref" href="additions.Monoid_instances.html#mod_eq"><span class="id" title="definition">mod_eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="mod_op" class="idref" href="#mod_op"><span class="id" title="instance">mod_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> := <a class="idref" href="additions.Monoid_instances.html#mult_mod"><span class="id" title="definition">mult_mod</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="mod_Equiv" class="idref" href="#mod_Equiv"><span class="id" title="instance">mod_Equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="additions.Monoid_instances.html#mod_equiv"><span class="id" title="instance">mod_equiv</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="mult_mod_proper" class="idref" href="#mult_mod_proper"><span class="id" title="instance">mult_mod_proper</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_instances.html#mod_equiv"><span class="id" title="instance">mod_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_instances.html#mod_equiv"><span class="id" title="instance">mod_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_instances.html#mod_equiv"><span class="id" title="instance">mod_equiv</span></a>) <a class="idref" href="additions.Monoid_instances.html#mod_op"><span class="id" title="instance">mod_op</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>]  <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="mult_mod_associative" class="idref" href="#mult_mod_associative"><span class="id" title="lemma">mult_mod_associative</span></a> :  <span class="id" title="keyword">∀</span> <a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a> <a id="y:35" class="idref" href="#y:35"><span class="id" title="binder">y</span></a> <a id="z:36" class="idref" href="#z:36"><span class="id" title="binder">z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Monoid_instances.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Monoid_instances.html#y:35"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#z:36"><span class="id" title="variable">z</span></a><a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Monoid_instances.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#y:35"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#z:36"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="one_mod_neutral_l" class="idref" href="#one_mod_neutral_l"><span class="id" title="lemma">one_mod_neutral_l</span></a>  : <span class="id" title="keyword">∀</span> <a id="x:37" class="idref" href="#x:37"><span class="id" title="binder">x</span></a>, 1 <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Monoid_instances.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Monoid_instances.html#x:37"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="one_mod_neutral_r" class="idref" href="#one_mod_neutral_r"><span class="id" title="lemma">one_mod_neutral_r</span></a>  : <span class="id" title="keyword">∀</span> <a id="x:38" class="idref" href="#x:38"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Monoid_instances.html#x:38"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> 1 <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Monoid_instances.html#x:38"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Nmod_Monoid" class="idref" href="#Nmod_Monoid"><span class="id" title="instance">Nmod_Monoid</span></a> : <a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="additions.Monoid_instances.html#mod_op"><span class="id" title="instance">mod_op</span></a> 1 <a class="idref" href="additions.Monoid_instances.html#mod_equiv"><span class="id" title="instance">mod_equiv</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Monoid_instances.html#Nmodulo"><span class="id" title="section">Nmodulo</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="S256" class="idref" href="#S256"><span class="id" title="section">S256</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="S256.mod256" class="idref" href="#S256.mod256"><span class="id" title="variable">mod256</span></a> :=  <a class="idref" href="additions.Monoid_instances.html#mod_op"><span class="id" title="instance">mod_op</span></a> 256.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">mod256</span> | 1.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Monoid_instances.html#S256"><span class="id" title="section">S256</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">N_scope</span>.<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.More_on_positive</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.More_on_positive</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">additions</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a> <a class="idref" href="additions.Compatibility.html#"><span class="id" title="library">Compatibility</span></a> <a class="idref" href="additions.Wf_transparent.html#"><span class="id" title="library">Wf_transparent</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="module">Monoid_def</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab387"></a><h1 class="section">Basic lemmas about positive and N</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pos_eq_dec" class="idref" href="#pos_eq_dec"><span class="id" title="definition">pos_eq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="p:1" class="idref" href="#p:1"><span class="id" title="binder">p</span></a> <a id="p':2" class="idref" href="#p':2"><span class="id" title="binder">p'</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="additions.More_on_positive.html#p:1"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.More_on_positive.html#p':2"><span class="id" title="variable">p'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="additions.More_on_positive.html#p:1"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.More_on_positive.html#p':2"><span class="id" title="variable">p'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_0_le_n" class="idref" href="#N_0_le_n"><span class="id" title="lemma">N_0_le_n</span></a>: <span class="id" title="keyword">∀</span> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>, (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#2ccca3280733654024e33c2005ae2530"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.More_on_positive.html#n:3"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_to_nat_neq_0" class="idref" href="#Pos_to_nat_neq_0"><span class="id" title="lemma">Pos_to_nat_neq_0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:4" class="idref" href="#p:4"><span class="id" title="binder">p</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.More_on_positive.html#p:4"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Pos_to_nat_neq_0</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab388"></a><h2 class="section">Relationship with <span class="inlinecode"><span class="id" title="var">nat</span></span> and <span class="inlinecode"><span class="id" title="var">N</span></span></h2>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Npos_diff_zero" class="idref" href="#Npos_diff_zero"><span class="id" title="lemma">Npos_diff_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Npos_gt_0" class="idref" href="#Npos_gt_0"><span class="id" title="lemma">Npos_gt_0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:6" class="idref" href="#p:6"><span class="id" title="binder">p</span></a>, (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:6"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Npos_diff_zero</span>  <span class="id" title="var">Npos_gt_0</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos2N_inj_lt" class="idref" href="#pos2N_inj_lt"><span class="id" title="lemma">pos2N_inj_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> <a id="p:8" class="idref" href="#p:8"><span class="id" title="binder">p</span></a>, (<a class="idref" href="additions.More_on_positive.html#n:7"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#p:8"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#n:7"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:8"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos2N_inj_add" class="idref" href="#pos2N_inj_add"><span class="id" title="lemma">pos2N_inj_add</span></a> : <span class="id" title="keyword">∀</span> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="p:10" class="idref" href="#p:10"><span class="id" title="binder">p</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.More_on_positive.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.More_on_positive.html#p:10"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:10"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">pos2nat_inj_tac</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.Pnat.html#Pos2Nat.inj_add"><span class="id" title="lemma">Pos2Nat.inj_add</span></a> || <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.Pnat.html#Pos2Nat.inj_mul"><span class="id" title="lemma">Pos2Nat.inj_mul</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.Pnat.html#Pos2Nat.inj_lt"><span class="id" title="lemma">Pos2Nat.inj_lt</span></a>  || <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.Pnat.html#Pos2Nat.inj_le"><span class="id" title="lemma">Pos2Nat.inj_le</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos2Nat_le_1_p" class="idref" href="#Pos2Nat_le_1_p"><span class="id" title="lemma">Pos2Nat_le_1_p</span></a> : <span class="id" title="keyword">∀</span> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a>, (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.More_on_positive.html#p:11"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_le_1_pos" class="idref" href="#N_le_1_pos"><span class="id" title="lemma">N_le_1_pos</span></a> : <span class="id" title="keyword">∀</span> <a id="p:12" class="idref" href="#p:12"><span class="id" title="binder">p</span></a>, (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#2ccca3280733654024e33c2005ae2530"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:12"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_le_mul" class="idref" href="#pos_le_mul"><span class="id" title="lemma">pos_le_mul</span></a> : <span class="id" title="keyword">∀</span> <a id="p:13" class="idref" href="#p:13"><span class="id" title="binder">p</span></a> <a id="q:14" class="idref" href="#q:14"><span class="id" title="binder">q</span></a> , (<a class="idref" href="additions.More_on_positive.html#p:13"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#c9d5bad8b60a20d83e777300ee498110"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.More_on_positive.html#p:13"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#q:14"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_lt_mul" class="idref" href="#pos_lt_mul"><span class="id" title="lemma">pos_lt_mul</span></a> : <span class="id" title="keyword">∀</span> <a id="p:15" class="idref" href="#p:15"><span class="id" title="binder">p</span></a> <a id="q:16" class="idref" href="#q:16"><span class="id" title="binder">q</span></a> , (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#q:16"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#p:15"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#p:15"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#q:16"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos2Nat_le_n_pn" class="idref" href="#Pos2Nat_le_n_pn"><span class="id" title="lemma">Pos2Nat_le_n_pn</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:17" class="idref" href="#p:17"><span class="id" title="binder">p</span></a> <a id="q:18" class="idref" href="#q:18"><span class="id" title="binder">q</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.More_on_positive.html#p:17"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.More_on_positive.html#p:17"><span class="id" title="variable">p</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.More_on_positive.html#q:18"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Pos2Nat_le_1_p</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab389"></a><h2 class="section">Surjection from <span class="inlinecode"><span class="id" title="var">N</span></span> into <span class="inlinecode"><span class="id" title="var">positive</span></span></h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N2pos" class="idref" href="#N2pos"><span class="id" title="definition">N2pos</span></a> (<a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.More_on_positive.html#n:19"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0%<span class="id" title="var">N</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span> ⇒ <span class="id" title="var">p</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N2pos_pos" class="idref" href="#N2pos_pos"><span class="id" title="lemma">N2pos_pos</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:21" class="idref" href="#i:21"><span class="id" title="binder">i</span></a>,  <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="additions.More_on_positive.html#i:21"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.More_on_positive.html#i:21"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_pos_N2pos" class="idref" href="#N_pos_N2pos"><span class="id" title="lemma">N_pos_N2pos</span></a> : <span class="id" title="keyword">∀</span> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>, 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="additions.More_on_positive.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="additions.More_on_positive.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#n:22"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N2pos_lt_switch" class="idref" href="#N2pos_lt_switch"><span class="id" title="lemma">N2pos_lt_switch</span></a> : <span class="id" title="keyword">∀</span>  <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> <a id="p:24" class="idref" href="#p:24"><span class="id" title="binder">p</span></a>, (0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#n:23"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:24"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#n:23"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="additions.More_on_positive.html#p:24"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#n:23"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">positive</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">N2pos_simpl</span> <span class="id" title="var">x</span> := <span class="id" title="tactic">simpl</span> (<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <span class="id" title="var">x</span>)) <span class="id" title="tactic">in</span> ×.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">N2pos_destruct</span> <span class="id" title="var">t</span> <span class="id" title="var">y</span> :=<br/>
+&nbsp;<span class="id" title="tactic">destruct</span> <span class="id" title="var">t</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span>] ; [<span class="id" title="tactic">try</span> (<span class="id" title="tactic">discriminate</span> || <span class="id" title="var">contradiction</span>) | <span class="id" title="var">N2pos_simpl</span> <span class="id" title="var">y</span>].<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N2pos_lt_switch2" class="idref" href="#N2pos_lt_switch2"><span class="id" title="lemma">N2pos_lt_switch2</span></a> : <span class="id" title="keyword">∀</span>  <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> <a id="p:26" class="idref" href="#p:26"><span class="id" title="binder">p</span></a>, (0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#n:25"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a>(<a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#n:25"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#p:26"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">positive</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="additions.More_on_positive.html#n:25"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:26"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_lt_wf" class="idref" href="#pos_lt_wf"><span class="id" title="lemma">pos_lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.lt"><span class="id" title="definition">Pos.lt</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Partial exact log2 function  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span>  <a id="exact_log2" class="idref" href="#exact_log2"><span class="id" title="definition">exact_log2</span></a>(<a id="p:31" class="idref" href="#p:31"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.More_on_positive.html#p:31"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 1%<span class="id" title="var">positive</span> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;| 2%<span class="id" title="var">positive</span> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="additions.More_on_positive.html#exact_log2:32"><span class="id" title="definition">exact_log2</span></a> <span class="id" title="var">q</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<span class="id" title="var">l</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">positive</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Compute exact_log2 16.
+ = Some 4
+     : option positive
+
+<div class="paragraph"> </div>
+
+Compute exact_log2 10.
+= None
+    : option positive
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exact_log2xOx0" class="idref" href="#exact_log2xOx0"><span class="id" title="lemma">exact_log2xOx0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:34" class="idref" href="#p:34"><span class="id" title="binder">p</span></a> <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a>, <a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.More_on_positive.html#p:34"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="additions.More_on_positive.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.More_on_positive.html#p:34"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="additions.More_on_positive.html#i:35"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exact_log2_spec" class="idref" href="#exact_log2_spec"><span class="id" title="lemma">exact_log2_spec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, <a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.More_on_positive.html#p:36"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="additions.More_on_positive.html#i:37"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#p:36"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#72c1e614b36b07dc51a712bc261ee813"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.More_on_positive.html#i:37"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Another induction principle for positive 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="positive_4step_ind" class="idref" href="#positive_4step_ind"><span class="id" title="lemma">positive_4step_ind</span></a> : <span class="id" title="keyword">∀</span> <a id="P:38" class="idref" href="#P:38"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> 1%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> 2%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> 3%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="p:39" class="idref" href="#p:39"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="additions.More_on_positive.html#p:39"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.More_on_positive.html#p:39"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="additions.More_on_positive.html#p:39"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="additions.More_on_positive.html#p:39"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="additions.More_on_positive.html#p:39"><span class="id" title="variable">p</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:40" class="idref" href="#p:40"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.More_on_positive.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="additions.More_on_positive.html#p:40"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_gt_3" class="idref" href="#pos_gt_3"><span class="id" title="lemma">pos_gt_3</span></a> : <span class="id" title="keyword">∀</span> <a id="p:41" class="idref" href="#p:41"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#p:41"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 1  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.More_on_positive.html#p:41"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="additions.More_on_positive.html#exact_log2"><span class="id" title="definition">exact_log2</span></a> <a class="idref" href="additions.More_on_positive.html#p:41"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#p:41"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">pos_gt_3</span> : <span class="id" title="var">chains</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab390"></a><h2 class="section">Lemmas on Euclidean division</h2>
+
+    N.pos_div_eucl (a:positive) (b:N) : N * N 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_div_eucl_quotient_pos" class="idref" href="#pos_div_eucl_quotient_pos"><span class="id" title="lemma">pos_div_eucl_quotient_pos</span></a> : <span class="id" title="keyword">∀</span> <a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a> <a id="b:43" class="idref" href="#b:43"><span class="id" title="binder">b</span></a> <a id="q:44" class="idref" href="#q:44"><span class="id" title="binder">q</span></a> <a id="r:45" class="idref" href="#r:45"><span class="id" title="binder">r</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.More_on_positive.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.More_on_positive.html#b:43"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.More_on_positive.html#q:44"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.More_on_positive.html#r:45"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.More_on_positive.html#b:43"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#2ccca3280733654024e33c2005ae2530"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#a:42"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#b:43"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.More_on_positive.html#q:44"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pos_div_eucl_quotient_lt" class="idref" href="#pos_div_eucl_quotient_lt"><span class="id" title="lemma">pos_div_eucl_quotient_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="a:46" class="idref" href="#a:46"><span class="id" title="binder">a</span></a> <a id="b:47" class="idref" href="#b:47"><span class="id" title="binder">b</span></a> <a id="q:48" class="idref" href="#q:48"><span class="id" title="binder">q</span></a> <a id="r:49" class="idref" href="#r:49"><span class="id" title="binder">r</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.More_on_positive.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.More_on_positive.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.More_on_positive.html#q:48"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.More_on_positive.html#r:49"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#b:47"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.More_on_positive.html#q:48"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#a:46"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_pos_div_eucl_divides" class="idref" href="#N_pos_div_eucl_divides"><span class="id" title="lemma">N_pos_div_eucl_divides</span></a> : <span class="id" title="keyword">∀</span> <a id="i:50" class="idref" href="#i:50"><span class="id" title="binder">i</span></a> <a id="b:51" class="idref" href="#b:51"><span class="id" title="binder">b</span></a> <a id="q:52" class="idref" href="#q:52"><span class="id" title="binder">q</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.More_on_positive.html#i:50"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#b:51"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.More_on_positive.html#q:52"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.More_on_positive.html#b:51"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#q:52"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.More_on_positive.html#i:50"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="N_pos_div_eucl_rest" class="idref" href="#N_pos_div_eucl_rest"><span class="id" title="lemma">N_pos_div_eucl_rest</span></a> : <span class="id" title="keyword">∀</span> <a id="i:53" class="idref" href="#i:53"><span class="id" title="binder">i</span></a> <a id="b:54" class="idref" href="#b:54"><span class="id" title="binder">b</span></a> <a id="q:55" class="idref" href="#q:55"><span class="id" title="binder">q</span></a> <a id="r:56" class="idref" href="#r:56"><span class="id" title="binder">r</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.More_on_positive.html#i:53"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#b:54"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.More_on_positive.html#q:55"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>  <a class="idref" href="additions.More_on_positive.html#r:56"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#r:56"><span class="id" title="variable">r</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#q:55"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.More_on_positive.html#b:54"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#q:55"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#r:56"><span class="id" title="variable">r</span></a>)%<span class="id" title="var">positive</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.More_on_positive.html#i:53"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="N_pos_div_eucl_q0" class="idref" href="#N_pos_div_eucl_q0"><span class="id" title="lemma">N_pos_div_eucl_q0</span></a> : <span class="id" title="keyword">∀</span> <a id="i:57" class="idref" href="#i:57"><span class="id" title="binder">i</span></a> <a id="b:58" class="idref" href="#b:58"><span class="id" title="binder">b</span></a>  <a id="r:59" class="idref" href="#r:59"><span class="id" title="binder">r</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.More_on_positive.html#i:57"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#b:58"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>   <a class="idref" href="additions.More_on_positive.html#r:59"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.More_on_positive.html#i:57"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.More_on_positive.html#N2pos"><span class="id" title="definition">N2pos</span></a> <a class="idref" href="additions.More_on_positive.html#r:59"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+An auxiliary lemma 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="lt_S_2i" class="idref" href="#lt_S_2i"><span class="id" title="lemma">lt_S_2i</span></a> : <br/>
+<span class="id" title="keyword">∀</span> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a> <a id="j:61" class="idref" href="#j:61"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="additions.More_on_positive.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.More_on_positive.html#j:61"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.More_on_positive.html#j:61"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_le_mul_pos" class="idref" href="#N_le_mul_pos"><span class="id" title="lemma">N_le_mul_pos</span></a>  : <span class="id" title="keyword">∀</span> <a id="q:62" class="idref" href="#q:62"><span class="id" title="binder">q</span></a> <a id="p:63" class="idref" href="#p:63"><span class="id" title="binder">p</span></a>, (<a class="idref" href="additions.More_on_positive.html#q:62"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#2ccca3280733654024e33c2005ae2530"><span class="id" title="notation">≤</span></a> <a class="idref" href="additions.More_on_positive.html#q:62"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="additions.More_on_positive.html#p:63"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">quotient_small</span> <span class="id" title="var">div_equation</span> <span class="id" title="var">H</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">div_equation</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> ?<span class="id" title="var">a</span> ?<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>?<span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>?<span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span>  (<span class="id" title="var">H</span> : (<span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <span class="id" title="var">a</span>)%<span class="id" title="var">N</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="tactic">apply</span> (<a class="idref" href="additions.More_on_positive.html#pos_div_eucl_quotient_lt"><span class="id" title="lemma">pos_div_eucl_quotient_lt</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">div_equation</span>); <span class="id" title="tactic">auto</span>|]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">rest_small</span> <span class="id" title="var">div_equation</span> <span class="id" title="var">H</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">div_equation</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> ?<span class="id" title="var">a</span> ?<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>?<span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>?<span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">H0</span> := <span class="id" title="tactic">fresh</span> "H" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span>  (<span class="id" title="var">H</span> : (<span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">b</span>)%<span class="id" title="var">N</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl_remainder"><span class="id" title="lemma">N.pos_div_eucl_remainder</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">intro</span>  <span class="id" title="var">H0</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">div_equation</span> <span class="id" title="tactic">in</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H0</span> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">discriminate</span>| ]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+</div>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Naive</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Naive</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab391"></a><h2 class="section">Naïve  and Monomorphic  Functions for Computing Powers.</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab392"></a><h3 class="section">Exponentiation of integers</h3>
+
+
+<div class="paragraph"> </div>
+
+Let us define a function <span class="inlinecode"><span class="id" title="var">Z.power</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">Z</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">Z</span></span> by structural recursion
+on the exponent <span class="inlinecode"><span class="id" title="var">n</span></span>:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Z" class="idref" href="#Z"><span class="id" title="module">Z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Z.power" class="idref" href="#Z.power"><span class="id" title="definition">power</span></a> (<a id="x:1" class="idref" href="#x:1"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)(<a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="additions.Naive.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+| 0%<span class="id" title="var">nat</span> ⇒ 1<br/>
+| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Naive.html#x:1"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Naive.html#power:3"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:1"><span class="id" title="variable">x</span></a> <span class="id" title="var">p</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Compute</span> <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> 2 10.<br/>
+</div>
+
+<div class="doc">
+
+
+<div class="paragraph"> </div>
+
+<a id="lab393"></a><h3 class="section">Some basic properties of <span class="inlinecode"><span class="id" title="var">Z.power</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z.power_S" class="idref" href="#Z.power_S"><span class="id" title="lemma">power_S</span></a>  : <span class="id" title="keyword">∀</span> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)(<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Naive.html#n:6"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Naive.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Naive.html#n:6"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z.power_of_plus" class="idref" href="#Z.power_of_plus"><span class="id" title="lemma">power_of_plus</span></a> (<a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:8" class="idref" href="#n:8"><span class="id" title="binder">n</span></a> <a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:7"><span class="id" title="variable">x</span></a> (<a class="idref" href="additions.Naive.html#n:8"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#p:9"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Naive.html#n:8"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#Z.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Naive.html#p:9"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Naive.html#Z"><span class="id" title="module">Z</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab394"></a><h3 class="section">Exponentiation modulo <span class="inlinecode"><span class="id" title="var">m</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+ Let <span class="inlinecode"><span class="id" title="var">m</span></span> be some natural number. We can compute for any number <span class="inlinecode"><span class="id" title="var">x</span></span>
+and exponent <span class="inlinecode"><span class="id" title="var">n</span></span> the number .
+
+<div class="paragraph"> </div>
+
+ Since <span class="inlinecode"><span class="id" title="var">m</span></span> and <span class="inlinecode"><span class="id" title="var">x</span></span> can be arbitrary large numbers, we give them the type 
+<span class="inlinecode"><span class="id" title="var">N</span></span> of binary natural number. Following our naïve approach, the exponent <span class="inlinecode"><span class="id" title="var">n</span></span>
+is still of type <span class="inlinecode"><span class="id" title="var">nat</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="N_mod" class="idref" href="#N_mod"><span class="id" title="module">N_mod</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">N_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="N_mod.m_fixed" class="idref" href="#N_mod.m_fixed"><span class="id" title="section">m_fixed</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="N_mod.m_fixed.m" class="idref" href="#N_mod.m_fixed.m"><span class="id" title="variable">m</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="N_mod.mult_mod" class="idref" href="#N_mod.mult_mod"><span class="id" title="definition">mult_mod</span></a> (<a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Naive.html#x:11"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#fe1417cc32a1703619b26940ce957c6a"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#y:12"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">mod</span></a> <a class="idref" href="additions.Naive.html#N_mod.m_fixed.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="N_mod.power" class="idref" href="#N_mod.power"><span class="id" title="definition">power</span></a> (<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Naive.html#n:14"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0%<span class="id" title="var">nat</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'mod'_x"><span class="id" title="notation">mod</span></a> <a class="idref" href="additions.Naive.html#N_mod.m_fixed.m"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Naive.html#N_mod.mult_mod"><span class="id" title="definition">mult_mod</span></a> <a class="idref" href="additions.Naive.html#x:13"><span class="id" title="variable">x</span></a> (<a class="idref" href="additions.Naive.html#power:15"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#x:13"><span class="id" title="variable">x</span></a> <span class="id" title="var">p</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Naive.html#N_mod.m_fixed"><span class="id" title="section">m_fixed</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Naive.html#N_mod"><span class="id" title="module">N_mod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Compute</span> <a class="idref" href="additions.Naive.html#power"><span class="id" title="definition">N_mod.power</span></a> 14555553%<span class="id" title="var">N</span> 5689%<span class="id" title="var">N</span> 27.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<br/>
+<span class="inlinecode">&nbsp;&nbsp;&nbsp;= 9086410<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;: <span class="id" title="var">N</span>
+<div class="paragraph"> </div>
+
+</span>
+
+<div class="paragraph"> </div>
+
+<a id="lab395"></a><h3 class="section">Exponentiation of  matrices</h3>
+
+
+<div class="paragraph"> </div>
+
+Our last example is a definition of  where  is a  matrix
+over any scalar type , assuming one can provide  with a semi-ring structure.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="M2" class="idref" href="#M2"><span class="id" title="module">M2</span></a>.<br/>
+<span class="id" title="keyword">Section</span> <a id="M2.Definitions" class="idref" href="#M2.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="M2.Definitions.A" class="idref" href="#M2.Definitions.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2.Definitions.zero" class="idref" href="#M2.Definitions.zero"><span class="id" title="variable">zero</span></a> <a id="M2.Definitions.one" class="idref" href="#M2.Definitions.one"><span class="id" title="variable">one</span></a> : <a class="idref" href="additions.Naive.html#A:17"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M2.Definitions.plus" class="idref" href="#M2.Definitions.plus"><span class="id" title="variable">plus</span></a> <a id="M2.Definitions.mult" class="idref" href="#M2.Definitions.mult"><span class="id" title="variable">mult</span></a>  : <a class="idref" href="additions.Naive.html#A:17"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Naive.html#A:17"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Naive.html#A:17"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="M2.Definitions.rt" class="idref" href="#M2.Definitions.rt"><span class="id" title="variable">rt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.Ring_theory.html#semi_ring_theory"><span class="id" title="record">semi_ring_theory</span></a>  <a class="idref" href="additions.Naive.html#M2.Definitions.zero"><span class="id" title="variable">zero</span></a> <a class="idref" href="additions.Naive.html#M2.Definitions.one"><span class="id" title="variable">one</span></a> <a class="idref" href="additions.Naive.html#M2.Definitions.plus"><span class="id" title="variable">plus</span></a> <a class="idref" href="additions.Naive.html#M2.Definitions.mult"><span class="id" title="variable">mult</span></a>   (@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="additions.Naive.html#M2.Definitions.A"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Add</span> <span class="id" title="var">Ring</span> <span class="id" title="var">Aring</span> : <a class="idref" href="additions.Naive.html#M2.Definitions.rt"><span class="id" title="variable">rt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2.Definitions.:::'0'" class="idref" href="#M2.Definitions.:::'0'"><span class="id" title="notation">&quot;</span></a>0" := <a class="idref" href="additions.Naive.html#M2.Definitions.zero"><span class="id" title="variable">zero</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="M2.Definitions.:::'1'" class="idref" href="#M2.Definitions.:::'1'"><span class="id" title="notation">&quot;</span></a>1" := <a class="idref" href="additions.Naive.html#M2.Definitions.one"><span class="id" title="variable">one</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="617a2cc97dac8fe5e951278b3bb11609" class="idref" href="#617a2cc97dac8fe5e951278b3bb11609"><span class="id" title="notation">&quot;</span></a>x + y" := (<a class="idref" href="additions.Naive.html#M2.Definitions.plus"><span class="id" title="variable">plus</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="4d985c683f0b20e66a34edc4883055ad" class="idref" href="#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">&quot;</span></a>x * y " := (<a class="idref" href="additions.Naive.html#M2.Definitions.mult"><span class="id" title="variable">mult</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Structure</span> <a id="M2.t" class="idref" href="#M2.t"><span class="id" title="record">t</span></a> : <span class="id" title="keyword">Type</span> := <a id="M2.mat" class="idref" href="#M2.mat"><span class="id" title="constructor">mat</span></a>{<a id="M2.c00" class="idref" href="#M2.c00"><span class="id" title="projection">c00</span></a> : <a class="idref" href="additions.Naive.html#M2.Definitions.A"><span class="id" title="variable">A</span></a>;  <a id="M2.c01" class="idref" href="#M2.c01"><span class="id" title="projection">c01</span></a> : <a class="idref" href="additions.Naive.html#M2.Definitions.A"><span class="id" title="variable">A</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="M2.c10" class="idref" href="#M2.c10"><span class="id" title="projection">c10</span></a> : <a class="idref" href="additions.Naive.html#M2.Definitions.A"><span class="id" title="variable">A</span></a>;  <a id="M2.c11" class="idref" href="#M2.c11"><span class="id" title="projection">c11</span></a> : <a class="idref" href="additions.Naive.html#M2.Definitions.A"><span class="id" title="variable">A</span></a>}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="M2.Id2" class="idref" href="#M2.Id2"><span class="id" title="definition">Id2</span></a> : <a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a> := <a class="idref" href="additions.Naive.html#M2.mat"><span class="id" title="constructor">mat</span></a> 1 0 0 1.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="M2.M2_mult" class="idref" href="#M2.M2_mult"><span class="id" title="definition">M2_mult</span></a> (<a id="M:28" class="idref" href="#M:28"><span class="id" title="binder">M</span></a> <a id="M':29" class="idref" href="#M':29"><span class="id" title="binder">M'</span></a>:<a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a>) : <a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Naive.html#M2.mat"><span class="id" title="constructor">mat</span></a> (<a class="idref" href="additions.Naive.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.Naive.html#617a2cc97dac8fe5e951278b3bb11609"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Naive.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.Naive.html#617a2cc97dac8fe5e951278b3bb11609"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Naive.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c00"><span class="id" title="projection">c00</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.Naive.html#617a2cc97dac8fe5e951278b3bb11609"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="additions.Naive.html#M2.c10"><span class="id" title="projection">c10</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c01"><span class="id" title="projection">c01</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.Naive.html#617a2cc97dac8fe5e951278b3bb11609"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Naive.html#M:28"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Naive.html#M2.c11"><span class="id" title="projection">c11</span></a> <a class="idref" href="additions.Naive.html#M':29"><span class="id" title="variable">M'</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="67a4b07b9250dfdc7f4120645cd84088" class="idref" href="#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">&quot;</span></a>**" := <a class="idref" href="additions.Naive.html#M2.M2_mult"><span class="id" title="definition">M2_mult</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="M2.power" class="idref" href="#M2.power"><span class="id" title="definition">power</span></a> (<a id="M:30" class="idref" href="#M:30"><span class="id" title="binder">M</span></a> : <a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a>) (<a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Naive.html#n:31"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0%<span class="id" title="var">nat</span> ⇒ <a class="idref" href="additions.Naive.html#M2.Id2"><span class="id" title="definition">Id2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒  <a class="idref" href="additions.Naive.html#M:30"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Naive.html#power:32"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#M:30"><span class="id" title="variable">M</span></a> <span class="id" title="var">p</span><a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab396"></a><h4 class="section"> </h4>
+
+   The <span class="inlinecode"><span class="id" title="tactic">ring</span></span> tactic, applied to inhabitants of type <span class="inlinecode"><span class="id" title="var">A</span></span>, allows us to
+   prove associativity of matrix multiplication, then a law of 
+   distributivity of <span class="inlinecode"><span class="id" title="var">power</span></span> upon <span class="inlinecode">**</span>
+   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="M2.Id2_neutral" class="idref" href="#M2.Id2_neutral"><span class="id" title="lemma">Id2_neutral</span></a> : <span class="id" title="keyword">∀</span> <a id="M:34" class="idref" href="#M:34"><span class="id" title="binder">M</span></a>:<a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a>, <a class="idref" href="additions.Naive.html#M2.Id2"><span class="id" title="definition">Id2</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#M:34"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Naive.html#M:34"><span class="id" title="variable">M</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="M2.M2_mult_assoc" class="idref" href="#M2.M2_mult_assoc"><span class="id" title="lemma">M2_mult_assoc</span></a> : <span class="id" title="keyword">∀</span> <a id="M:35" class="idref" href="#M:35"><span class="id" title="binder">M</span></a> <a id="M':36" class="idref" href="#M':36"><span class="id" title="binder">M'</span></a> <a id="M'':37" class="idref" href="#M'':37"><span class="id" title="binder">M''</span></a>:<a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a>,  <a class="idref" href="additions.Naive.html#M:35"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Naive.html#M':36"><span class="id" title="variable">M'</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#M'':37"><span class="id" title="variable">M''</span></a><a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Naive.html#M:35"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#M':36"><span class="id" title="variable">M'</span></a><a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a> <a class="idref" href="additions.Naive.html#M'':37"><span class="id" title="variable">M''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="M2.power_of_plus" class="idref" href="#M2.power_of_plus"><span class="id" title="lemma">power_of_plus</span></a> (<a id="M:38" class="idref" href="#M:38"><span class="id" title="binder">M</span></a>:<a class="idref" href="additions.Naive.html#M2.t"><span class="id" title="record">t</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> <a id="p:40" class="idref" href="#p:40"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Naive.html#M2.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#M:38"><span class="id" title="variable">M</span></a> (<a class="idref" href="additions.Naive.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Naive.html#p:40"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="additions.Naive.html#M2.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#M:38"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088"><span class="id" title="notation">**</span></a>  <a class="idref" href="additions.Naive.html#M2.power"><span class="id" title="definition">power</span></a> <a class="idref" href="additions.Naive.html#M:38"><span class="id" title="variable">M</span></a> <a class="idref" href="additions.Naive.html#p:40"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Naive.html#M2.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Naive.html#M2"><span class="id" title="module">M2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fibonacci" class="idref" href="#fibonacci"><span class="id" title="definition">fibonacci</span></a> (<a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Naive.html#c00"><span class="id" title="projection">M2.c00</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> (<a class="idref" href="additions.Naive.html#power"><span class="id" title="definition">M2.power</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> 0%<span class="id" title="var">N</span> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#Nplus"><span class="id" title="abbreviation">Nplus</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#Nmult"><span class="id" title="abbreviation">Nmult</span></a> (<a class="idref" href="additions.Naive.html#mat"><span class="id" title="constructor">M2.mat</span></a> <span class="id" title="var">_</span> 1 1 1 0)%<span class="id" title="var">N</span>  <a class="idref" href="additions.Naive.html#n:41"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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@@ -0,0 +1,469 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Pow</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Pow</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab397"></a><h1 class="section">Exponentiation   in a Monoid</h1>
+
+
+<div class="paragraph"> </div>
+
+In this section, we give two polymorphic functions for computing
+ : the  naive ( linear) one and the aforementionned 
+binary method, that takes less than  multiplications.
+
+<div class="paragraph"> </div>
+
+ Both functions require an instance of <span class="inlinecode"><span class="id" title="var">EMonoid</span></span>. Their code use the 
+ multiplication of the monoid, and sometimes  its  unity. 
+  Correctness proofs require the "axioms" of monoid structure.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Div2.html#"><span class="id" title="library">Div2</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="library">Morphisms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab398"></a><h2 class="section">Two functions for computing powers</h2>
+
+
+<div class="paragraph"> </div>
+
+ The  module defines two functions for exponentiation on any <span class="inlinecode"><span class="id" title="var">Emonoid</span></span> 
+  on carrier .
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> The function <span class="inlinecode"><span class="id" title="var">power</span></span> has type <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">A</span></span>; it is linear with respect to 
+    the exponent. Its simplicity and the fact that the exponent has type <span class="inlinecode"><span class="id" title="var">nat</span></span> 
+    make it adequate for being the reference for any other definition, and for 
+    easily proving laws like .
+    
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> The function <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> has type <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">positive</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">A</span></span> and is logarithmic
+   with respect to its exponent. This function should be used for effective
+   computations. Its variant <span class="inlinecode"><span class="id" title="var">N_bpow</span></span> allows the exponent to be .
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab399"></a><h3 class="section">The "naive" reference function</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Generalizable Variables</span>  <span class="id" title="var">A</span> <span class="id" title="var">E_op</span> <span class="id" title="var">E_one</span> <span class="id" title="var">E_eq</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="power" class="idref" href="#power"><span class="id" title="definition">power</span></a> `{<a id="M:5" class="idref" href="#M:5"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>  <a id="E_op:2" class="idref" href="#E_op:2"><span class="id" title="binder">E_op</span></a> <a id="E_one:3" class="idref" href="#E_one:3"><span class="id" title="binder">E_one</span></a> <a id="E_eq:4" class="idref" href="#E_eq:4"><span class="id" title="binder">E_eq</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#A:1"><span class="id" title="variable">A</span></a>)(<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow.html#n:7"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0%<span class="id" title="var">nat</span> ⇒ <a class="idref" href="additions.Pow.html#E_one:3"><span class="id" title="variable">E_one</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒   <a class="idref" href="additions.Pow.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="765592cb239fbb605ff794a9833fef51" class="idref" href="#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">&quot;</span></a>x ^ n" := (<a class="idref" href="additions.Pow.html#power:8"><span class="id" title="definition">power</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span>) : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq1" class="idref" href="#power_eq1"><span class="id" title="lemma">power_eq1</span></a>  `{<a id="M:14" class="idref" href="#M:14"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:10" class="idref" href="#A:10"><span class="id" title="binder">A</span></a>  <a id="E_op:11" class="idref" href="#E_op:11"><span class="id" title="binder">E_op</span></a> <a id="E_one:12" class="idref" href="#E_one:12"><span class="id" title="binder">E_one</span></a> <a id="E_eq:13" class="idref" href="#E_eq:13"><span class="id" title="binder">E_eq</span></a>}(<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#A:10"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#E_one:12"><span class="id" title="variable">E_one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq2" class="idref" href="#power_eq2"><span class="id" title="lemma">power_eq2</span></a>  `{<a id="M:20" class="idref" href="#M:20"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a>  <a id="E_op:17" class="idref" href="#E_op:17"><span class="id" title="binder">E_op</span></a> <a id="E_one:18" class="idref" href="#E_one:18"><span class="id" title="binder">E_one</span></a> <a id="E_eq:19" class="idref" href="#E_eq:19"><span class="id" title="binder">E_eq</span></a>}(<a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#A:16"><span class="id" title="variable">A</span></a>) (<a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="additions.Pow.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Pow.html#n:22"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:22"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq3" class="idref" href="#power_eq3"><span class="id" title="lemma">power_eq3</span></a>  `{<a id="M:27" class="idref" href="#M:27"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:23" class="idref" href="#A:23"><span class="id" title="binder">A</span></a>  <a id="E_op:24" class="idref" href="#E_op:24"><span class="id" title="binder">E_op</span></a> <a id="E_one:25" class="idref" href="#E_one:25"><span class="id" title="binder">E_one</span></a> <a id="E_eq:26" class="idref" href="#E_eq:26"><span class="id" title="binder">E_eq</span></a>}(<a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#A:23"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;<a class="idref" href="additions.Pow.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 1 <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:28"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab400"></a><h3 class="section">The binary exponentiation function (exponents in  <span class="inlinecode"><span class="id" title="var">positive</span></span>)</h3>
+
+<div class="paragraph"> </div>
+
+<a id="lab401"></a><h3 class="section"> </h3>
+
+
+<div class="paragraph"> </div>
+
+The auxiliary function below computes  , where
+the "accumulator" <span class="inlinecode"><span class="id" title="var">acc</span></span> is intented to be an already computed power of <span class="inlinecode"><span class="id" title="var">x</span></span>:
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="binary_power_mult" class="idref" href="#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  `{<a id="M:33" class="idref" href="#M:33"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:29" class="idref" href="#A:29"><span class="id" title="binder">A</span></a> <a id="E_op:30" class="idref" href="#E_op:30"><span class="id" title="binder">E_op</span></a> <a id="E_one:31" class="idref" href="#E_one:31"><span class="id" title="binder">E_one</span></a> <a id="E_eq:32" class="idref" href="#E_eq:32"><span class="id" title="binder">E_eq</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a> <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Pow.html#A:29"><span class="id" title="variable">A</span></a>)(<a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Pow.html#A:29"><span class="id" title="variable">A</span></a> <br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow.html#p:36"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒  <a class="idref" href="additions.Pow.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow.html#binary_power_mult:37"><span class="id" title="definition">binary_power_mult</span></a> (<a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Pow.html#a:35"><span class="id" title="variable">a</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒  <a class="idref" href="additions.Pow.html#binary_power_mult:37"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.Pow.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:34"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab402"></a><h3 class="section"> </h3>
+
+
+<div class="paragraph"> </div>
+
+The  following function decomposes the exponent <span class="inlinecode"><span class="id" title="var">p</span></span>
+  into  , then calls <span class="inlinecode"><span class="id" title="var">binary_power_mult</span></span>
+   with  and .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Pos_bpow" class="idref" href="#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a>  `{<a id="M:43" class="idref" href="#M:43"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a> <a id="E_op:40" class="idref" href="#E_op:40"><span class="id" title="binder">E_op</span></a> <a id="E_one:41" class="idref" href="#E_one:41"><span class="id" title="binder">E_one</span></a> <a id="E_eq:42" class="idref" href="#E_eq:42"><span class="id" title="binder">E_eq</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#A:39"><span class="id" title="variable">A</span></a>)(<a id="p:45" class="idref" href="#p:45"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow.html#p:45"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow.html#Pos_bpow:46"><span class="id" title="definition">Pos_bpow</span></a>  (<a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Pow.html#x:44"><span class="id" title="variable">x</span></a> <span class="id" title="var">q</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+***
+  It is straightforward to adapt <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span>
+ for accepting exponents of type <span class="inlinecode"><span class="id" title="var">N</span></span> :
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N_bpow" class="idref" href="#N_bpow"><span class="id" title="definition">N_bpow</span></a>  `{<a id="M:52" class="idref" href="#M:52"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:48" class="idref" href="#A:48"><span class="id" title="binder">A</span></a> <a id="E_op:49" class="idref" href="#E_op:49"><span class="id" title="binder">E_op</span></a> <a id="E_one:50" class="idref" href="#E_one:50"><span class="id" title="binder">E_one</span></a> <a id="E_eq:51" class="idref" href="#E_eq:51"><span class="id" title="binder">E_eq</span></a>} <a id="x:53" class="idref" href="#x:53"><span class="id" title="binder">x</span></a> (<a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow.html#n:54"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| 0%<span class="id" title="var">N</span> ⇒ <a class="idref" href="additions.Pow.html#E_one:50"><span class="id" title="variable">E_one</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:53"><span class="id" title="variable">x</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="0ad68a69d54362abc023eb138c77275f" class="idref" href="#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">&quot;</span></a>^b" := <a class="idref" href="additions.Pow.html#N_bpow"><span class="id" title="definition">N_bpow</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab403"></a><h2 class="section">Properties of the power function</h2>
+
+
+<div class="paragraph"> </div>
+
+ Taking <span class="inlinecode"><span class="id" title="var">power</span></span> as a reference, it remains to prove two kinds of properties
+<ul class="doclist">
+<li> Mathematical properties of exponentiation, <i>i.e</i> the function <span class="inlinecode"><span class="id" title="var">power</span></span>,
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> proving  correctness of functions <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> and <span class="inlinecode"><span class="id" title="var">N_bpow</span></span>
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+First, let us consider some <span class="inlinecode"><span class="id" title="var">Emonoid</span></span> and define some useful notations and tactics:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M_given" class="idref" href="#M_given"><span class="id" title="section">M_given</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="M_given.A" class="idref" href="#M_given.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="M_given.E_one" class="idref" href="#M_given.E_one"><span class="id" title="variable">E_one</span></a>:<a class="idref" href="additions.Pow.html#A:56"><span class="id" title="variable">A</span></a>) .<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="E_op:58" class="idref" href="#E_op:58"><span class="id" title="binder">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Pow.html#M_given.A"><span class="id" title="variable">A</span></a>) (<a id="E_eq:59" class="idref" href="#E_eq:59"><span class="id" title="binder">E_eq</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Pow.html#M_given.A"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="M:60" class="idref" href="#M:60"><span class="id" title="binder">M</span></a>:<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="additions.Pow.html#E_op:58"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Pow.html#M_given.E_one"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Pow.html#E_eq:59"><span class="id" title="variable">E_eq</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Eop_proper" class="idref" href="#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Pow.html#M_given.E_op"><span class="id" title="variable">E_op</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_rw</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a>  ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a>  || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_simpl</span> := <span class="id" title="tactic">repeat</span> <span class="id" title="var">monoid_rw</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="power_proper" class="idref" href="#power_proper"><span class="id" title="instance">power_proper</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_plus" class="idref" href="#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:61" class="idref" href="#x:61"><span class="id" title="binder">x</span></a> <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> <a id="p:63" class="idref" href="#p:63"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Pow.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Pow.html#p:63"><span class="id" title="variable">p</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:63"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">power_simpl</span> := <span class="id" title="tactic">repeat</span> (<span class="id" title="var">monoid_rw</span> || <span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_commute" class="idref" href="#power_commute"><span class="id" title="lemma">power_commute</span></a> <a id="x:64" class="idref" href="#x:64"><span class="id" title="binder">x</span></a> <a id="n:65" class="idref" href="#n:65"><span class="id" title="binder">n</span></a> <a id="p:66" class="idref" href="#p:66"><span class="id" title="binder">p</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:66"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:66"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:65"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_commute_with_x" class="idref" href="#power_commute_with_x"><span class="id" title="lemma">power_commute_with_x</span></a> <a id="x:67" class="idref" href="#x:67"><span class="id" title="binder">x</span></a> <a id="n:68" class="idref" href="#n:68"><span class="id" title="binder">n</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:68"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:68"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:67"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_power" class="idref" href="#power_of_power"><span class="id" title="lemma">power_of_power</span></a> <a id="x:69" class="idref" href="#x:69"><span class="id" title="binder">x</span></a> <a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a> <a id="p:71" class="idref" href="#p:71"><span class="id" title="binder">p</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#x:69"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:70"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:71"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:69"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#p:71"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#n:70"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_power_comm" class="idref" href="#power_of_power_comm"><span class="id" title="lemma">power_of_power_comm</span></a> <a id="x:72" class="idref" href="#x:72"><span class="id" title="binder">x</span></a> <a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> <a id="p:74" class="idref" href="#p:74"><span class="id" title="binder">p</span></a> : <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#x:72"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:73"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:74"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#x:72"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:74"><span class="id" title="variable">p</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:73"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sqr_eqn" class="idref" href="#sqr_eqn"><span class="id" title="lemma">sqr_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 2 <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:75"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">factorize</span> := <span class="id" title="tactic">repeat</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow.html#power_commute_with_x"><span class="id" title="lemma">power_commute_with_x</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span>  &lt;- <a class="idref" href="additions.Pow.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>  ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow.html#sqr_eqn"><span class="id" title="lemma">sqr_eqn</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow.html#power_eq2"><span class="id" title="lemma">power_eq2</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Pow.html#power_of_power"><span class="id" title="lemma">power_of_power</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_square" class="idref" href="#power_of_square"><span class="id" title="lemma">power_of_square</span></a> <a id="x:76" class="idref" href="#x:76"><span class="id" title="binder">x</span></a> <a id="n:77" class="idref" href="#n:77"><span class="id" title="binder">n</span></a> : <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#x:76"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:76"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:77"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#x:76"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:77"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:76"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:77"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab404"></a><h2 class="section">Correctness of the binary algorithm</h2>
+
+
+<div class="paragraph"> </div>
+
+Correctness of the "concrete" functions <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> and <span class="inlinecode"><span class="id" title="var">N_bpow</span></span>
+with respect to the more abstract function <span class="inlinecode"><span class="id" title="var">power</span></span> is expressed 
+by  extensional equalities, taking into account the conversion between 
+various representations of natural numbers.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="binary_power_mult_ok" class="idref" href="#binary_power_mult_ok"><span class="id" title="lemma">binary_power_mult_ok</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:78" class="idref" href="#p:78"><span class="id" title="binder">p</span></a> <a id="a:79" class="idref" href="#a:79"><span class="id" title="binder">a</span></a> <a id="x:80" class="idref" href="#x:80"><span class="id" title="binder">x</span></a>,   <a class="idref" href="additions.Pow.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  <a class="idref" href="additions.Pow.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow.html#p:78"><span class="id" title="variable">p</span></a>  <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Pow.html#p:78"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_ok" class="idref" href="#Pos_bpow_ok"><span class="id" title="lemma">Pos_bpow_ok</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:81" class="idref" href="#p:81"><span class="id" title="binder">p</span></a> <a id="x:82" class="idref" href="#x:82"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:82"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#p:81"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:82"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Pow.html#p:81"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Pos_bpow_proper" class="idref" href="#Pos_bpow_proper"><span class="id" title="instance">Pos_bpow_proper</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a>  (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_ok" class="idref" href="#N_bpow_ok"><span class="id" title="lemma">N_bpow_ok</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:83" class="idref" href="#n:83"><span class="id" title="binder">n</span></a> <a id="x:84" class="idref" href="#x:84"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow.html#x:84"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow.html#n:83"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:84"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="additions.Pow.html#n:83"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_ok_R" class="idref" href="#N_bpow_ok_R"><span class="id" title="lemma">N_bpow_ok_R</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a> <a id="x:86" class="idref" href="#x:86"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a>  <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.of_nat"><span class="id" title="definition">N.of_nat</span></a> <a class="idref" href="additions.Pow.html#n:85"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:85"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_ok_R" class="idref" href="#Pos_bpow_ok_R"><span class="id" title="lemma">Pos_bpow_ok_R</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:87" class="idref" href="#p:87"><span class="id" title="binder">p</span></a> <a id="x:88" class="idref" href="#x:88"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow.html#p:87"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:88"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> <a class="idref" href="additions.Pow.html#p:87"><span class="id" title="variable">p</span></a>) <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#x:88"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#p:87"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_commute" class="idref" href="#N_bpow_commute"><span class="id" title="lemma">N_bpow_commute</span></a> : <span class="id" title="keyword">∀</span> <a id="x:89" class="idref" href="#x:89"><span class="id" title="binder">x</span></a> <a id="n:90" class="idref" href="#n:90"><span class="id" title="binder">n</span></a> <a id="p:91" class="idref" href="#p:91"><span class="id" title="binder">p</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow.html#n:90"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow.html#p:91"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow.html#p:91"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow.html#n:90"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_of_plus" class="idref" href="#Pos_bpow_of_plus"><span class="id" title="lemma">Pos_bpow_of_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="x:92" class="idref" href="#x:92"><span class="id" title="binder">x</span></a> <a id="n:93" class="idref" href="#n:93"><span class="id" title="binder">n</span></a> <a id="p:94" class="idref" href="#p:94"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:92"><span class="id" title="variable">x</span></a>  (<a class="idref" href="additions.Pow.html#n:93"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Pow.html#p:94"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">positive</span> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:92"><span class="id" title="variable">x</span></a>  <a class="idref" href="additions.Pow.html#n:93"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:92"><span class="id" title="variable">x</span></a>  <a class="idref" href="additions.Pow.html#p:94"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_of_bpow" class="idref" href="#Pos_bpow_of_bpow"><span class="id" title="lemma">Pos_bpow_of_bpow</span></a> : <span class="id" title="keyword">∀</span> (<a id="x:95" class="idref" href="#x:95"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#M_given.A"><span class="id" title="variable">A</span></a>) <a id="n:96" class="idref" href="#n:96"><span class="id" title="binder">n</span></a> <a id="p:97" class="idref" href="#p:97"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> (<a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:95"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#n:96"><span class="id" title="variable">n</span></a>) <a class="idref" href="additions.Pow.html#p:97"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow.html#x:95"><span class="id" title="variable">x</span></a> (<a class="idref" href="additions.Pow.html#p:97"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#83db3fa9f059c10cb623a122cf097cd0"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#n:96"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab405"></a><h2 class="section">Remark</h2>
+
+
+<div class="paragraph"> </div>
+
+If we normalize exponentiation functions with a given exponent, we notice
+that the obtained functions do not execute the same computations, but it is
+hard to visualize why the binary method is more efficient than the naive one.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span> <span class="id" title="keyword">fun</span> (<a id="x:98" class="idref" href="#x:98"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#M_given.A"><span class="id" title="variable">A</span></a>) ⇒ <a class="idref" href="additions.Pow.html#x:98"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> 17.<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span>  <span class="id" title="keyword">fun</span> <a id="x:99" class="idref" href="#x:99"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="additions.Pow.html#x:99"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 17.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pow_17" class="idref" href="#pow_17"><span class="id" title="definition">pow_17</span></a>  (<a id="x:100" class="idref" href="#x:100"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow.html#M_given.A"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x2:101" class="idref" href="#x2:101"><span class="id" title="binder">x2</span></a> := <a class="idref" href="additions.Pow.html#x:100"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:100"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x4:102" class="idref" href="#x4:102"><span class="id" title="binder">x4</span></a> := <a class="idref" href="additions.Pow.html#x2:101"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x2:101"><span class="id" title="variable">x2</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x8:103" class="idref" href="#x8:103"><span class="id" title="binder">x8</span></a> := <a class="idref" href="additions.Pow.html#x4:102"><span class="id" title="variable">x4</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x4:102"><span class="id" title="variable">x4</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x16:104" class="idref" href="#x16:104"><span class="id" title="binder">x16</span></a> := <a class="idref" href="additions.Pow.html#x8:103"><span class="id" title="variable">x8</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x8:103"><span class="id" title="variable">x8</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow.html#x16:104"><span class="id" title="variable">x16</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#x:100"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">cbv</span>  <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">pow_17</span>]  <span class="id" title="tactic">in</span>  <a class="idref" href="additions.Pow.html#pow_17"><span class="id" title="definition">pow_17</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ = fun x : A =&gt;
+       x * x * (x * x) * (x * x * (x * x)) *
+       (x * x * (x * x) * (x * x * (x * x))) * x
+     : A -&gt; A
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+In order to compare the real computations needed to raise some  to its -th
+power, we need to make more explicit how intermediate values are used during 
+some computation. 
+This is described in the module  (see ).
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab406"></a><h2 class="section">Properties of Abelian Monoids</h2>
+
+
+<div class="paragraph"> </div>
+
+ Some equalities hold in the restricted context of abelian (a.k.a. commutative)
+ monoids. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M_given.Power_of_op" class="idref" href="#M_given.Power_of_op"><span class="id" title="section">Power_of_op</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Context</span>  {<a id="AM:105" class="idref" href="#AM:105"><span class="id" title="binder">AM</span></a>:<a class="idref" href="additions.Monoid_def.html#Abelian_EMonoid"><span class="id" title="class">Abelian_EMonoid</span></a> <a class="idref" href="additions.Pow.html#M_given.M"><span class="id" title="variable">M</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="power_of_mult" class="idref" href="#power_of_mult"><span class="id" title="lemma">power_of_mult</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:106" class="idref" href="#n:106"><span class="id" title="binder">n</span></a> <a id="x:107" class="idref" href="#x:107"><span class="id" title="binder">x</span></a> <a id="y:108" class="idref" href="#y:108"><span class="id" title="binder">y</span></a>,  <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow.html#y:108"><span class="id" title="variable">y</span></a><a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>  <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:106"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:106"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow.html#y:108"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Pow.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow.html#n:106"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Pow.html#M_given.Power_of_op"><span class="id" title="section">Power_of_op</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Pow.html#M_given"><span class="id" title="section">M_given</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="765592cb239fbb605ff794a9833fef51" class="idref" href="#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">&quot;</span></a>^" := <a class="idref" href="additions.Pow.html#power"><span class="id" title="definition">power</span></a> : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_simpl</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">generalize</span> (<a class="idref" href="additions.Pow.html#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span>; <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> ( <span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a> ) || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a>  ) || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a> )).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">power_simpl</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">generalize</span> (<a class="idref" href="additions.Pow.html#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span>; <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> ( <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a>  ||  <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a> ||  <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Pow.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Pow_variant.html b/theories/html/additions.Pow_variant.html
new file mode 100644
index 00000000..02c0cc0b
--- /dev/null
+++ b/theories/html/additions.Pow_variant.html
@@ -0,0 +1,477 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Pow_variant</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Pow_variant</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab407"></a><h1 class="section">Exponentiation   in a Monoid</h1>
+
+
+<div class="paragraph"> </div>
+
+In this section, we give two polymorphic functions for computing
+ : the  naive ( linear) one and the aforementionned 
+binary method, that takes less than  multiplications.
+
+<div class="paragraph"> </div>
+
+ Both functions require an instance of <span class="inlinecode"><span class="id" title="var">EMonoid</span></span>. Their code use the multiplication of the monoid, and sometimes  its  unity. Correctness proofs require 
+the "axioms" of monoid structure.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Div2.html#"><span class="id" title="library">Div2</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="library">Morphisms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Monoid_def.html#"><span class="id" title="library">Monoid_def</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable Variables</span> <span class="id" title="var">A</span> <span class="id" title="var">op</span> <span class="id" title="var">one</span> <span class="id" title="var">Aeq</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab408"></a><h2 class="section">Two functions for computing powers</h2>
+
+
+<div class="paragraph"> </div>
+
+ The  module defines two functions for exponentiation on any <span class="inlinecode"><span class="id" title="var">Emonoid</span></span> 
+  on carrier .
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> The function <span class="inlinecode"><span class="id" title="var">power</span></span> has type <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">A</span></span>; it is linear with respect to 
+    the exponent. Its simplicity and the fact that the exponent has type <span class="inlinecode"><span class="id" title="var">nat</span></span> 
+    make it adequate for being the reference for any other definition, and for 
+    easily proving laws like .
+    
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> The function <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> has type <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">positive</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">A</span></span> and is logarithmic
+   with respect to its exponent. This function should be used for effective
+   computations. Its variant <span class="inlinecode"><span class="id" title="var">N_bpow</span></span> allows the exponent to be .
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab409"></a><h3 class="section">The "naive" reference function</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Generalizable Variables</span>  <span class="id" title="var">E_op</span> <span class="id" title="var">E_one</span> <span class="id" title="var">E_eq</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="power" class="idref" href="#power"><span class="id" title="definition">power</span></a> `{<a id="M:5" class="idref" href="#M:5"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>  <a id="E_op:2" class="idref" href="#E_op:2"><span class="id" title="binder">E_op</span></a> <a id="E_one:3" class="idref" href="#E_one:3"><span class="id" title="binder">E_one</span></a> <a id="E_eq:4" class="idref" href="#E_eq:4"><span class="id" title="binder">E_eq</span></a>}(<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#A:1"><span class="id" title="variable">A</span></a>)(<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow_variant.html#n:7"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0%<span class="id" title="var">nat</span> ⇒ <a class="idref" href="additions.Pow_variant.html#E_one:3"><span class="id" title="variable">E_one</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒   <a class="idref" href="additions.Pow_variant.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow_variant.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="765592cb239fbb605ff794a9833fef51" class="idref" href="#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">&quot;</span></a>x ^ n" := (<a class="idref" href="additions.Pow_variant.html#power:8"><span class="id" title="definition">power</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span>) : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq1" class="idref" href="#power_eq1"><span class="id" title="lemma">power_eq1</span></a> {<a id="A:10" class="idref" href="#A:10"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} `{<a id="M:14" class="idref" href="#M:14"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Pow_variant.html#A:10"><span class="id" title="variable">A</span></a>  <a id="E_op:11" class="idref" href="#E_op:11"><span class="id" title="binder">E_op</span></a> <a id="E_one:12" class="idref" href="#E_one:12"><span class="id" title="binder">E_one</span></a> <a id="E_eq:13" class="idref" href="#E_eq:13"><span class="id" title="binder">E_eq</span></a>}(<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#A:10"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow_variant.html#E_one:12"><span class="id" title="variable">E_one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq2" class="idref" href="#power_eq2"><span class="id" title="lemma">power_eq2</span></a> {<a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} `{<a id="M:20" class="idref" href="#M:20"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Pow_variant.html#A:16"><span class="id" title="variable">A</span></a>  <a id="E_op:17" class="idref" href="#E_op:17"><span class="id" title="binder">E_op</span></a> <a id="E_one:18" class="idref" href="#E_one:18"><span class="id" title="binder">E_one</span></a> <a id="E_eq:19" class="idref" href="#E_eq:19"><span class="id" title="binder">E_eq</span></a>}(<a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#A:16"><span class="id" title="variable">A</span></a>) (<a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="additions.Pow_variant.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="additions.Pow_variant.html#n:22"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Pow_variant.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:22"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_eq3" class="idref" href="#power_eq3"><span class="id" title="lemma">power_eq3</span></a> {<a id="A:23" class="idref" href="#A:23"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} `{<a id="M:27" class="idref" href="#M:27"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a class="idref" href="additions.Pow_variant.html#A:23"><span class="id" title="variable">A</span></a>  <a id="E_op:24" class="idref" href="#E_op:24"><span class="id" title="binder">E_op</span></a> <a id="E_one:25" class="idref" href="#E_one:25"><span class="id" title="binder">E_one</span></a> <a id="E_eq:26" class="idref" href="#E_eq:26"><span class="id" title="binder">E_eq</span></a>}(<a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#A:23"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;<a class="idref" href="additions.Pow_variant.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 1 <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow_variant.html#x:28"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab410"></a><h3 class="section">The binary exponentiation function (exponents in  <span class="inlinecode"><span class="id" title="var">positive</span></span>)</h3>
+
+<div class="paragraph"> </div>
+
+<a id="lab411"></a><h3 class="section"> </h3>
+
+
+<div class="paragraph"> </div>
+
+The auxiliary function below computes  , where
+the "accumulator" <span class="inlinecode"><span class="id" title="var">acc</span></span> is intented to be an already computed power of <span class="inlinecode"><span class="id" title="var">x</span></span>:
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="binary_power_mult" class="idref" href="#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  `{<a id="M:33" class="idref" href="#M:33"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:29" class="idref" href="#A:29"><span class="id" title="binder">A</span></a> <a id="E_op:30" class="idref" href="#E_op:30"><span class="id" title="binder">E_op</span></a> <a id="E_one:31" class="idref" href="#E_one:31"><span class="id" title="binder">E_one</span></a> <a id="E_eq:32" class="idref" href="#E_eq:32"><span class="id" title="binder">E_eq</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a> <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a>:<a class="idref" href="additions.Pow_variant.html#A:29"><span class="id" title="variable">A</span></a>)(<a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="additions.Pow_variant.html#A:29"><span class="id" title="variable">A</span></a> <br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow_variant.html#p:36"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒  <a class="idref" href="additions.Pow_variant.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow_variant.html#binary_power_mult:37"><span class="id" title="definition">binary_power_mult</span></a> (<a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Pow_variant.html#a:35"><span class="id" title="variable">a</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒  <a class="idref" href="additions.Pow_variant.html#binary_power_mult:37"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.Pow_variant.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:34"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab412"></a><h3 class="section"> </h3>
+
+
+<div class="paragraph"> </div>
+
+The  following function decomposes the exponent <span class="inlinecode"><span class="id" title="var">p</span></span>
+  into  , then calls <span class="inlinecode"><span class="id" title="var">binary_power_mult</span></span>
+   with  and .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Pos_bpow" class="idref" href="#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a>  `{<a id="M:43" class="idref" href="#M:43"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a> <a id="E_op:40" class="idref" href="#E_op:40"><span class="id" title="binder">E_op</span></a> <a id="E_one:41" class="idref" href="#E_one:41"><span class="id" title="binder">E_one</span></a> <a id="E_eq:42" class="idref" href="#E_eq:42"><span class="id" title="binder">E_eq</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#A:39"><span class="id" title="variable">A</span></a>)(<a id="p:45" class="idref" href="#p:45"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow_variant.html#p:45"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow_variant.html#Pos_bpow:46"><span class="id" title="definition">Pos_bpow</span></a>  (<a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.Pow_variant.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  (<a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.Pow_variant.html#x:44"><span class="id" title="variable">x</span></a> <span class="id" title="var">q</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+***
+  It is straightforward to adapt <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span>
+ for accepting exponents of type <span class="inlinecode"><span class="id" title="var">N</span></span> :
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N_bpow" class="idref" href="#N_bpow"><span class="id" title="definition">N_bpow</span></a>  `{<a id="M:52" class="idref" href="#M:52"><span class="id" title="binder">M</span></a>: @<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a> <a id="A:48" class="idref" href="#A:48"><span class="id" title="binder">A</span></a> <a id="E_op:49" class="idref" href="#E_op:49"><span class="id" title="binder">E_op</span></a> <a id="E_one:50" class="idref" href="#E_one:50"><span class="id" title="binder">E_one</span></a> <a id="E_eq:51" class="idref" href="#E_eq:51"><span class="id" title="binder">E_eq</span></a>} <a id="x:53" class="idref" href="#x:53"><span class="id" title="binder">x</span></a> (<a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Pow_variant.html#n:54"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| 0%<span class="id" title="var">N</span> ⇒ <a class="idref" href="additions.Pow_variant.html#E_one:50"><span class="id" title="variable">E_one</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.Pow_variant.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow_variant.html#x:53"><span class="id" title="variable">x</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="0ad68a69d54362abc023eb138c77275f" class="idref" href="#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">&quot;</span></a>^b" := <a class="idref" href="additions.Pow_variant.html#N_bpow"><span class="id" title="definition">N_bpow</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab413"></a><h2 class="section">Properties of the power function</h2>
+
+
+<div class="paragraph"> </div>
+
+ Taking <span class="inlinecode"><span class="id" title="var">power</span></span> as a reference, it remains to prove two kinds of properties
+<ul class="doclist">
+<li> Mathematical properties of exponentiation, <i>i.e</i> the function <span class="inlinecode"><span class="id" title="var">power</span></span>,
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> proving  correctness of functions <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> and <span class="inlinecode"><span class="id" title="var">N_bpow</span></span>
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+First, let us consider some <span class="inlinecode"><span class="id" title="var">Emonoid</span></span> and define some useful notations and tactics:
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M_given" class="idref" href="#M_given"><span class="id" title="section">M_given</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="M_given.A" class="idref" href="#M_given.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="M_given.E_op" class="idref" href="#M_given.E_op"><span class="id" title="variable">E_op</span></a> : <a class="idref" href="additions.Monoid_def.html#Mult_op"><span class="id" title="class">Mult_op</span></a> <a class="idref" href="additions.Pow_variant.html#A:56"><span class="id" title="variable">A</span></a>)(<a id="M_given.E_one" class="idref" href="#M_given.E_one"><span class="id" title="variable">E_one</span></a>:<a class="idref" href="additions.Pow_variant.html#A:56"><span class="id" title="variable">A</span></a>) (<a id="M_given.E_eq" class="idref" href="#M_given.E_eq"><span class="id" title="variable">E_eq</span></a> : <a class="idref" href="additions.Monoid_def.html#Equiv"><span class="id" title="class">Equiv</span></a> <a class="idref" href="additions.Pow_variant.html#A:56"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Context</span> (<a id="M:60" class="idref" href="#M:60"><span class="id" title="binder">M</span></a>:<a class="idref" href="additions.Monoid_def.html#EMonoid"><span class="id" title="class">EMonoid</span></a>  <a class="idref" href="additions.Pow_variant.html#M_given.E_op"><span class="id" title="variable">E_op</span></a> <a class="idref" href="additions.Pow_variant.html#M_given.E_one"><span class="id" title="variable">E_one</span></a> <a class="idref" href="additions.Pow_variant.html#M_given.E_eq"><span class="id" title="variable">E_eq</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Eop_proper" class="idref" href="#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Pow_variant.html#M_given.E_op"><span class="id" title="variable">E_op</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_rw</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a>  ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a>  || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_simpl</span> := <span class="id" title="tactic">repeat</span> <span class="id" title="var">monoid_rw</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M_given.About_power" class="idref" href="#M_given.About_power"><span class="id" title="section">About_power</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="power_proper" class="idref" href="#power_proper"><span class="id" title="instance">power_proper</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="additions.Monoid_def.html#equiv"><span class="id" title="definition">equiv</span></a>) <a class="idref" href="additions.Pow_variant.html#power"><span class="id" title="definition">power</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_plus" class="idref" href="#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="x:61" class="idref" href="#x:61"><span class="id" title="binder">x</span></a> <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> <a id="p:63" class="idref" href="#p:63"><span class="id" title="binder">p</span></a>, <a class="idref" href="additions.Pow_variant.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow_variant.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.Pow_variant.html#p:63"><span class="id" title="variable">p</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow_variant.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#p:63"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">power_simpl</span> := <span class="id" title="tactic">repeat</span> (<span class="id" title="var">monoid_rw</span> || <span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow_variant.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_commute" class="idref" href="#power_commute"><span class="id" title="lemma">power_commute</span></a> : <span class="id" title="keyword">∀</span> <a id="x:64" class="idref" href="#x:64"><span class="id" title="binder">x</span></a> <a id="n:65" class="idref" href="#n:65"><span class="id" title="binder">n</span></a> <a id="p:66" class="idref" href="#p:66"><span class="id" title="binder">p</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#p:66"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#p:66"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:65"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_commute_with_x" class="idref" href="#power_commute_with_x"><span class="id" title="lemma">power_commute_with_x</span></a> : <span class="id" title="keyword">∀</span> <a id="x:67" class="idref" href="#x:67"><span class="id" title="binder">x</span></a> <a id="n:68" class="idref" href="#n:68"><span class="id" title="binder">n</span></a>,  <a class="idref" href="additions.Pow_variant.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:68"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow_variant.html#x:67"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:68"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:67"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_power" class="idref" href="#power_of_power"><span class="id" title="lemma">power_of_power</span></a> : <span class="id" title="keyword">∀</span> <a id="x:69" class="idref" href="#x:69"><span class="id" title="binder">x</span></a> <a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a> <a id="p:71" class="idref" href="#p:71"><span class="id" title="binder">p</span></a>,  <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow_variant.html#x:69"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:70"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#p:71"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow_variant.html#x:69"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow_variant.html#p:71"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#n:70"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sqr_eqn" class="idref" href="#sqr_eqn"><span class="id" title="lemma">sqr_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="x:72" class="idref" href="#x:72"><span class="id" title="binder">x</span></a>, <a class="idref" href="additions.Pow_variant.html#x:72"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 2 <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:72"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:72"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">factorize</span> := <span class="id" title="tactic">repeat</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow_variant.html#power_commute_with_x"><span class="id" title="lemma">power_commute_with_x</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span>  &lt;- <a class="idref" href="additions.Pow_variant.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>  ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow_variant.html#sqr_eqn"><span class="id" title="lemma">sqr_eqn</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="additions.Pow_variant.html#power_eq2"><span class="id" title="lemma">power_eq2</span></a> ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Pow_variant.html#power_of_power"><span class="id" title="lemma">power_of_power</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_square" class="idref" href="#power_of_square"><span class="id" title="lemma">power_of_square</span></a> : <span class="id" title="keyword">∀</span> <a id="x:73" class="idref" href="#x:73"><span class="id" title="binder">x</span></a> <a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a>, <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow_variant.html#x:73"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:73"><span class="id" title="variable">x</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:73"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:73"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:74"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab414"></a><h2 class="section">Correctness of the binary algorithm</h2>
+
+
+<div class="paragraph"> </div>
+
+Correctness of the "concrete" functions <span class="inlinecode"><span class="id" title="var">Pos_bpow</span></span> and <span class="inlinecode"><span class="id" title="var">N_bpow</span></span>
+with respect to the more abstract function <span class="inlinecode"><span class="id" title="var">power</span></span> is expressed 
+by  extensional equalities, taking into account the conversion between 
+various representations of natural numbers.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="binary_power_mult_ok" class="idref" href="#binary_power_mult_ok"><span class="id" title="lemma">binary_power_mult_ok</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:75" class="idref" href="#p:75"><span class="id" title="binder">p</span></a> <a id="a:76" class="idref" href="#a:76"><span class="id" title="binder">a</span></a> <a id="x:77" class="idref" href="#x:77"><span class="id" title="binder">x</span></a>,   <a class="idref" href="additions.Pow_variant.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a>  <a class="idref" href="additions.Pow_variant.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#a:76"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Pow_variant.html#p:75"><span class="id" title="variable">p</span></a>  <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#a:76"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Pow_variant.html#p:75"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_ok" class="idref" href="#Pos_bpow_ok"><span class="id" title="lemma">Pos_bpow_ok</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="p:78" class="idref" href="#p:78"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)(<a id="x:79" class="idref" href="#x:79"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>), <a class="idref" href="additions.Pow_variant.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow_variant.html#x:79"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#p:78"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow_variant.html#x:79"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.Pow_variant.html#p:78"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_ok" class="idref" href="#N_bpow_ok"><span class="id" title="lemma">N_bpow_ok</span></a> : <br/>
+<span class="id" title="keyword">∀</span> (<a id="x:80" class="idref" href="#x:80"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>) (<a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),   <a class="idref" href="additions.Pow_variant.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow_variant.html#n:81"><span class="id" title="variable">n</span></a>  <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a> <a class="idref" href="additions.Pow_variant.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="additions.Pow_variant.html#n:81"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_ok_R" class="idref" href="#N_bpow_ok_R"><span class="id" title="lemma">N_bpow_ok_R</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="x:82" class="idref" href="#x:82"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>) (<a id="n:83" class="idref" href="#n:83"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="additions.Pow_variant.html#x:82"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a>  <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.of_nat"><span class="id" title="definition">N.of_nat</span></a> <a class="idref" href="additions.Pow_variant.html#n:83"><span class="id" title="variable">n</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">)</span></a>   <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:82"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a>  <a class="idref" href="additions.Pow_variant.html#n:83"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Pos_bpow_ok_R" class="idref" href="#Pos_bpow_ok_R"><span class="id" title="lemma">Pos_bpow_ok_R</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="x:84" class="idref" href="#x:84"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>) (<a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="additions.Pow_variant.html#n:85"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#Pos_bpow"><span class="id" title="definition">Pos_bpow</span></a> <a class="idref" href="additions.Pow_variant.html#x:84"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.of_nat"><span class="id" title="definition">Pos.of_nat</span></a> <a class="idref" href="additions.Pow_variant.html#n:85"><span class="id" title="variable">n</span></a>)   <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:84"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a>  <a class="idref" href="additions.Pow_variant.html#n:85"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Pow_variant.html#M_given.About_power"><span class="id" title="section">About_power</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="N_bpow_commute" class="idref" href="#N_bpow_commute"><span class="id" title="lemma">N_bpow_commute</span></a> : <span class="id" title="keyword">∀</span> <a id="x:86" class="idref" href="#x:86"><span class="id" title="binder">x</span></a> <a id="n:87" class="idref" href="#n:87"><span class="id" title="binder">n</span></a> <a id="p:88" class="idref" href="#p:88"><span class="id" title="binder">p</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow_variant.html#n:87"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow_variant.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow_variant.html#p:88"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow_variant.html#p:88"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow_variant.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> <a class="idref" href="additions.Pow_variant.html#n:87"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab415"></a><h2 class="section">Remark</h2>
+
+
+<div class="paragraph"> </div>
+
+Iw we normalize exponentiation functions with a given exponent, we notice
+that the obtained functions do not execute the same computations, but it is
+hard to visualize why the binary method is more efficient than the na\u00efve one.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span>     <span class="id" title="keyword">fun</span> (<a id="x:89" class="idref" href="#x:89"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>) ⇒ <a class="idref" href="additions.Pow_variant.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">^</span></a><a class="idref" href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f"><span class="id" title="notation">b</span></a> 17.<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode">[</span> 
+  <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="keyword">fun</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">⇒</span>
+       <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">×</span>
+       <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span>))</span> <span class="inlinecode">×</span>
+        <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">x</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x</span>))))</span>
+     <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">A</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">A</span></span>
+<span class="inlinecode">]</span>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span>   <span class="id" title="keyword">fun</span> <a id="x:90" class="idref" href="#x:90"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="additions.Pow_variant.html#x:90"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> 17.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<br/>
+<span class="inlinecode">= <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> : <span class="id" title="var">A</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> ×  (<span class="id" title="var">x</span> ×  (<span class="id" title="var">x</span> ×  (<span class="id" title="var">x</span> ×   (<span class="id" title="var">x</span> ×  (<span class="id" title="var">x</span> ×  (<span class="id" title="var">x</span> ×<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × (<span class="id" title="var">x</span> × <span class="id" title="var">one</span>))))))))))))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;: <span class="id" title="var">A</span> → <span class="id" title="var">A</span>
+<div class="paragraph"> </div>
+
+</span>
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pow_17" class="idref" href="#pow_17"><span class="id" title="definition">pow_17</span></a>  (<a id="x:91" class="idref" href="#x:91"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Pow_variant.html#M_given.A"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x2:92" class="idref" href="#x2:92"><span class="id" title="binder">x2</span></a> := <a class="idref" href="additions.Pow_variant.html#x:91"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:91"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x4:93" class="idref" href="#x4:93"><span class="id" title="binder">x4</span></a> := <a class="idref" href="additions.Pow_variant.html#x2:92"><span class="id" title="variable">x2</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x2:92"><span class="id" title="variable">x2</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x8:94" class="idref" href="#x8:94"><span class="id" title="binder">x8</span></a> := <a class="idref" href="additions.Pow_variant.html#x4:93"><span class="id" title="variable">x4</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x4:93"><span class="id" title="variable">x4</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x16:95" class="idref" href="#x16:95"><span class="id" title="binder">x16</span></a> := <a class="idref" href="additions.Pow_variant.html#x8:94"><span class="id" title="variable">x8</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x8:94"><span class="id" title="variable">x8</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.Pow_variant.html#x16:95"><span class="id" title="variable">x16</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#x:91"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">cbv</span>  <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">pow_17</span>]  <span class="id" title="tactic">in</span>  <a class="idref" href="additions.Pow_variant.html#pow_17"><span class="id" title="definition">pow_17</span></a>.<br/>
+</div>
+
+<div class="doc">
+ = fun x : A =&gt;
+       x * x * (x * x) * (x * x * (x * x)) *
+       (x * x * (x * x) * (x * x * (x * x))) * x
+     : A -&gt; A
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+In order to compare the real computations needed to raise some  to its -th
+power, we need to make more explicit how intermediate values are used during 
+some computation. 
+This is described in the module  (see ).
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab416"></a><h2 class="section">Properties of Abelian Monoids</h2>
+
+
+<div class="paragraph"> </div>
+
+ Some equalities hold in the restricted context of abelian (a.k.a. commutative)
+ monoids. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="M_given.Power_of_op" class="idref" href="#M_given.Power_of_op"><span class="id" title="section">Power_of_op</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Context</span>  {<a id="AM:96" class="idref" href="#AM:96"><span class="id" title="binder">AM</span></a>:<a class="idref" href="additions.Monoid_def.html#Abelian_EMonoid"><span class="id" title="class">Abelian_EMonoid</span></a> <a class="idref" href="additions.Pow_variant.html#M_given.M"><span class="id" title="variable">M</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="power_of_mult" class="idref" href="#power_of_mult"><span class="id" title="lemma">power_of_mult</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a> <a id="x:98" class="idref" href="#x:98"><span class="id" title="binder">x</span></a> <a id="y:99" class="idref" href="#y:99"><span class="id" title="binder">y</span></a>,  <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Pow_variant.html#x:98"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Pow_variant.html#y:99"><span class="id" title="variable">y</span></a><a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">)</span></a>  <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390"><span class="id" title="notation">==</span></a>  <a class="idref" href="additions.Pow_variant.html#x:98"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3"><span class="id" title="notation">×</span></a>  <a class="idref" href="additions.Pow_variant.html#y:99"><span class="id" title="variable">y</span></a> <a class="idref" href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">^</span></a> <a class="idref" href="additions.Pow_variant.html#n:97"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Pow_variant.html#M_given.Power_of_op"><span class="id" title="section">Power_of_op</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Pow_variant.html#M_given"><span class="id" title="section">M_given</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="765592cb239fbb605ff794a9833fef51" class="idref" href="#765592cb239fbb605ff794a9833fef51"><span class="id" title="notation">&quot;</span></a>^" := <a class="idref" href="additions.Pow_variant.html#power"><span class="id" title="definition">power</span></a> : <span class="id" title="var">M_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">monoid_simpl</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">generalize</span> (<a class="idref" href="additions.Pow_variant.html#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span>; <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> ( <span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a> ) || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a>  ) || <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a> )).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">power_simpl</span> <span class="id" title="var">M</span> := <span class="id" title="tactic">generalize</span> (<a class="idref" href="additions.Pow_variant.html#Eop_proper"><span class="id" title="instance">Eop_proper</span></a> <span class="id" title="var">M</span>); <span class="id" title="tactic">intro</span>; <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> ( <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_left"><span class="id" title="method">Eone_left</span></a>  ||  <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eone_right"><span class="id" title="method">Eone_right</span></a> ||  <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Monoid_def.html#Eop_assoc"><span class="id" title="method">Eop_assoc</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|| <span class="id" title="tactic">rewrite</span> <a class="idref" href="additions.Pow_variant.html#power_of_plus"><span class="id" title="lemma">power_of_plus</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Strategies.html b/theories/html/additions.Strategies.html
new file mode 100644
index 00000000..cb439291
--- /dev/null
+++ b/theories/html/additions.Strategies.html
@@ -0,0 +1,69 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Strategies</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Strategies</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+A <i>strategy</i> is any function  of type <span class="inlinecode"><span class="id" title="var">positive</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">positive</span></span> 
+such that, if , then .
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a> <a class="idref" href="additions.Pow.html#"><span class="id" title="library">Pow</span></a> <a class="idref" href="additions.Compatibility.html#"><span class="id" title="library">Compatibility</span></a> <a class="idref" href="additions.More_on_positive.html#"><span class="id" title="library">More_on_positive</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+<span class="id" title="keyword">Class</span> <a id="Strategy" class="idref" href="#Strategy"><span class="id" title="record">Strategy</span></a> (<a id="gamma:1" class="idref" href="#gamma:1"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>):=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;<a id="gamma_lt" class="idref" href="#gamma_lt"><span class="id" title="projection">gamma_lt</span></a> :<span class="id" title="keyword">∀</span> <a id="p:3" class="idref" href="#p:3"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#p:3"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Strategies.html#gamma:1"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="additions.Strategies.html#p:3"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#p:3"><span class="id" title="variable">p</span></a>;<br/>
+&nbsp;&nbsp;<a id="gamma_gt" class="idref" href="#gamma_gt"><span class="id" title="projection">gamma_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#gamma:1"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="additions.Strategies.html#p:5"><span class="id" title="variable">p</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">gamma_bounds</span> <span class="id" title="var">gamma</span> <span class="id" title="var">i</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">H1</span> : 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">gamma</span> <span class="id" title="var">i</span>) <span class="id" title="tactic">by</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="additions.Strategies.html#gamma_gt"><span class="id" title="method">gamma_gt</span></a>;<span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">chains</span>);<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">H2</span> : <span class="id" title="var">gamma</span> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">i</span>) <span class="id" title="tactic">by</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="additions.Strategies.html#gamma_lt"><span class="id" title="method">gamma_lt</span></a>; <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">chains</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div_gamma_pos" class="idref" href="#div_gamma_pos"><span class="id" title="lemma">div_gamma_pos</span></a> {<a id="gamma:7" class="idref" href="#gamma:7"><span class="id" title="binder">gamma</span></a>}{<a id="Hgamma:8" class="idref" href="#Hgamma:8"><span class="id" title="binder">Hgamma</span></a> : <a class="idref" href="additions.Strategies.html#Strategy"><span class="id" title="class">Strategy</span></a> <a class="idref" href="additions.Strategies.html#gamma:7"><span class="id" title="variable">gamma</span></a>}<br/>
+: <span class="id" title="keyword">∀</span> (<a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) <a id="q:10" class="idref" href="#q:10"><span class="id" title="binder">q</span></a> <a id="r:11" class="idref" href="#r:11"><span class="id" title="binder">r</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos_div_eucl"><span class="id" title="definition">N.pos_div_eucl</span></a> <a class="idref" href="additions.Strategies.html#p:9"><span class="id" title="variable">p</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> (<a class="idref" href="additions.Strategies.html#gamma:7"><span class="id" title="variable">gamma</span></a> <a class="idref" href="additions.Strategies.html#p:9"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="additions.Strategies.html#q:10"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="additions.Strategies.html#r:11"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#::positive_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#p:9"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="additions.Strategies.html#q:10"><span class="id" title="variable">q</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Trace_exercise.html b/theories/html/additions.Trace_exercise.html
new file mode 100644
index 00000000..16228079
--- /dev/null
+++ b/theories/html/additions.Trace_exercise.html
@@ -0,0 +1,99 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Trace_exercise</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Trace_exercise</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Solution to an exercise 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="additions.Addition_Chains.html#"><span class="id" title="library">Addition_Chains</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.PArith.html#"><span class="id" title="library">PArith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fusion" class="idref" href="#fusion"><span class="id" title="definition">fusion</span></a>  {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>} (<a id="compare:2" class="idref" href="#compare:2"><span class="id" title="binder">compare</span></a> : <a class="idref" href="additions.Trace_exercise.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Trace_exercise.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a>)(<a id="l1:3" class="idref" href="#l1:3"><span class="id" title="binder">l1</span></a> <a id="l2:4" class="idref" href="#l2:4"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="additions.Trace_exercise.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="aux:10" class="idref" href="#aux:10"><span class="id" title="binder">aux</span></a> <a id="l2:6" class="idref" href="#l2:6"><span class="id" title="binder">l2</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">l2</span>}:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Trace_exercise.html#l1:3"><span class="id" title="variable">l1</span></a>,<a class="idref" href="additions.Trace_exercise.html#l2:6"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>,<span class="id" title="var">_</span> ⇒ <a class="idref" href="additions.Trace_exercise.html#l2:6"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="additions.Trace_exercise.html#l1:3"><span class="id" title="variable">l1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l1'</span>,<span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l2'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Trace_exercise.html#compare:2"><span class="id" title="variable">compare</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <span class="id" title="var">a2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.Trace_exercise.html#aux:7"><span class="id" title="variable">aux</span></a>  <span class="id" title="var">l2'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.Trace_exercise.html#fusion:5"><span class="id" title="definition">fusion</span></a> <a class="idref" href="additions.Trace_exercise.html#compare:2"><span class="id" title="variable">compare</span></a> <span class="id" title="var">l1'</span> <span class="id" title="var">l2'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.Trace_exercise.html#fusion:5"><span class="id" title="definition">fusion</span></a> <a class="idref" href="additions.Trace_exercise.html#compare:2"><span class="id" title="variable">compare</span></a> <span class="id" title="var">l1'</span> <a class="idref" href="additions.Trace_exercise.html#l2:6"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="additions.Trace_exercise.html#aux:10"><span class="id" title="variable">aux</span></a> <a class="idref" href="additions.Trace_exercise.html#l2:4"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">positive_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Traces with full information 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="info" class="idref" href="#info"><span class="id" title="inductive">info</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="Init" class="idref" href="#Init"><span class="id" title="constructor">Init</span></a><br/>
+| <a id="Add" class="idref" href="#Add"><span class="id" title="constructor">Add</span></a> (<a id="p:13" class="idref" href="#p:13"><span class="id" title="binder">p</span></a> <a id="q:14" class="idref" href="#q:14"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="trace_compare" class="idref" href="#trace_compare"><span class="id" title="definition">trace_compare</span></a> (<a id="t:15" class="idref" href="#t:15"><span class="id" title="binder">t</span></a> <a id="t':16" class="idref" href="#t':16"><span class="id" title="binder">t'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Trace_exercise.html#info"><span class="id" title="inductive">info</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Trace_exercise.html#t:15"><span class="id" title="variable">t</span></a>, <a class="idref" href="additions.Trace_exercise.html#t':16"><span class="id" title="variable">t'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.compare"><span class="id" title="definition">Pos.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="trace_mult" class="idref" href="#trace_mult"><span class="id" title="definition">trace_mult</span></a> (<a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a> <a id="l':20" class="idref" href="#l':20"><span class="id" title="binder">l'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.Trace_exercise.html#info"><span class="id" title="inductive">info</span></a>)):=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.Trace_exercise.html#l:19"><span class="id" title="variable">l</span></a>, <a class="idref" href="additions.Trace_exercise.html#l':20"><span class="id" title="variable">l'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <span class="id" title="var">_</span> | <span class="id" title="var">_</span>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="additions.Trace_exercise.html#l:19"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l1</span>),(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l'1</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#50efadf7a9f9d41adb2b162cbba3bda7"><span class="id" title="notation">+</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.Trace_exercise.html#Add"><span class="id" title="constructor">Add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="additions.Trace_exercise.html#fusion"><span class="id" title="definition">fusion</span></a> <a class="idref" href="additions.Trace_exercise.html#trace_compare"><span class="id" title="definition">trace_compare</span></a> <a class="idref" href="additions.Trace_exercise.html#l:19"><span class="id" title="variable">l</span></a> <a class="idref" href="additions.Trace_exercise.html#l':20"><span class="id" title="variable">l'</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="chain_trace" class="idref" href="#chain_trace"><span class="id" title="definition">chain_trace</span></a> <a id="c:23" class="idref" href="#c:23"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="additions.Addition_Chains.html#chain_execute"><span class="id" title="definition">chain_execute</span></a> <a class="idref" href="additions.Trace_exercise.html#c:23"><span class="id" title="variable">c</span></a>  <a class="idref" href="additions.Trace_exercise.html#trace_mult"><span class="id" title="definition">trace_mult</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="additions.Trace_exercise.html#Init"><span class="id" title="constructor">Init</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="exponents" class="idref" href="#exponents"><span class="id" title="definition">exponents</span></a> <a id="c:24" class="idref" href="#c:24"><span class="id" title="binder">c</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="additions.Trace_exercise.html#chain_trace"><span class="id" title="definition">chain_trace</span></a> <a class="idref" href="additions.Trace_exercise.html#c:24"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/additions.Wf_transparent.html b/theories/html/additions.Wf_transparent.html
new file mode 100644
index 00000000..cf65c156
--- /dev/null
+++ b/theories/html/additions.Wf_transparent.html
@@ -0,0 +1,69 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.Wf_transparent</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.Wf_transparent</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Transparent versions of wf_incl and wf_inverse_image 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#"><span class="id" title="library">Relation_Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wf_incl_transparent" class="idref" href="#wf_incl_transparent"><span class="id" title="lemma">wf_incl_transparent</span></a>  :<br/>
+<span class="id" title="keyword">∀</span> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="R1:2" class="idref" href="#R1:2"><span class="id" title="binder">R1</span></a> <a id="R2:3" class="idref" href="#R2:3"><span class="id" title="binder">R2</span></a> : <a class="idref" href="additions.Wf_transparent.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Wf_transparent.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
+<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#inclusion"><span class="id" title="definition">Relation_Definitions.inclusion</span></a> <a class="idref" href="additions.Wf_transparent.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="additions.Wf_transparent.html#R1:2"><span class="id" title="variable">R1</span></a> <a class="idref" href="additions.Wf_transparent.html#R2:3"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="additions.Wf_transparent.html#R2:3"><span class="id" title="variable">R2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="additions.Wf_transparent.html#R1:2"><span class="id" title="variable">R1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Inverse_Image_transp" class="idref" href="#Inverse_Image_transp"><span class="id" title="section">Inverse_Image_transp</span></a>. 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Inverse_Image_transp.A" class="idref" href="#Inverse_Image_transp.A"><span class="id" title="variable">A</span></a> <a id="Inverse_Image_transp.B" class="idref" href="#Inverse_Image_transp.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Inverse_Image_transp.R" class="idref" href="#Inverse_Image_transp.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Inverse_Image_transp.f" class="idref" href="#Inverse_Image_transp.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Inverse_Image_transp.Rof" class="idref" href="#Inverse_Image_transp.Rof"><span class="id" title="variable">Rof</span></a> (<a id="x:8" class="idref" href="#x:8"><span class="id" title="binder">x</span></a> <a id="y:9" class="idref" href="#y:9"><span class="id" title="binder">y</span></a>:<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Wf_transparent.html#x:8"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Wf_transparent.html#y:9"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Acc_lemma" class="idref" href="#Acc_lemma"><span class="id" title="lemma">Acc_lemma</span></a> : <span class="id" title="keyword">∀</span> <a id="y:10" class="idref" href="#y:10"><span class="id" title="binder">y</span></a>:<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.B"><span class="id" title="variable">B</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.R"><span class="id" title="variable">R</span></a> <a class="idref" href="additions.Wf_transparent.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="additions.Wf_transparent.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Wf_transparent.html#x:11"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.Rof"><span class="id" title="variable">Rof</span></a> <a class="idref" href="additions.Wf_transparent.html#x:11"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Acc_inverse_image" class="idref" href="#Acc_inverse_image"><span class="id" title="lemma">Acc_inverse_image</span></a> : <span class="id" title="keyword">∀</span> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a>:<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.Wf_transparent.html#x:12"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.Rof"><span class="id" title="variable">Rof</span></a> <a class="idref" href="additions.Wf_transparent.html#x:12"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="wf_inverse_image_transparent" class="idref" href="#wf_inverse_image_transparent"><span class="id" title="lemma">wf_inverse_image_transparent</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp.Rof"><span class="id" title="variable">Rof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.Wf_transparent.html#Inverse_Image_transp"><span class="id" title="section">Inverse_Image_transp</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>additions.fib</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library additions.fib</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+author Yves Bertot, Inria  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.extraction.Extraction.html#"><span class="id" title="library">Extraction</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">all_algebra</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fib" class="idref" href="#fib"><span class="id" title="definition">fib</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.fib.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">as</span> <span class="id" title="var">p'</span>) <span class="id" title="keyword">then</span> <a class="idref" href="additions.fib.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">p</span> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">else</span> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fibt" class="idref" href="#fibt"><span class="id" title="definition">fibt</span></a> (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="acc1:5" class="idref" href="#acc1:5"><span class="id" title="binder">acc1</span></a> <a id="acc2:6" class="idref" href="#acc2:6"><span class="id" title="binder">acc2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.fib.html#n:4"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">then</span> <a class="idref" href="additions.fib.html#fibt:7"><span class="id" title="definition">fibt</span></a> <span class="id" title="var">p</span> (<a class="idref" href="additions.fib.html#acc2:6"><span class="id" title="variable">acc2</span></a> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#acc1:5"><span class="id" title="variable">acc1</span></a>) <a class="idref" href="additions.fib.html#acc1:5"><span class="id" title="variable">acc1</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="additions.fib.html#acc1:5"><span class="id" title="variable">acc1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Zfibt" class="idref" href="#Zfibt"><span class="id" title="definition">Zfibt</span></a> (<a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="acc1:10" class="idref" href="#acc1:10"><span class="id" title="binder">acc1</span></a> <a id="acc2:11" class="idref" href="#acc2:11"><span class="id" title="binder">acc2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="additions.fib.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">then</span> <a class="idref" href="additions.fib.html#Zfibt:12"><span class="id" title="definition">Zfibt</span></a> <span class="id" title="var">p</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> <a class="idref" href="additions.fib.html#acc2:11"><span class="id" title="variable">acc2</span></a> <a class="idref" href="additions.fib.html#acc1:10"><span class="id" title="variable">acc1</span></a>) <a class="idref" href="additions.fib.html#acc1:10"><span class="id" title="variable">acc1</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="additions.fib.html#acc1:10"><span class="id" title="variable">acc1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fibt_aux" class="idref" href="#fibt_aux"><span class="id" title="lemma">fibt_aux</span></a> (<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> <a id="k:15" class="idref" href="#k:15"><span class="id" title="binder">k</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.fib.html#fibt"><span class="id" title="definition">fibt</span></a> <a class="idref" href="additions.fib.html#n:14"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#k:15"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span>) (<a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#k:15"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#n:14"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#k:15"><span class="id" title="variable">k</span></a><span class="id" title="notation">).+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fibtP" class="idref" href="#fibtP"><span class="id" title="lemma">fibtP</span></a> (<a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="additions.fib.html#fibt"><span class="id" title="definition">fibt</span></a> <a class="idref" href="additions.fib.html#n:16"><span class="id" title="variable">n</span></a> 1 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#n:16"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="bits" class="idref" href="#bits"><span class="id" title="definition">bits</span></a> <a id="p:17" class="idref" href="#p:17"><span class="id" title="binder">p</span></a> <a id="acc:18" class="idref" href="#acc:18"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#p:17"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="notation">::</span> <a class="idref" href="additions.fib.html#acc:18"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.fib.html#bits:19"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:17"><span class="id" title="variable">p</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="notation">::</span> <a class="idref" href="additions.fib.html#acc:18"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.fib.html#bits:19"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:17"><span class="id" title="variable">p</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="notation">::</span> <a class="idref" href="additions.fib.html#acc:18"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bits_cat" class="idref" href="#bits_cat"><span class="id" title="lemma">bits_cat</span></a> <a id="p:21" class="idref" href="#p:21"><span class="id" title="binder">p</span></a> <a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a> : <a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.fib.html#a:22"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:21"><span class="id" title="variable">p</span></a> <span class="id" title="notation">[::]</span> <span class="id" title="notation">++</span> <a class="idref" href="additions.fib.html#a:22"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bitsP" class="idref" href="#bitsP"><span class="id" title="lemma">bitsP</span></a> <a id="p:23" class="idref" href="#p:23"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.fib.html#p:23"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="i:26" class="idref" href="#i:26"><span class="id" title="binder">i</span></a> <span class="id" title="notation">&lt;</span> <span class="id" title="definition">size</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:23"><span class="id" title="variable">p</span></a> <span class="id" title="notation">[::]</span>)<span class="id" title="notation">)</span> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:23"><span class="id" title="variable">p</span></a> <span class="id" title="notation">[::]</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="definition">size</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:23"><span class="id" title="variable">p</span></a> <span class="id" title="notation">[::]</span>) <span class="id" title="notation">-</span> 1 <span class="id" title="notation">-</span> <a class="idref" href="additions.fib.html#i:24"><span class="id" title="variable">i</span></a>) <span class="id" title="notation">×</span> 2 <span class="id" title="notation">^</span> <a class="idref" href="additions.fib.html#i:24"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="with_matrices" class="idref" href="#with_matrices"><span class="id" title="section">with_matrices</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="with_matrices.R" class="idref" href="#with_matrices.R"><span class="id" title="variable">R</span></a> : <span class="id" title="abbreviation">comUnitRingType</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">GRing.Theory</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ZtoR" class="idref" href="#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<a id="x:32" class="idref" href="#x:32"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="additions.fib.html#x:32"><span class="id" title="variable">x</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">-</span><span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.abs_nat"><span class="id" title="definition">Z.abs_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#::Z_scope:'-'_x"><span class="id" title="notation">-</span></a><a class="idref" href="additions.fib.html#x:32"><span class="id" title="variable">x</span></a>)<span class="id" title="notation">)%:</span><span class="id" title="notation">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.abs_nat"><span class="id" title="definition">Z.abs_nat</span></a> <a class="idref" href="additions.fib.html#x:32"><span class="id" title="variable">x</span></a><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ZtoRD" class="idref" href="#ZtoRD"><span class="id" title="lemma">ZtoRD</span></a> (<a id="x:33" class="idref" href="#x:33"><span class="id" title="binder">x</span></a> <a id="y:34" class="idref" href="#y:34"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<a class="idref" href="additions.fib.html#x:33"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.fib.html#y:34"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#x:33"><span class="id" title="variable">x</span></a> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#y:34"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ZtoRM" class="idref" href="#ZtoRM"><span class="id" title="lemma">ZtoRM</span></a> <a id="x:37" class="idref" href="#x:37"><span class="id" title="binder">x</span></a> <a id="y:38" class="idref" href="#y:38"><span class="id" title="binder">y</span></a> : <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<a class="idref" href="additions.fib.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#y:38"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#x:37"><span class="id" title="variable">x</span></a> <span class="id" title="notation">×</span> <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#y:38"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iter_mul" class="idref" href="#iter_mul"><span class="id" title="lemma">iter_mul</span></a> (<a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="m:40" class="idref" href="#m:40"><span class="id" title="binder">m</span></a> <a id="v:41" class="idref" href="#v:41"><span class="id" title="binder">v</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">eta</span></a> <span class="id" title="notation">*%</span><span class="id" title="notation">R</span> <span class="id" title="notation">m</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">]</span></a> <a class="idref" href="additions.fib.html#v:41"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#m:40"><span class="id" title="variable">m</span></a> <span class="id" title="notation">^+</span> <a class="idref" href="additions.fib.html#n:39"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="additions.fib.html#v:41"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fibm" class="idref" href="#fibm"><span class="id" title="definition">fibm</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:42" class="idref" href="#i:42"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:43" class="idref" href="#j:43"><span class="id" title="binder">j</span></a><span class="id" title="notation">)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#i:42"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#j:43"><span class="id" title="variable">j</span></a> <span class="id" title="notation">==</span> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> 0%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> 1%<span class="id" title="var">R</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fibmP" class="idref" href="#fibmP"><span class="id" title="lemma">fibmP</span></a> <a id="n:44" class="idref" href="#n:44"><span class="id" title="binder">n</span></a> : <a class="idref" href="additions.fib.html#fibm"><span class="id" title="definition">fibm</span></a> <span class="id" title="notation">^+</span> <a class="idref" href="additions.fib.html#n:44"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:46" class="idref" href="#j:46"><span class="id" title="binder">j</span></a><span class="id" title="notation">)</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#i:45"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#j:46"><span class="id" title="variable">j</span></a> <span class="id" title="notation">==</span> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#n:44"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a>(<span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#i:45"><span class="id" title="variable">i</span></a> <span class="id" title="notation">+</span> <span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#j:46"><span class="id" title="variable">j</span></a> <span class="id" title="notation">==</span> 1)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#n:44"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="additions.fib.html#n:44"><span class="id" title="variable">n</span></a><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m4lval" class="idref" href="#m4lval"><span class="id" title="definition">m4lval</span></a> (<a id="l:50" class="idref" href="#l:50"><span class="id" title="binder">l</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a> <a id="j:52" class="idref" href="#j:52"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l:50"><span class="id" title="variable">l</span></a> (<a class="idref" href="additions.fib.html#j:52"><span class="id" title="variable">j</span></a> <span class="id" title="notation">+</span> 2 <span class="id" title="notation">×</span> <a class="idref" href="additions.fib.html#i:51"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m4lmx" class="idref" href="#m4lmx"><span class="id" title="definition">m4lmx</span></a> (<a id="l:53" class="idref" href="#l:53"><span class="id" title="binder">l</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:54" class="idref" href="#i:54"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:55" class="idref" href="#j:55"><span class="id" title="binder">j</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l:53"><span class="id" title="variable">l</span></a> <a class="idref" href="additions.fib.html#i:54"><span class="id" title="variable">i</span></a> <a class="idref" href="additions.fib.html#j:55"><span class="id" title="variable">j</span></a>)<span class="id" title="notation">)</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m4lmul" class="idref" href="#m4lmul"><span class="id" title="definition">m4lmul</span></a> (<a id="l1:56" class="idref" href="#l1:56"><span class="id" title="binder">l1</span></a> <a id="l2:57" class="idref" href="#l2:57"><span class="id" title="binder">l2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 0 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 0 0) (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 0 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 1 0)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span><br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 0 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 0 1) (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 0 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 1 1)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span><br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 1 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 0 0) (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 1 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 1 0)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span><br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 1 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 0 1) (<a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l1:56"><span class="id" title="variable">l1</span></a> 1 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#m4lval"><span class="id" title="definition">m4lval</span></a> <a class="idref" href="additions.fib.html#l2:57"><span class="id" title="variable">l2</span></a> 1 1)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="m4lfib" class="idref" href="#m4lfib"><span class="id" title="definition">m4lfib</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">[::</span> 1<span class="id" title="notation">;</span> 1<span class="id" title="notation">;</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1<span class="id" title="notation">;</span> 0<span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m3lmx" class="idref" href="#m3lmx"><span class="id" title="definition">m3lmx</span></a> (<a id="l:58" class="idref" href="#l:58"><span class="id" title="binder">l</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:59" class="idref" href="#i:59"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:60" class="idref" href="#j:60"><span class="id" title="binder">j</span></a><span class="id" title="notation">)</span><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.fib.html#i:59"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> <a class="idref" href="additions.fib.html#j:60"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l:58"><span class="id" title="variable">l</span></a> <a class="idref" href="additions.fib.html#i:59"><span class="id" title="variable">i</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l:58"><span class="id" title="variable">l</span></a> 2).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m3lmul" class="idref" href="#m3lmul"><span class="id" title="definition">m3lmul</span></a> (<a id="l1:61" class="idref" href="#l1:61"><span class="id" title="binder">l1</span></a> <a id="l2:62" class="idref" href="#l2:62"><span class="id" title="binder">l2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 0) (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 2)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span><br/>
+&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 2) (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 1)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span><br/>
+<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 2) (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:61"><span class="id" title="variable">l1</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:62"><span class="id" title="variable">l2</span></a> 1)<span class="id" title="notation">)</span> <span class="id" title="notation">::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m3lfib" class="idref" href="#m3lfib"><span class="id" title="definition">m3lfib</span></a> := <span class="id" title="notation">[::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pos"><span class="id" title="abbreviation">Z.pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a><span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pos"><span class="id" title="abbreviation">Z.pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fastexp" class="idref" href="#fastexp"><span class="id" title="definition">fastexp</span></a> (<a id="m:63" class="idref" href="#m:63"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="p:64" class="idref" href="#p:64"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#p:64"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.fib.html#m:63"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">p</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="r:67" class="idref" href="#r:67"><span class="id" title="binder">r</span></a> := <a class="idref" href="additions.fib.html#fastexp:65"><span class="id" title="definition">fastexp</span></a> <a class="idref" href="additions.fib.html#m:63"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:64"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#r:67"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.fib.html#r:67"><span class="id" title="variable">r</span></a>) <a class="idref" href="additions.fib.html#m:63"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">p</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="r:68" class="idref" href="#r:68"><span class="id" title="binder">r</span></a> := <a class="idref" href="additions.fib.html#fastexp:65"><span class="id" title="definition">fastexp</span></a> <a class="idref" href="additions.fib.html#m:63"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:64"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#r:68"><span class="id" title="variable">r</span></a> <a class="idref" href="additions.fib.html#r:68"><span class="id" title="variable">r</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fastexp2" class="idref" href="#fastexp2"><span class="id" title="definition">fastexp2</span></a> (<a id="m:69" class="idref" href="#m:69"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="p:70" class="idref" href="#p:70"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<a id="acc:71" class="idref" href="#acc:71"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#p:70"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.fib.html#acc:71"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.fib.html#fastexp2:72"><span class="id" title="definition">fastexp2</span></a> <a class="idref" href="additions.fib.html#m:69"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:70"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#acc:71"><span class="id" title="variable">acc</span></a> <a class="idref" href="additions.fib.html#acc:71"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="additions.fib.html#fastexp2:72"><span class="id" title="definition">fastexp2</span></a> <a class="idref" href="additions.fib.html#m:69"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:70"><span class="id" title="variable">p</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#m:69"><span class="id" title="variable">m</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#acc:71"><span class="id" title="variable">acc</span></a> <a class="idref" href="additions.fib.html#acc:71"><span class="id" title="variable">acc</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fastexp3" class="idref" href="#fastexp3"><span class="id" title="definition">fastexp3</span></a> {<a id="A:74" class="idref" href="#A:74"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} (<a id="mul:75" class="idref" href="#mul:75"><span class="id" title="binder">mul</span></a> : <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;(<a id="m:76" class="idref" href="#m:76"><span class="id" title="binder">m</span></a> : <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">f</span> (<a id="l:77" class="idref" href="#l:77"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="acc:78" class="idref" href="#acc:78"><span class="id" title="binder">acc</span></a> : <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a>) : <a class="idref" href="additions.fib.html#A:74"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#l:77"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="additions.fib.html#acc:78"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="notation">::</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="additions.fib.html#f:79"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.fib.html#l:77"><span class="id" title="variable">l</span></a> (<a class="idref" href="additions.fib.html#mul:75"><span class="id" title="variable">mul</span></a> (<a class="idref" href="additions.fib.html#mul:75"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#acc:78"><span class="id" title="variable">acc</span></a> <a class="idref" href="additions.fib.html#acc:78"><span class="id" title="variable">acc</span></a>) <a class="idref" href="additions.fib.html#m:76"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="notation">::</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="additions.fib.html#f:79"><span class="id" title="variable">f</span></a> <a class="idref" href="additions.fib.html#l:77"><span class="id" title="variable">l</span></a> (<a class="idref" href="additions.fib.html#mul:75"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#acc:78"><span class="id" title="variable">acc</span></a> <a class="idref" href="additions.fib.html#acc:78"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="my_pow" class="idref" href="#my_pow"><span class="id" title="definition">my_pow</span></a> {<a id="A:81" class="idref" href="#A:81"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} (<a id="mul:82" class="idref" href="#mul:82"><span class="id" title="binder">mul</span></a> : <a class="idref" href="additions.fib.html#A:81"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:81"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:81"><span class="id" title="variable">A</span></a>) (<a id="m:83" class="idref" href="#m:83"><span class="id" title="binder">m</span></a> : <a class="idref" href="additions.fib.html#A:81"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:84" class="idref" href="#p:84"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) <br/>
+&nbsp;&nbsp;: <a class="idref" href="additions.fib.html#A:81"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#fastexp3"><span class="id" title="definition">fastexp3</span></a> <a class="idref" href="additions.fib.html#mul:82"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#m:83"><span class="id" title="variable">m</span></a> (<span class="id" title="definition">behead</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:84"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="additions.fib.html#m:83"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m3lid" class="idref" href="#m3lid"><span class="id" title="definition">m3lid</span></a> := <span class="id" title="notation">[::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pos"><span class="id" title="abbreviation">Z.pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a><span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a><span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.pos"><span class="id" title="abbreviation">Z.pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="slowexp" class="idref" href="#slowexp"><span class="id" title="definition">slowexp</span></a> (<a id="m:85" class="idref" href="#m:85"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) <a id="p:86" class="idref" href="#p:86"><span class="id" title="binder">p</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.iter"><span class="id" title="definition">Pos.iter</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#m:85"><span class="id" title="variable">m</span></a>) <a class="idref" href="additions.fib.html#m3lid"><span class="id" title="definition">m3lid</span></a> <a class="idref" href="additions.fib.html#p:86"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="binary_power_mult" class="idref" href="#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a> (<a id="mul:88" class="idref" href="#mul:88"><span class="id" title="binder">mul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">f</span> (<a id="x:89" class="idref" href="#x:89"><span class="id" title="binder">x</span></a> <a id="a:90" class="idref" href="#a:90"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="p:91" class="idref" href="#p:91"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#p:91"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.fib.html#mul:88"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.fib.html#f:92"><span class="id" title="variable">f</span></a> (<a class="idref" href="additions.fib.html#mul:88"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.fib.html#a:90"><span class="id" title="variable">a</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.fib.html#f:92"><span class="id" title="variable">f</span></a> (<a class="idref" href="additions.fib.html#mul:88"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a>) (<a class="idref" href="additions.fib.html#mul:88"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="additions.fib.html#x:89"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fastexp4" class="idref" href="#fastexp4"><span class="id" title="definition">fastexp4</span></a> (<a id="mul:94" class="idref" href="#mul:94"><span class="id" title="binder">mul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">f</span> (<a id="x:95" class="idref" href="#x:95"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="p:96" class="idref" href="#p:96"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="additions.fib.html#p:96"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.fib.html#f:97"><span class="id" title="variable">f</span></a> (<a class="idref" href="additions.fib.html#mul:94"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a>) <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="additions.fib.html#binary_power_mult"><span class="id" title="definition">binary_power_mult</span></a> <a class="idref" href="additions.fib.html#mul:94"><span class="id" title="variable">mul</span></a> (<a class="idref" href="additions.fib.html#mul:94"><span class="id" title="variable">mul</span></a> <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a> <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a>) <a class="idref" href="additions.fib.html#x:95"><span class="id" title="variable">x</span></a> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m3lmulP" class="idref" href="#m3lmulP"><span class="id" title="lemma">m3lmulP</span></a> <a id="l1:99" class="idref" href="#l1:99"><span class="id" title="binder">l1</span></a> <a id="l2:100" class="idref" href="#l2:100"><span class="id" title="binder">l2</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">GRing.comm</span> (<a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#l1:99"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#l2:100"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> (<a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#l1:99"><span class="id" title="variable">l1</span></a> <a class="idref" href="additions.fib.html#l2:100"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#l1:99"><span class="id" title="variable">l1</span></a> <span class="id" title="notation">×</span> <a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#l2:100"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m3lfibP" class="idref" href="#m3lfibP"><span class="id" title="lemma">m3lfibP</span></a> : <a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#m3lfib"><span class="id" title="definition">m3lfib</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#fibm"><span class="id" title="definition">fibm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iter_comm" class="idref" href="#iter_comm"><span class="id" title="lemma">iter_comm</span></a> {<a id="A:101" class="idref" href="#A:101"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} (<a id="op:102" class="idref" href="#op:102"><span class="id" title="binder">op</span></a> : <a class="idref" href="additions.fib.html#A:101"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:101"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:101"><span class="id" title="variable">A</span></a>) (<a id="e1:103" class="idref" href="#e1:103"><span class="id" title="binder">e1</span></a> <a id="e2:104" class="idref" href="#e2:104"><span class="id" title="binder">e2</span></a> : <a class="idref" href="additions.fib.html#A:101"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;(<a id="assoc:105" class="idref" href="#assoc:105"><span class="id" title="binder">assoc</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a>)(<a id="cm:106" class="idref" href="#cm:106"><span class="id" title="binder">cm</span></a> : <a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a> <a class="idref" href="additions.fib.html#e2:104"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e2:104"><span class="id" title="variable">e2</span></a> <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a>) <a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a> (<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#n:107"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a>) <a class="idref" href="additions.fib.html#e2:104"><span class="id" title="variable">e2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> (<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#n:107"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.fib.html#op:102"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a>) <a class="idref" href="additions.fib.html#e2:104"><span class="id" title="variable">e2</span></a>) <a class="idref" href="additions.fib.html#e1:103"><span class="id" title="variable">e1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iter_combine" class="idref" href="#iter_combine"><span class="id" title="lemma">iter_combine</span></a>  {<a id="A:108" class="idref" href="#A:108"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} (<a id="op:109" class="idref" href="#op:109"><span class="id" title="binder">op</span></a> : <a class="idref" href="additions.fib.html#A:108"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:108"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:108"><span class="id" title="variable">A</span></a>) (<a id="e1:110" class="idref" href="#e1:110"><span class="id" title="binder">e1</span></a> <a id="e2:111" class="idref" href="#e2:111"><span class="id" title="binder">e2</span></a> : <a class="idref" href="additions.fib.html#A:108"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;(<a id="assoc:112" class="idref" href="#assoc:112"><span class="id" title="binder">assoc</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a>)(<a id="cm:113" class="idref" href="#cm:113"><span class="id" title="binder">cm</span></a> : <a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:110"><span class="id" title="variable">e1</span></a> <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a> <a class="idref" href="additions.fib.html#e1:110"><span class="id" title="variable">e1</span></a>) <a id="n:114" class="idref" href="#n:114"><span class="id" title="binder">n</span></a> <a id="m:115" class="idref" href="#m:115"><span class="id" title="binder">m</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> (<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#n:114"><span class="id" title="variable">n</span></a> (<a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:110"><span class="id" title="variable">e1</span></a>) <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a>) (<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#m:115"><span class="id" title="variable">m</span></a> (<a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:110"><span class="id" title="variable">e1</span></a>) <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="definition">iter</span> (<a class="idref" href="additions.fib.html#n:114"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#m:115"><span class="id" title="variable">m</span></a>) (<a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e1:110"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="additions.fib.html#op:109"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a> <a class="idref" href="additions.fib.html#e2:111"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fastexp3P" class="idref" href="#fastexp3P"><span class="id" title="lemma">fastexp3P</span></a> {<a id="A:116" class="idref" href="#A:116"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} (<a id="op:117" class="idref" href="#op:117"><span class="id" title="binder">op</span></a> : <a class="idref" href="additions.fib.html#A:116"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:116"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="additions.fib.html#A:116"><span class="id" title="variable">A</span></a>) (<a id="id:118" class="idref" href="#id:118"><span class="id" title="binder">id</span></a> <a id="e:119" class="idref" href="#e:119"><span class="id" title="binder">e</span></a> : <a class="idref" href="additions.fib.html#A:116"><span class="id" title="variable">A</span></a>) (<a id="h:120" class="idref" href="#h:120"><span class="id" title="binder">h</span></a> <a id="l:121" class="idref" href="#l:121"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;(<a id="v:122" class="idref" href="#v:122"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="assoc:123" class="idref" href="#assoc:123"><span class="id" title="binder">assoc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a>) (<a id="cm:124" class="idref" href="#cm:124"><span class="id" title="binder">cm</span></a> : <a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#id:118"><span class="id" title="variable">id</span></a> <a class="idref" href="additions.fib.html#e:119"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e:119"><span class="id" title="variable">e</span></a> <a class="idref" href="additions.fib.html#id:118"><span class="id" title="variable">id</span></a>)<br/>
+&nbsp;&nbsp;(<a id="idn:126" class="idref" href="#idn:126"><span class="id" title="binder">idn</span></a> : <span class="id" title="keyword">∀</span> <a id="a:125" class="idref" href="#a:125"><span class="id" title="binder">a</span></a>, <a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#id:118"><span class="id" title="variable">id</span></a> <a class="idref" href="additions.fib.html#a:125"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#a:125"><span class="id" title="variable">a</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#l:121"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="i:129" class="idref" href="#i:129"><span class="id" title="binder">i</span></a> <span class="id" title="notation">&lt;</span> <span class="id" title="definition">size</span> <a class="idref" href="additions.fib.html#v:122"><span class="id" title="variable">v</span></a><span class="id" title="notation">)</span> <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="additions.fib.html#v:122"><span class="id" title="variable">v</span></a> (<span class="id" title="definition">size</span> <a class="idref" href="additions.fib.html#v:122"><span class="id" title="variable">v</span></a> <span class="id" title="notation">-</span> 1 <span class="id" title="notation">-</span> <a class="idref" href="additions.fib.html#i:127"><span class="id" title="variable">i</span></a>) <span class="id" title="notation">×</span> 2 <span class="id" title="notation">^</span> <a class="idref" href="additions.fib.html#i:127"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="definition">iter</span> (<a class="idref" href="additions.fib.html#h:120"><span class="id" title="variable">h</span></a> <span class="id" title="notation">×</span> 2 <span class="id" title="notation">^</span> <span class="id" title="definition">size</span> <a class="idref" href="additions.fib.html#v:122"><span class="id" title="variable">v</span></a> <span class="id" title="notation">+</span> <a class="idref" href="additions.fib.html#l:121"><span class="id" title="variable">l</span></a>) (<a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e:119"><span class="id" title="variable">e</span></a>) <a class="idref" href="additions.fib.html#id:118"><span class="id" title="variable">id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#fastexp3"><span class="id" title="definition">fastexp3</span></a> <a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e:119"><span class="id" title="variable">e</span></a> <a class="idref" href="additions.fib.html#v:122"><span class="id" title="variable">v</span></a> (<span class="id" title="definition">iter</span> <a class="idref" href="additions.fib.html#h:120"><span class="id" title="variable">h</span></a> (<a class="idref" href="additions.fib.html#op:117"><span class="id" title="variable">op</span></a> <a class="idref" href="additions.fib.html#e:119"><span class="id" title="variable">e</span></a>) <a class="idref" href="additions.fib.html#id:118"><span class="id" title="variable">id</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="headbits" class="idref" href="#headbits"><span class="id" title="lemma">headbits</span></a> <a id="p:134" class="idref" href="#p:134"><span class="id" title="binder">p</span></a> <a id="l:135" class="idref" href="#l:135"><span class="id" title="binder">l</span></a> : <a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:134"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.fib.html#l:135"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="notation">::</span> <span class="id" title="definition">behead</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#p:134"><span class="id" title="variable">p</span></a> <a class="idref" href="additions.fib.html#l:135"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="my_powP" class="idref" href="#my_powP"><span class="id" title="lemma">my_powP</span></a> <a id="m:136" class="idref" href="#m:136"><span class="id" title="binder">m</span></a> <a id="p:137" class="idref" href="#p:137"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;@<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span> <span class="id" title="notation">*%</span><span class="id" title="notation">R</span> <a class="idref" href="additions.fib.html#m:136"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:137"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#m:136"><span class="id" title="variable">m</span></a> <span class="id" title="notation">^</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.fib.html#p:137"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="my_pow_m3lmul" class="idref" href="#my_pow_m3lmul"><span class="id" title="lemma">my_pow_m3lmul</span></a> <a id="m:139" class="idref" href="#m:139"><span class="id" title="binder">m</span></a> <a id="p:140" class="idref" href="#p:140"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> (<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#m:139"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:140"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <span class="id" title="notation">*%</span><span class="id" title="notation">R</span> (<a class="idref" href="additions.fib.html#m3lmx"><span class="id" title="definition">m3lmx</span></a> <a class="idref" href="additions.fib.html#m:139"><span class="id" title="variable">m</span></a>) <a class="idref" href="additions.fib.html#p:140"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m2lmul" class="idref" href="#m2lmul"><span class="id" title="definition">m2lmul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> <a id="l1:141" class="idref" href="#l1:141"><span class="id" title="binder">l1</span></a> <a id="l2:142" class="idref" href="#l2:142"><span class="id" title="binder">l2</span></a> ⇒<br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="a:143" class="idref" href="#a:143"><span class="id" title="binder">a</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:141"><span class="id" title="variable">l1</span></a> 0 <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="b:144" class="idref" href="#b:144"><span class="id" title="binder">b</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l1:141"><span class="id" title="variable">l1</span></a> 1 <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="c:145" class="idref" href="#c:145"><span class="id" title="binder">c</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:142"><span class="id" title="variable">l2</span></a> 0 <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="d:146" class="idref" href="#d:146"><span class="id" title="binder">d</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l2:142"><span class="id" title="variable">l2</span></a> 1 <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="notation">[::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#a:143"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">(</span></a><a class="idref" href="additions.fib.html#c:145"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.fib.html#d:146"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">)</span></a>) (<a class="idref" href="additions.fib.html#b:144"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#c:145"><span class="id" title="variable">c</span></a>) <span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.add"><span class="id" title="definition">Z.add</span></a> (<a class="idref" href="additions.fib.html#a:143"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#c:145"><span class="id" title="variable">c</span></a>) (<a class="idref" href="additions.fib.html#b:144"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#d3ea1afb7d81263dc23c4eb0da2a3c4e"><span class="id" title="notation">×</span></a> <a class="idref" href="additions.fib.html#d:146"><span class="id" title="variable">d</span></a>)<span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m2lmx" class="idref" href="#m2lmx"><span class="id" title="definition">m2lmx</span></a> (<a id="l:147" class="idref" href="#l:147"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="additions.fib.html#with_matrices.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_2</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="a:148" class="idref" href="#a:148"><span class="id" title="binder">a</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l:147"><span class="id" title="variable">l</span></a> 0 <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="b:149" class="idref" href="#b:149"><span class="id" title="binder">b</span></a> := <span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> <a class="idref" href="additions.fib.html#l:147"><span class="id" title="variable">l</span></a> 1 <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:150" class="idref" href="#i:150"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:151" class="idref" href="#j:151"><span class="id" title="binder">j</span></a><span class="id" title="notation">)</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#i:150"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="projection">val</span> <a class="idref" href="additions.fib.html#j:151"><span class="id" title="variable">j</span></a> <span class="id" title="notation">==</span> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<a class="idref" href="additions.fib.html#a:148"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="additions.fib.html#b:149"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="additions.fib.html#i:150"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> <a class="idref" href="additions.fib.html#j:151"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#b:149"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> <a class="idref" href="additions.fib.html#a:148"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m2lfib" class="idref" href="#m2lfib"><span class="id" title="definition">m2lfib</span></a> := <span class="id" title="notation">[::</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a><span class="id" title="notation">;</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a><span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m2lfibP" class="idref" href="#m2lfibP"><span class="id" title="lemma">m2lfibP</span></a> : <a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> <a class="idref" href="additions.fib.html#m2lfib"><span class="id" title="definition">m2lfib</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#fibm"><span class="id" title="definition">fibm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m2lmulP" class="idref" href="#m2lmulP"><span class="id" title="lemma">m2lmulP</span></a> <a id="m1:152" class="idref" href="#m1:152"><span class="id" title="binder">m1</span></a> <a id="m2:153" class="idref" href="#m2:153"><span class="id" title="binder">m2</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> (<a class="idref" href="additions.fib.html#m2lmul"><span class="id" title="definition">m2lmul</span></a> <a class="idref" href="additions.fib.html#m1:152"><span class="id" title="variable">m1</span></a> <a class="idref" href="additions.fib.html#m2:153"><span class="id" title="variable">m2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> <a class="idref" href="additions.fib.html#m1:152"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">×</span> <a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> <a class="idref" href="additions.fib.html#m2:153"><span class="id" title="variable">m2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="my_pow_m2lmul" class="idref" href="#my_pow_m2lmul"><span class="id" title="lemma">my_pow_m2lmul</span></a> <a id="m:154" class="idref" href="#m:154"><span class="id" title="binder">m</span></a> <a id="p:155" class="idref" href="#p:155"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> (<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <a class="idref" href="additions.fib.html#m2lmul"><span class="id" title="definition">m2lmul</span></a> <a class="idref" href="additions.fib.html#m:154"><span class="id" title="variable">m</span></a> <a class="idref" href="additions.fib.html#p:155"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <span class="id" title="notation">*%</span><span class="id" title="notation">R</span> (<a class="idref" href="additions.fib.html#m2lmx"><span class="id" title="definition">m2lmx</span></a> <a class="idref" href="additions.fib.html#m:154"><span class="id" title="variable">m</span></a>) <a class="idref" href="additions.fib.html#p:155"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fibZ3P" class="idref" href="#fibZ3P"><span class="id" title="lemma">fibZ3P</span></a> <a id="p:156" class="idref" href="#p:156"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> (<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#m3lfib"><span class="id" title="definition">m3lfib</span></a> <a class="idref" href="additions.fib.html#p:156"><span class="id" title="variable">p</span></a>) 0) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.fib.html#p:156"><span class="id" title="variable">p</span></a>)<span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fibZ2P" class="idref" href="#fibZ2P"><span class="id" title="lemma">fibZ2P</span></a> <a id="p:157" class="idref" href="#p:157"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#ZtoR"><span class="id" title="definition">ZtoR</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> (<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <a class="idref" href="additions.fib.html#m2lmul"><span class="id" title="definition">m2lmul</span></a> <a class="idref" href="additions.fib.html#m2lfib"><span class="id" title="definition">m2lfib</span></a> <a class="idref" href="additions.fib.html#p:157"><span class="id" title="variable">p</span></a>) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">nth</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> (<a class="idref" href="additions.fib.html#my_pow"><span class="id" title="definition">my_pow</span></a> <a class="idref" href="additions.fib.html#m2lmul"><span class="id" title="definition">m2lmul</span></a> <a class="idref" href="additions.fib.html#m2lfib"><span class="id" title="definition">m2lfib</span></a> <a class="idref" href="additions.fib.html#p:157"><span class="id" title="variable">p</span></a>) 1) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="additions.fib.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.BinPos.html#Pos.to_nat"><span class="id" title="definition">Pos.to_nat</span></a> <a class="idref" href="additions.fib.html#p:157"><span class="id" title="variable">p</span></a>)<span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m3lpow" class="idref" href="#m3lpow"><span class="id" title="definition">m3lpow</span></a> (<a id="m:158" class="idref" href="#m:158"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="n:159" class="idref" href="#n:159"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#fastexp3"><span class="id" title="definition">fastexp3</span></a> <a class="idref" href="additions.fib.html#m3lmul"><span class="id" title="definition">m3lmul</span></a> <a class="idref" href="additions.fib.html#m:158"><span class="id" title="variable">m</span></a> (<span class="id" title="definition">behead</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#n:159"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="additions.fib.html#m:158"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="m2lpow" class="idref" href="#m2lpow"><span class="id" title="definition">m2lpow</span></a> (<a id="m:160" class="idref" href="#m:160"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="n:161" class="idref" href="#n:161"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="additions.fib.html#fastexp3"><span class="id" title="definition">fastexp3</span></a> <a class="idref" href="additions.fib.html#m2lmul"><span class="id" title="definition">m2lmul</span></a> <a class="idref" href="additions.fib.html#m:160"><span class="id" title="variable">m</span></a> (<span class="id" title="definition">behead</span> (<a class="idref" href="additions.fib.html#bits"><span class="id" title="definition">bits</span></a> <a class="idref" href="additions.fib.html#n:161"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="additions.fib.html#m:160"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="additions.fib.html#with_matrices"><span class="id" title="section">with_matrices</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bigarg" class="idref" href="#bigarg"><span class="id" title="definition">bigarg</span></a> := 30000%<span class="id" title="var">positive</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Extraction</span> "bigfib" <span class="id" title="var">Z.ltb</span> <span class="id" title="var">Z.div_eucl</span> <span class="id" title="var">Pos.iter</span> <span class="id" title="var">Z.log2</span> <span class="id" title="var">fastexp4</span><br/>
+&nbsp;&nbsp;<span class="id" title="var">my_pow</span> <span class="id" title="var">m4lmul</span> <span class="id" title="var">m3lmul</span> <span class="id" title="var">m2lmul</span> <span class="id" title="var">m4lfib</span> <span class="id" title="var">m3lfib</span> <span class="id" title="var">m2lfib</span> <span class="id" title="var">bigarg</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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diff --git a/theories/html/gaia_hydras.GCanon.html b/theories/html/gaia_hydras.GCanon.html
new file mode 100644
index 00000000..44ba1f7c
--- /dev/null
+++ b/theories/html/gaia_hydras.GCanon.html
@@ -0,0 +1,204 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GCanon</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GCanon</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Gaia-compatible canonical sequences  
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.Canon</span></span>) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">CantorOrdinal</span>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Importation of Ketonen-Solovay's  machinery into gaia's world
+    (work in progress) 
+
+<div class="paragraph"> </div>
+
+  Note that a lemma <span class="inlinecode"><span class="id" title="var">Foo</span></span> may mask <span class="inlinecode"><span class="id" title="var">Canon.Foo</span></span>  
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hcanon" class="idref" href="#hcanon"><span class="id" title="abbreviation">hcanon</span></a> := <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">Epsilon0.Canon.canon</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="canon" class="idref" href="#canon"><span class="id" title="definition">canon</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a>: <span class="id" title="inductive">T1</span>) (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="inductive">T1</span> :=  <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#hcanon"><span class="id" title="abbreviation">hcanon</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:1"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:2"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="canonS" class="idref" href="#canonS"><span class="id" title="abbreviation">canonS</span></a> <span class="id" title="var">a</span>  := (<span class="id" title="keyword">fun</span> <a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a> ⇒  <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <span class="id" title="var">a</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:3"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex0" class="idref" href="#ex0"><span class="id" title="definition">ex0</span></a>: <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">phi0</span> <span class="id" title="definition">T1omega</span>) 6 <span class="id" title="notation">==</span> <span class="id" title="definition">phi0</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 6).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex1" class="idref" href="#ex1"><span class="id" title="definition">ex1</span></a>: <a class="idref" href="gaia_hydras.GCanon.html#canonS"><span class="id" title="abbreviation">canonS</span></a> (<span class="id" title="definition">phi0</span> <span class="id" title="definition">T1omega</span>) 6 <span class="id" title="notation">==</span> <span class="id" title="definition">phi0</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 7).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+rewriting lemmas 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gcanon_zero" class="idref" href="#gcanon_zero"><span class="id" title="lemma">gcanon_zero</span></a> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a>:  <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.GCanon.html#i:4"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_canon" class="idref" href="#g2h_canon"><span class="id" title="lemma">g2h_canon</span></a> <a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="i:6" class="idref" href="#i:6"><span class="id" title="binder">i</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:6"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GCanon.html#hcanon"><span class="id" title="abbreviation">hcanon</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:5"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:6"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_succ" class="idref" href="#canon_succ"><span class="id" title="lemma">canon_succ</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> <a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a> (<a id="Ha:9" class="idref" href="#Ha:9"><span class="id" title="binder">Ha</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:8"><span class="id" title="variable">a</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.GCanon.html#a:8"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:8"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_LE" class="idref" href="#canonS_LE"><span class="id" title="lemma">canonS_LE</span></a> <a id="a:10" class="idref" href="#a:10"><span class="id" title="binder">a</span></a> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:11"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:11"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_phi0_succE" class="idref" href="#canon0_phi0_succE"><span class="id" title="lemma">canon0_phi0_succE</span></a> <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">phi0</span> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.GCanon.html#a:12"><span class="id" title="variable">a</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_lt" class="idref" href="#canon0_lt"><span class="id" title="lemma">canon0_lt</span></a> <a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1lt</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:13"><span class="id" title="variable">a</span></a> 0) <a class="idref" href="gaia_hydras.GCanon.html#a:13"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_lt" class="idref" href="#canonS_lt"><span class="id" title="lemma">canonS_lt</span></a>  (<a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) [<a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>]:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1lt</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:14"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span>) <a class="idref" href="gaia_hydras.GCanon.html#a:15"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lt" class="idref" href="#canon_lt"><span class="id" title="lemma">canon_lt</span></a> (<a id="i:16" class="idref" href="#i:16"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) [<a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>]: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:16"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GCanon.html#a:17"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_cons_not_zero" class="idref" href="#canonS_cons_not_zero"><span class="id" title="lemma">canonS_cons_not_zero</span></a>  (<a id="i:18" class="idref" href="#i:18"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:19" class="idref" href="#a:19"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>) (<a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a> : <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.GCanon.html#b:21"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:18"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_canonS_not_zero" class="idref" href="#limit_canonS_not_zero"><span class="id" title="lemma">limit_canonS_not_zero</span></a> <a id="i:22" class="idref" href="#i:22"><span class="id" title="binder">i</span></a> <a id="lambda:23" class="idref" href="#lambda:23"><span class="id" title="binder">lambda</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:22"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_phi0_succE" class="idref" href="#canonS_phi0_succE"><span class="id" title="lemma">canonS_phi0_succE</span></a> (<a id="i:24" class="idref" href="#i:24"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="gamma:25" class="idref" href="#gamma:25"><span class="id" title="binder">gamma</span></a> : <span class="id" title="inductive">T1</span>): <br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:25"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">phi0</span> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:25"><span class="id" title="variable">gamma</span></a>)) <a class="idref" href="gaia_hydras.GCanon.html#i:24"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:25"><span class="id" title="variable">gamma</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:24"><span class="id" title="variable">i</span></a> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_SSn_zero" class="idref" href="#canon_SSn_zero"><span class="id" title="lemma">canon_SSn_zero</span></a> (<a id="i:26" class="idref" href="#i:26"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>) (<a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:28"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <span class="id" title="constructor">zero</span>) <a class="idref" href="gaia_hydras.GCanon.html#i:26"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:28"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.GCanon.html#a:27"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:26"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_zero_inv" class="idref" href="#canonS_zero_inv"><span class="id" title="lemma">canonS_zero_inv</span></a>  (<a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>) (<a id="i:30" class="idref" href="#i:30"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:30"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">one</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim1" class="idref" href="#canon_lim1"><span class="id" title="lemma">canon_lim1</span></a> <a id="i:31" class="idref" href="#i:31"><span class="id" title="binder">i</span></a> (<a id="lambda:32" class="idref" href="#lambda:32"><span class="id" title="binder">lambda</span></a>: <span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:32"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:32"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:32"><span class="id" title="variable">lambda</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:31"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">phi0</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:32"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:31"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_tail" class="idref" href="#canon_tail"><span class="id" title="lemma">canon_tail</span></a> <a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a> (<a id="n:34" class="idref" href="#n:34"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="b:35" class="idref" href="#b:35"><span class="id" title="binder">b</span></a> (<a id="i:36" class="idref" href="#i:36"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:34"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.GCanon.html#b:35"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:34"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.GCanon.html#b:35"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:36"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:34"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:36"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_ocons_succE" class="idref" href="#canonS_ocons_succE"><span class="id" title="lemma">canonS_ocons_succE</span></a> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a> <a id="n:38" class="idref" href="#n:38"><span class="id" title="binder">n</span></a> (<a id="gamma:39" class="idref" href="#gamma:39"><span class="id" title="binder">gamma</span></a>: <span class="id" title="inductive">T1</span>)(<a id="Hnf:40" class="idref" href="#Hnf:40"><span class="id" title="binder">Hnf</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:39"><span class="id" title="variable">gamma</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:39"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#n:38"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <span class="id" title="constructor">zero</span>) <a class="idref" href="gaia_hydras.GCanon.html#i:37"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">cons</span> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:39"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#n:38"><span class="id" title="variable">n</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#gamma:39"><span class="id" title="variable">gamma</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:37"><span class="id" title="variable">i</span></a> <span class="id" title="constructor">zero</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim2" class="idref" href="#canon_lim2"><span class="id" title="lemma">canon_lim2</span></a> <a id="i:41" class="idref" href="#i:41"><span class="id" title="binder">i</span></a> <a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a> (<a id="lambda:43" class="idref" href="#lambda:43"><span class="id" title="binder">lambda</span></a> : <span class="id" title="inductive">T1</span>) (<a id="Hnf:44" class="idref" href="#Hnf:44"><span class="id" title="binder">Hnf</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:43"><span class="id" title="variable">lambda</span></a>) (<a id="Hlim:45" class="idref" href="#Hlim:45"><span class="id" title="binder">Hlim</span></a>: <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:43"><span class="id" title="variable">lambda</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:43"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:42"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <span class="id" title="constructor">zero</span>) <a class="idref" href="gaia_hydras.GCanon.html#i:41"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:43"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:42"><span class="id" title="variable">n</span></a> (<span class="id" title="definition">phi0</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:43"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:41"><span class="id" title="variable">i</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim3" class="idref" href="#canon_lim3"><span class="id" title="lemma">canon_lim3</span></a> <a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a> <a id="n:47" class="idref" href="#n:47"><span class="id" title="binder">n</span></a> <a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="lambda:49" class="idref" href="#lambda:49"><span class="id" title="binder">lambda</span></a> (<a id="Ha:50" class="idref" href="#Ha:50"><span class="id" title="binder">Ha</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:48"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hlambda:51" class="idref" href="#Hlambda:51"><span class="id" title="binder">Hlambda</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:49"><span class="id" title="variable">lambda</span></a>) (<a id="Hlim:52" class="idref" href="#Hlim:52"><span class="id" title="binder">Hlim</span></a> :<span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:49"><span class="id" title="variable">lambda</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:47"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:49"><span class="id" title="variable">lambda</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:46"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.GCanon.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#n:47"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:49"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:46"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_limit_strong" class="idref" href="#canon_limit_strong"><span class="id" title="lemma">canon_limit_strong</span></a> <a id="lambda:53" class="idref" href="#lambda:53"><span class="id" title="binder">lambda</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:53"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:53"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:54" class="idref" href="#b:54"><span class="id" title="binder">b</span></a>, <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#b:54"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.GCanon.html#b:54"><span class="id" title="variable">b</span></a>  <a class="idref" href="gaia_hydras.GCanon.html#lambda:53"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.GCanon.html#b:54"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:53"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:55"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1nf_canon" class="idref" href="#T1nf_canon"><span class="id" title="lemma">T1nf_canon</span></a> <a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a> <a id="i:57" class="idref" href="#i:57"><span class="id" title="binder">i</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:57"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gcanon_limit_v2" class="idref" href="#gcanon_limit_v2"><span class="id" title="lemma">gcanon_limit_v2</span></a> (<a id="lambda:58" class="idref" href="#lambda:58"><span class="id" title="binder">lambda</span></a>: <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:58"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:58"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">limit_v2</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:58"><span class="id" title="variable">lambda</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#lambda:58"><span class="id" title="variable">lambda</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="canon_limit_mono" class="idref" href="#canon_limit_mono"><span class="id" title="lemma">canon_limit_mono</span></a> <a id="lambda:59" class="idref" href="#lambda:59"><span class="id" title="binder">lambda</span></a> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a> <a id="j:61" class="idref" href="#j:61"><span class="id" title="binder">j</span></a> (<a id="Hnf:62" class="idref" href="#Hnf:62"><span class="id" title="binder">Hnf</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:59"><span class="id" title="variable">lambda</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hlim:63" class="idref" href="#Hlim:63"><span class="id" title="binder">Hlim</span></a> : <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:59"><span class="id" title="variable">lambda</span></a>) (<a id="Hij:64" class="idref" href="#Hij:64"><span class="id" title="binder">Hij</span></a> : (<a class="idref" href="gaia_hydras.GCanon.html#i:60"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GCanon.html#j:61"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:59"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:60"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:59"><span class="id" title="variable">lambda</span></a> <a class="idref" href="gaia_hydras.GCanon.html#j:61"><span class="id" title="variable">j</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="canon_limit_of" class="idref" href="#canon_limit_of"><span class="id" title="lemma">canon_limit_of</span></a> <a id="lambda:65" class="idref" href="#lambda:65"><span class="id" title="binder">lambda</span></a> (<a id="Hnf:66" class="idref" href="#Hnf:66"><span class="id" title="binder">Hnf</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:65"><span class="id" title="variable">lambda</span></a>) (<a id="Hlim:67" class="idref" href="#Hlim:67"><span class="id" title="binder">Hlim</span></a> : <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GCanon.html#lambda:65"><span class="id" title="variable">lambda</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="definition">limit_of</span> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#lambda:65"><span class="id" title="variable">lambda</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#lambda:65"><span class="id" title="variable">lambda</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab465"></a><h1 class="section">Adaptation of <span class="inlinecode"><span class="id" title="var">canon</span></span> to type E0</h1>
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0Canon" class="idref" href="#E0Canon"><span class="id" title="definition">E0Canon</span></a> (<a id="a:68" class="idref" href="#a:68"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) (<a id="i:69" class="idref" href="#i:69"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;@<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:68"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#i:69"><span class="id" title="variable">i</span></a>)  <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_canon_lt" class="idref" href="#E0_canon_lt"><span class="id" title="lemma">E0_canon_lt</span></a> (<a id="a:70" class="idref" href="#a:70"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) <a id="i:71" class="idref" href="#i:71"><span class="id" title="binder">i</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:70"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#E0Canon"><span class="id" title="definition">E0Canon</span></a> <a class="idref" href="gaia_hydras.GCanon.html#a:70"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GCanon.html#i:71"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.GCanon.html#a:70"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GF_alpha.html b/theories/html/gaia_hydras.GF_alpha.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/gaia_hydras.GF_alpha.html
@@ -0,0 +1,169 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GF_alpha</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GF_alpha</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Gaia-compatible <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> fast growing functions  
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.F_alpha</span></span> )
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Epsilon0.F_alpha.html#"><span class="id" title="library">F_alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#"><span class="id" title="library">F_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a> <a class="idref" href="gaia_hydras.GHprime.html#"><span class="id" title="library">GHprime</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab466"></a><h1 class="section">Rapidly growing functions</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hF_" class="idref" href="#hF_"><span class="id" title="abbreviation">hF_</span></a> := <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_alpha.F_</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F_" class="idref" href="#F_"><span class="id" title="definition">F_</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>)  := <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_alpha.F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:1"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>]<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="T1F_" class="idref" href="#T1F_"><span class="id" title="definition">T1F_</span></a> (<a id="a:2" class="idref" href="#a:2"><span class="id" title="binder">a</span></a> : <span class="id" title="inductive">T1</span>)(<a id="Hnf:3" class="idref" href="#Hnf:3"><span class="id" title="binder">Hnf</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GF_alpha.html#a:2"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>:=<br/>
+&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (@<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#a:2"><span class="id" title="variable">a</span></a> <span class="id" title="var">_</span>) <a class="idref" href="gaia_hydras.GF_alpha.html#n:4"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Please note that a lemma <span class="inlinecode"><span class="id" title="var">foo</span></span> may mask <span class="inlinecode"><span class="id" title="var">F_alpha.Foo</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_gt" class="idref" href="#F_alpha_gt"><span class="id" title="lemma">F_alpha_gt</span></a> (<a id="alpha:5" class="idref" href="#alpha:5"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) (<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): (<a class="idref" href="gaia_hydras.GF_alpha.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:5"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:6"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_mono" class="idref" href="#F_alpha_mono"><span class="id" title="lemma">F_alpha_mono</span></a> (<a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:7"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_dom" class="idref" href="#F_alpha_dom"><span class="id" title="lemma">F_alpha_dom</span></a> <a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:8"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:8"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_Succ_le" class="idref" href="#F_alpha_Succ_le"><span class="id" title="lemma">F_alpha_Succ_le</span></a> <a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a>:<br/>
+&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#fun_le"><span class="id" title="definition">fun_le</span></a> (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:9"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:9"><span class="id" title="variable">alpha</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_positive" class="idref" href="#F_alpha_positive"><span class="id" title="lemma">F_alpha_positive</span></a> (<a id="alpha:10" class="idref" href="#alpha:10"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hE0"><span class="id" title="abbreviation">hE0</span></a>) (<a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): (0 <span class="id" title="notation">&lt;</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_alpha.F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:10"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:11"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_zeroE" class="idref" href="#F_zeroE"><span class="id" title="lemma">F_zeroE</span></a> <a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a>: <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:12"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:12"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_mono_l" class="idref" href="#F_mono_l"><span class="id" title="lemma">F_mono_l</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#beta:14"><span class="id" title="variable">beta</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#dominates"><span class="id" title="definition">dominates</span></a> (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:13"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#beta:14"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_0_eq" class="idref" href="#F_alpha_0_eq"><span class="id" title="lemma">F_alpha_0_eq</span></a> (<a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:15"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_succE" class="idref" href="#F_succE"><span class="id" title="lemma">F_succE</span></a> <a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a> <a id="i:17" class="idref" href="#i:17"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:16"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GF_alpha.html#i:17"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">Iterates.iterate</span></a> (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:16"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GF_alpha.html#i:17"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GF_alpha.html#i:17"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_limE" class="idref" href="#F_limE"><span class="id" title="lemma">F_limE</span></a> <a id="alpha:18" class="idref" href="#alpha:18"><span class="id" title="binder">alpha</span></a> <a id="i:19" class="idref" href="#i:19"><span class="id" title="binder">i</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1limit</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:18"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:19"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#E0Canon"><span class="id" title="definition">E0Canon</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:19"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.GF_alpha.html#i:19"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ numerical examples 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF1" class="idref" href="#LF1"><span class="id" title="lemma">LF1</span></a> <a id="i:20" class="idref" href="#i:20"><span class="id" title="binder">i</span></a>: <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 1) <a class="idref" href="gaia_hydras.GF_alpha.html#i:20"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<span class="id" title="notation">(</span>2 <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.GF_alpha.html#i:20"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">.+1</span>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF2" class="idref" href="#LF2"><span class="id" title="lemma">LF2</span></a> <a id="i:21" class="idref" href="#i:21"><span class="id" title="binder">i</span></a>:  (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">Exp2.exp2</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:21"><span class="id" title="variable">i</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.GF_alpha.html#i:21"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 2) <a class="idref" href="gaia_hydras.GF_alpha.html#i:21"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF2'" class="idref" href="#LF2'"><span class="id" title="lemma">LF2'</span></a> <a id="i:22" class="idref" href="#i:22"><span class="id" title="binder">i</span></a>:  (1 <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GF_alpha.html#i:22"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">Exp2.exp2</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#i:22"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 2) <a class="idref" href="gaia_hydras.GF_alpha.html#i:22"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF3_2" class="idref" href="#LF3_2"><span class="id" title="lemma">LF3_2</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">Iterates.dominates_from</span></a> 2 (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 3))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">Iterates.iterate</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">Exp2.exp2</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:23"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GF_alpha.html#n:23"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Canon_plus" class="idref" href="#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> <a id="alpha:25" class="idref" href="#alpha:25"><span class="id" title="binder">alpha</span></a> <a id="beta:26" class="idref" href="#beta:26"><span class="id" title="binder">beta</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Paths.Canon_plus</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:24"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:25"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#beta:26"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_restricted_mono_l" class="idref" href="#F_restricted_mono_l"><span class="id" title="lemma">F_restricted_mono_l</span></a> <a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a> <a id="beta:28" class="idref" href="#beta:28"><span class="id" title="binder">beta</span></a> <a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GF_alpha.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:29"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#beta:28"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#beta:28"><span class="id" title="variable">beta</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:29"><span class="id" title="variable">n</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:29"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_F" class="idref" href="#H'_F"><span class="id" title="lemma">H'_F</span></a> <a id="alpha:30" class="idref" href="#alpha:30"><span class="id" title="binder">alpha</span></a> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> : (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:30"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#n:31"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_phi0"><span class="id" title="definition">E0_phi0</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:30"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GF_alpha.html#n:31"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_not_PR_E0" class="idref" href="#F_alpha_not_PR_E0"><span class="id" title="lemma">F_alpha_not_PR_E0</span></a>  <a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0le"><span class="id" title="definition">E0le</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_omega"><span class="id" title="definition">E0_omega</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="gaia_hydras.GF_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:32"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_not_PR" class="idref" href="#F_alpha_not_PR"><span class="id" title="lemma">F_alpha_not_PR</span></a> <a id="alpha:33" class="idref" href="#alpha:33"><span class="id" title="binder">alpha</span></a> (<a id="Hnf:34" class="idref" href="#Hnf:34"><span class="id" title="binder">Hnf</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:33"><span class="id" title="variable">alpha</span></a> <span class="id" title="notation">==</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#LE"><span class="id" title="abbreviation">LE</span></a> <span class="id" title="definition">T1omega</span> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:33"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (@<a class="idref" href="gaia_hydras.GF_alpha.html#T1F_"><span class="id" title="definition">T1F_</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#alpha:33"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GF_alpha.html#Hnf:34"><span class="id" title="variable">Hnf</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GHessenberg.html b/theories/html/gaia_hydras.GHessenberg.html
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@@ -0,0 +1,150 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GHessenberg</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GHessenberg</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Hessenberg (commutative) sum 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="hoplus" class="idref" href="#hoplus"><span class="id" title="abbreviation">hoplus</span></a> := <a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">Epsilon0.Hessenberg.oplus</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="oplus" class="idref" href="#oplus"><span class="id" title="definition">oplus</span></a> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> <a id="beta:2" class="idref" href="#beta:2"><span class="id" title="binder">beta</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.GHessenberg.html#hoplus"><span class="id" title="abbreviation">hoplus</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#alpha:1"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#beta:2"><span class="id" title="variable">beta</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="9b335537bf17336701138ce9fa631c98" class="idref" href="#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">&quot;</span></a>o+" := <a class="idref" href="gaia_hydras.GHessenberg.html#oplus"><span class="id" title="definition">oplus</span></a>: <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="o_finite_mult" class="idref" href="#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.GHessenberg.html#n:3"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">p</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.GHessenberg.html#alpha:4"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#o_finite_mult:5"><span class="id" title="definition">o_finite_mult</span></a> <span class="id" title="var">p</span> <a class="idref" href="gaia_hydras.GHessenberg.html#alpha:4"><span class="id" title="variable">alpha</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="FixpointDef" class="idref" href="#FixpointDef"><span class="id" title="module">FixpointDef</span></a>.<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="FixpointDef.oplus" class="idref" href="#FixpointDef.oplus"><span class="id" title="definition">oplus</span></a> (<a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> <a id="beta:8" class="idref" href="#beta:8"><span class="id" title="binder">beta</span></a> : <span class="id" title="inductive">T1</span>) : <span class="id" title="inductive">T1</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="oplus_aux:14" class="idref" href="#oplus_aux:14"><span class="id" title="binder">oplus_aux</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="keyword">beta</span>} :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.GHessenberg.html#alpha:7"><span class="id" title="variable">alpha</span></a>, <a class="idref" href="gaia_hydras.GHessenberg.html#beta:10"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="constructor">zero</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="gaia_hydras.GHessenberg.html#beta:10"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="constructor">zero</span> ⇒ <a class="idref" href="gaia_hydras.GHessenberg.html#alpha:7"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> <span class="id" title="var">b1</span>, <span class="id" title="constructor">cons</span> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> <span class="id" title="var">b2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> (<a class="idref" href="gaia_hydras.GHessenberg.html#oplus:9"><span class="id" title="definition">oplus</span></a> <span class="id" title="var">b1</span> <a class="idref" href="gaia_hydras.GHessenberg.html#beta:10"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> (<a class="idref" href="gaia_hydras.GHessenberg.html#oplus_aux:11"><span class="id" title="variable">oplus_aux</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n1</span> <span class="id" title="notation">+</span> <span class="id" title="var">n2</span>)%<span class="id" title="var">nat</span>) (<a class="idref" href="gaia_hydras.GHessenberg.html#oplus:9"><span class="id" title="definition">oplus</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.GHessenberg.html#oplus_aux:14"><span class="id" title="variable">oplus_aux</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#beta:8"><span class="id" title="variable">beta</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.GHessenberg.html#FixpointDef"><span class="id" title="module">FixpointDef</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplusE" class="idref" href="#oplusE"><span class="id" title="lemma">oplusE</span></a> (<a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a> <a id="b:16" class="idref" href="#b:16"><span class="id" title="binder">b</span></a> :<span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GHessenberg.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:16"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:15"><span class="id" title="variable">a</span></a>, <a class="idref" href="gaia_hydras.GHessenberg.html#b:16"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="constructor">zero</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="gaia_hydras.GHessenberg.html#b:16"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="constructor">zero</span> ⇒ <a class="idref" href="gaia_hydras.GHessenberg.html#a:15"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> <span class="id" title="var">b1</span>, <span class="id" title="constructor">cons</span> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> <span class="id" title="var">b2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>  <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> (<span class="id" title="var">b1</span> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:16"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n1</span> <span class="id" title="notation">+</span> <span class="id" title="var">n2</span>)) (<span class="id" title="var">b1</span> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <span class="id" title="constructor">cons</span> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> (<a class="idref" href="gaia_hydras.GHessenberg.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Equations for oplus 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus0b" class="idref" href="#oplus0b"><span class="id" title="lemma">oplus0b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.GHessenberg.html#oplus"><span class="id" title="definition">oplus</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplusa0" class="idref" href="#oplusa0"><span class="id" title="lemma">oplusa0</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.GHessenberg.html#oplus"><span class="id" title="definition">oplus</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="oplus_nf" class="idref" href="#oplus_nf"><span class="id" title="lemma">oplus_nf</span></a> (<a id="a:19" class="idref" href="#a:19"><span class="id" title="binder">a</span></a>  <a id="b:20" class="idref" href="#b:20"><span class="id" title="binder">b</span></a> : <span class="id" title="inductive">T1</span>) : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:20"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> (<a class="idref" href="gaia_hydras.GHessenberg.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:20"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplusC" class="idref" href="#oplusC"><span class="id" title="lemma">oplusC</span></a> (<a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a> <a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>):  <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:21"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplusA" class="idref" href="#oplusA"><span class="id" title="lemma">oplusA</span></a>  (<a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a> <a id="c:25" class="idref" href="#c:25"><span class="id" title="binder">c</span></a>: <span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:25"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#c:25"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#c:25"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt1" class="idref" href="#oplus_lt1"><span class="id" title="lemma">oplus_lt1</span></a> (<a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a>:<span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="constructor">zero</span> <span class="id" title="notation">&lt;</span>  <a class="idref" href="gaia_hydras.GHessenberg.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#b:27"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#a:26"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt2" class="idref" href="#oplus_lt2"><span class="id" title="lemma">oplus_lt2</span></a> (<a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="constructor">zero</span> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:28"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:28"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_l" class="idref" href="#oplus_strict_mono_l"><span class="id" title="lemma">oplus_strict_mono_l</span></a> (<a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a> <a id="c:32" class="idref" href="#c:32"><span class="id" title="binder">c</span></a>: <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:32"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#a:30"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#c:32"><span class="id" title="variable">c</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:31"><span class="id" title="variable">b</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#c:32"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_r" class="idref" href="#oplus_strict_mono_r"><span class="id" title="lemma">oplus_strict_mono_r</span></a> (<a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a> <a id="b:34" class="idref" href="#b:34"><span class="id" title="binder">b</span></a> <a id="c:35" class="idref" href="#c:35"><span class="id" title="binder">c</span></a>: <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:35"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#b:34"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:35"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#b:34"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:33"><span class="id" title="variable">a</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#c:35"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt_phi0" class="idref" href="#oplus_lt_phi0"><span class="id" title="lemma">oplus_lt_phi0</span></a> (<a id="a:36" class="idref" href="#a:36"><span class="id" title="binder">a</span></a> <a id="b:37" class="idref" href="#b:37"><span class="id" title="binder">b</span></a> <a id="c:38" class="idref" href="#c:38"><span class="id" title="binder">c</span></a>: <span class="id" title="inductive">T1</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:37"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:38"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GHessenberg.html#a:36"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:38"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHessenberg.html#b:37"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:38"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.GHessenberg.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.GHessenberg.html#b:37"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.GHessenberg.html#c:38"><span class="id" title="variable">c</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GHprime</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GHprime</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Gaia-compatible <span class="inlinecode"><span class="id" title="var">H'_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> fast growing functions  
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.Hprime</span></span> )
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#"><span class="id" title="library">Hprime</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#"><span class="id" title="library">L_alpha</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#"><span class="id" title="library">GPaths</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="H'_" class="idref" href="#H'_"><span class="id" title="definition">H'_</span></a> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> <a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a> := <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">Hprime.H'_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:1"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GHprime.html#i:2"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab467"></a><h2 class="section">Equations for <span class="inlinecode"><span class="id" title="var">H'_</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq1" class="idref" href="#H'_eq1"><span class="id" title="lemma">H'_eq1</span></a> (<a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="gaia_hydras.GHprime.html#i:3"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GHprime.html#i:3"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq2" class="idref" href="#H'_eq2"><span class="id" title="lemma">H'_eq2</span></a>  <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:4"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GHprime.html#i:5"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:4"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GHprime.html#i:5"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq3" class="idref" href="#H'_eq3"><span class="id" title="lemma">H'_eq3</span></a> <a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:6"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:6"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHprime.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#E0Canon"><span class="id" title="definition">E0Canon</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:6"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GHprime.html#i:7"><span class="id" title="variable">i</span></a>)) <a class="idref" href="gaia_hydras.GHprime.html#i:7"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab468"></a><h2 class="section">Examples</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega" class="idref" href="#H'_omega"><span class="id" title="lemma">H'_omega</span></a> <a id="k:8" class="idref" href="#k:8"><span class="id" title="binder">k</span></a> : <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_omega"><span class="id" title="definition">E0_omega</span></a> <a class="idref" href="gaia_hydras.GHprime.html#k:8"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span>2 <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.GHprime.html#k:8"><span class="id" title="variable">k</span></a><span class="id" title="notation">).+1</span> %<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_double" class="idref" href="#H'_omega_double"><span class="id" title="lemma">H'_omega_double</span></a> (<a id="k:9" class="idref" href="#k:9"><span class="id" title="binder">k</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0mul"><span class="id" title="definition">E0mul</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_omega"><span class="id" title="definition">E0_omega</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 2)) <a class="idref" href="gaia_hydras.GHprime.html#k:9"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  (4 <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.GHprime.html#k:9"><span class="id" title="variable">k</span></a> <span class="id" title="notation">+</span> 3)%<span class="id" title="var">coq_nat</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+TODO
+    import more abstract properties of H' 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_dom" class="idref" href="#H'_dom"><span class="id" title="lemma">H'_dom</span></a> <a id="alpha:10" class="idref" href="#alpha:10"><span class="id" title="binder">alpha</span></a> <a id="beta:11" class="idref" href="#beta:11"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:10"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHprime.html#beta:11"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> (<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#beta:11"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:10"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_alpha_mono" class="idref" href="#H'_alpha_mono"><span class="id" title="lemma">H'_alpha_mono</span></a> (<a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:12"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="H'_alpha_gt" class="idref" href="#H'_alpha_gt"><span class="id" title="lemma">H'_alpha_gt</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> (<a id="Halpha:14" class="idref" href="#Halpha:14"><span class="id" title="binder">Halpha</span></a>: <a class="idref" href="gaia_hydras.GHprime.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a>) <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.GHprime.html#n:15"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GHprime.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHprime.html#n:15"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_cube_min" class="idref" href="#H'_omega_cube_min"><span class="id" title="lemma">H'_omega_cube_min</span></a> :<br/>
+<span class="id" title="keyword">∀</span> <a id="k:16" class="idref" href="#k:16"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.GHprime.html#k:16"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> <a class="idref" href="gaia_hydras.GHprime.html#k:16"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span>  <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_phi0"><span class="id" title="definition">E0_phi0</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> 3)) <a class="idref" href="gaia_hydras.GHprime.html#k:16"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">N</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GHydra.html b/theories/html/gaia_hydras.GHydra.html
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+++ b/theories/html/gaia_hydras.GHydra.html
@@ -0,0 +1,115 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GHydra</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GHydra</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#"><span class="id" title="library">Hydra_Theorems</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="library">Hydra_Definitions</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Termination.html#"><span class="id" title="library">Hydra_Termination</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#"><span class="id" title="library">Battle_length</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#"><span class="id" title="library">Hydra_Examples</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#"><span class="id" title="library">Epsilon0_Needed_Free</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#"><span class="id" title="library">Epsilon0_Needed_Std</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#"><span class="id" title="library">GHessenberg</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#"><span class="id" title="library">GL_alpha</span></a> <a class="idref" href="gaia_hydras.GPrelude.html#"><span class="id" title="library">GPrelude</span></a>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="module">Hydra_Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="m" class="idref" href="#m"><span class="id" title="definition">m</span></a> (<a id="h:1" class="idref" href="#h:1"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <span class="id" title="inductive">T1</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.GHydra.html#h:1"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">is</span>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">hs</span>) <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.GHydra.html#ms:4"><span class="id" title="definition">ms</span></a> <span class="id" title="var">hs</span> <span class="id" title="keyword">else</span> <span class="id" title="constructor">zero</span><br/>
+<span class="id" title="keyword">with</span> <a id="ms" class="idref" href="#ms"><span class="id" title="definition">ms</span></a> (<a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) : <span class="id" title="inductive">T1</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.GHydra.html#s:2"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="definition">phi0</span> (<a class="idref" href="gaia_hydras.GHydra.html#m:3"><span class="id" title="definition">m</span></a> <span class="id" title="var">h</span>) <a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">o</span></a><a class="idref" href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ms:4"><span class="id" title="definition">ms</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m_ref" class="idref" href="#m_ref"><span class="id" title="lemma">m_ref</span></a> <a id="h:7" class="idref" href="#h:7"><span class="id" title="binder">h</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.GHydra.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="gaia_hydras.GHydra.html#h:7"><span class="id" title="variable">h</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">Hydra_Termination.m</span></a> <a class="idref" href="gaia_hydras.GHydra.html#h:7"><span class="id" title="variable">h</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m_nf" class="idref" href="#m_nf"><span class="id" title="lemma">m_nf</span></a> <a id="h:10" class="idref" href="#h:10"><span class="id" title="binder">h</span></a> : <span class="id" title="definition">T1nf</span> (<a class="idref" href="gaia_hydras.GHydra.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="gaia_hydras.GHydra.html#h:10"><span class="id" title="variable">h</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mVariant" class="idref" href="#mVariant"><span class="id" title="lemma">mVariant</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a>  <span class="id" title="lemma">nf_Wf</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="gaia_hydras.GHydra.html#m"><span class="id" title="definition">m</span></a> .<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="every_battle_terminates" class="idref" href="#every_battle_terminates"><span class="id" title="lemma">every_battle_terminates</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Termination"><span class="id" title="definition">Termination</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="l_std" class="idref" href="#l_std"><span class="id" title="definition">l_std</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="k:16" class="idref" href="#k:16"><span class="id" title="binder">k</span></a> := (<a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GHydra.html#alpha:15"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GHydra.html#k:16"><span class="id" title="variable">k</span></a>) <span class="id" title="notation">-</span> <a class="idref" href="gaia_hydras.GHydra.html#k:16"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="l_stdE" class="idref" href="#l_stdE"><span class="id" title="lemma">l_stdE</span></a> <a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a> <a id="k:18" class="idref" href="#k:18"><span class="id" title="binder">k</span></a> : <a class="idref" href="gaia_hydras.GHydra.html#l_std"><span class="id" title="definition">l_std</span></a> <a class="idref" href="gaia_hydras.GHydra.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHydra.html#k:18"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#l_std"><span class="id" title="definition">Battle_length.l_std</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GHydra.html#alpha:17"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GHydra.html#k:18"><span class="id" title="variable">k</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1toH" class="idref" href="#T1toH"><span class="id" title="definition">T1toH</span></a> (<a id="a:19" class="idref" href="#a:19"><span class="id" title="binder">a</span></a>: <span class="id" title="inductive">T1</span>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> := <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">O2H.iota</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GHydra.html#a:19"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="l_std_ok" class="idref" href="#l_std_ok"><span class="id" title="lemma">l_std_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GHydra.html#alpha:20"><span class="id" title="variable">alpha</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:21" class="idref" href="#k:21"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1 <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GHydra.html#k:21"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_length"><span class="id" title="definition">battle_length</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="gaia_hydras.GHydra.html#k:21"><span class="id" title="variable">k</span></a> (<a class="idref" href="gaia_hydras.GHydra.html#T1toH"><span class="id" title="definition">T1toH</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.GHydra.html#alpha:20"><span class="id" title="variable">alpha</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.GHydra.html#l_std"><span class="id" title="definition">l_std</span></a> <a class="idref" href="gaia_hydras.GHydra.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GHydra.html#k:21"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ImpossibilityProof" class="idref" href="#ImpossibilityProof"><span class="id" title="section">ImpossibilityProof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ImpossibilityProof.mu" class="idref" href="#ImpossibilityProof.mu"><span class="id" title="variable">mu</span></a>:<span class="id" title="inductive">T1</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="ImpossibilityProof.nfMu" class="idref" href="#ImpossibilityProof.nfMu"><span class="id" title="variable">nfMu</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.mu"><span class="id" title="variable">mu</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ImpossibilityProof.m" class="idref" href="#ImpossibilityProof.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="inductive">T1</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="ImpossibilityProof.mh" class="idref" href="#ImpossibilityProof.mh"><span class="id" title="variable">mh</span></a> (<a id="h:26" class="idref" href="#h:26"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) := <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.GHydra.html#h:26"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span>  (<a id="Var:27" class="idref" href="#Var:27"><span class="id" title="binder">Var</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="lemma">nf_Wf</span> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.b"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.m"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="BVar:28" class="idref" href="#BVar:28"><span class="id" title="binder">BVar</span></a>:  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="gaia_hydras.GHydra.html#Var:27"><span class="id" title="variable">Var</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.mu"><span class="id" title="variable">mu</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="hVar" class="idref" href="#hVar"><span class="id" title="instance">hVar</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.b"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.mh"><span class="id" title="variable">mh</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="bVar" class="idref" href="#bVar"><span class="id" title="instance">bVar</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="gaia_hydras.GHydra.html#hVar"><span class="id" title="instance">hVar</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof.mu"><span class="id" title="variable">mu</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.GHydra.html#ImpossibilityProof"><span class="id" title="section">ImpossibilityProof</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Impossibility_free" class="idref" href="#Impossibility_free"><span class="id" title="lemma">Impossibility_free</span></a> <a id="mu:29" class="idref" href="#mu:29"><span class="id" title="binder">mu</span></a> <a id="m:30" class="idref" href="#m:30"><span class="id" title="binder">m</span></a> (<a id="Var:31" class="idref" href="#Var:31"><span class="id" title="binder">Var</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="lemma">nf_Wf</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="gaia_hydras.GHydra.html#m:30"><span class="id" title="variable">m</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="gaia_hydras.GHydra.html#Var:31"><span class="id" title="variable">Var</span></a> <a class="idref" href="gaia_hydras.GHydra.html#mu:29"><span class="id" title="variable">mu</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Impossibility_std" class="idref" href="#Impossibility_std"><span class="id" title="lemma">Impossibility_std</span></a> <a id="mu:32" class="idref" href="#mu:32"><span class="id" title="binder">mu</span></a>  <a id="m:33" class="idref" href="#m:33"><span class="id" title="binder">m</span></a> (<a id="Var:34" class="idref" href="#Var:34"><span class="id" title="binder">Var</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="lemma">nf_Wf</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="gaia_hydras.GHydra.html#m:33"><span class="id" title="variable">m</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="gaia_hydras.GHydra.html#Var:34"><span class="id" title="variable">Var</span></a> <a class="idref" href="gaia_hydras.GHydra.html#mu:32"><span class="id" title="variable">mu</span></a>.<br/>
+ 
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GL_alpha.html b/theories/html/gaia_hydras.GL_alpha.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/gaia_hydras.GL_alpha.html
@@ -0,0 +1,116 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GL_alpha</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GL_alpha</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Gaia-compatible <span class="inlinecode"><span class="id" title="var">L_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> fast growing functions  
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.Hprime</span></span> )
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#"><span class="id" title="library">L_alpha</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Epsilon0.L_alpha.html#"><span class="id" title="library">L_alpha</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#"><span class="id" title="library">GPaths</span></a> <a class="idref" href="gaia_hydras.GHprime.html#"><span class="id" title="library">GHprime</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_" class="idref" href="#L_"><span class="id" title="definition">L_</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_alpha.L_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:1"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GL_alpha.html#i:2"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_zeroE" class="idref" href="#L_zeroE"><span class="id" title="lemma">L_zeroE</span></a> <a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a> :  <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:3"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:3"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_eq2" class="idref" href="#L_eq2"><span class="id" title="lemma">L_eq2</span></a> <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:4"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:4"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:5"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:4"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:5"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_succE" class="idref" href="#L_succE"><span class="id" title="lemma">L_succE</span></a> <a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> : <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:6"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GL_alpha.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:6"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:7"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_limE" class="idref" href="#L_limE"><span class="id" title="lemma">L_limE</span></a> <a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> <a id="i:9" class="idref" href="#i:9"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:9"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#E0Canon"><span class="id" title="definition">E0Canon</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:9"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:9"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_finite" class="idref" href="#L_finite"><span class="id" title="lemma">L_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a> <a id="k:11" class="idref" href="#k:11"><span class="id" title="binder">k</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a>  <a class="idref" href="gaia_hydras.GL_alpha.html#i:10"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.GL_alpha.html#k:11"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="gaia_hydras.GL_alpha.html#i:10"><span class="id" title="variable">i</span></a><span class="id" title="notation">+</span><a class="idref" href="gaia_hydras.GL_alpha.html#k:11"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega" class="idref" href="#L_omega"><span class="id" title="lemma">L_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="k:12" class="idref" href="#k:12"><span class="id" title="binder">k</span></a>, <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_omega"><span class="id" title="definition">E0_omega</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:12"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.GL_alpha.html#k:12"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_ge_id" class="idref" href="#L_ge_id"><span class="id" title="lemma">L_ge_id</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a>,  (<a class="idref" href="gaia_hydras.GL_alpha.html#i:14"><span class="id" title="variable">i</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:14"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_ge_S" class="idref" href="#L_ge_S"><span class="id" title="lemma">L_ge_S</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="i:16" class="idref" href="#i:16"><span class="id" title="binder">i</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GL_alpha.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.GL_alpha.html#i:16"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span>  <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:16"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_spec" class="idref" href="#L_spec"><span class="id" title="definition">L_spec</span></a> (<a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a>:<span class="id" title="inductive">T1</span>) (<a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">Large_Sets.L_spec</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#alpha:17"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="gaia_hydras.GL_alpha.html#f:18"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_spec0" class="idref" href="#L_spec0"><span class="id" title="lemma">L_spec0</span></a> (<a id="f:19" class="idref" href="#f:19"><span class="id" title="binder">f</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.GL_alpha.html#L_spec"><span class="id" title="definition">L_spec</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.GL_alpha.html#f:19"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <span class="id" title="keyword">∀</span> <a id="k:20" class="idref" href="#k:20"><span class="id" title="binder">k</span></a>, <a class="idref" href="gaia_hydras.GL_alpha.html#f:19"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:20"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:20"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_spec1" class="idref" href="#L_spec1"><span class="id" title="lemma">L_spec1</span></a> (<a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a>:<span class="id" title="inductive">T1</span>) (<a id="f:22" class="idref" href="#f:22"><span class="id" title="binder">f</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GL_alpha.html#a:21"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GL_alpha.html#L_spec"><span class="id" title="definition">L_spec</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#f:22"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="k:23" class="idref" href="#k:23"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">Large_Sets.mlarge</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:21"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;( <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">MoreLists.interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:23"><span class="id" title="variable">k</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="gaia_hydras.GL_alpha.html#f:22"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:23"><span class="id" title="variable">k</span></a>))))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> .<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_pos_inv" class="idref" href="#L_pos_inv"><span class="id" title="lemma">L_pos_inv</span></a> <a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="f:25" class="idref" href="#f:25"><span class="id" title="binder">f</span></a> :<br/>
+&nbsp;<a class="idref" href="gaia_hydras.GL_alpha.html#a:24"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#L_spec"><span class="id" title="definition">L_spec</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:26" class="idref" href="#k:26"><span class="id" title="binder">k</span></a>, (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:26"><span class="id" title="variable">k</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GL_alpha.html#f:25"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#k:26"><span class="id" title="variable">k</span></a>))%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="L_correct" class="idref" href="#L_correct"><span class="id" title="lemma">L_correct</span></a> <a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.GL_alpha.html#L_spec"><span class="id" title="definition">L_spec</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:27"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:27"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="H'_L_" class="idref" href="#H'_L_"><span class="id" title="lemma">H'_L_</span></a> <a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="i:29" class="idref" href="#i:29"><span class="id" title="binder">i</span></a>: (<a class="idref" href="gaia_hydras.GHprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:29"><span class="id" title="variable">i</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GL_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GL_alpha.html#i:29"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span>)%<span class="id" title="var">N</span>.<br/>
+ </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GLarge_Sets.html b/theories/html/gaia_hydras.GLarge_Sets.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/gaia_hydras.GLarge_Sets.html
@@ -0,0 +1,88 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GLarge_Sets</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GLarge_Sets</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Gaia-compatible large sequences 
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.Large_Sets</span></span> )
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="library">Large_Sets</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#"><span class="id" title="library">GPaths</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">CantorOrdinal</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hmlarge" class="idref" href="#hmlarge"><span class="id" title="abbreviation">hmlarge</span></a>  := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="hmlargeS" class="idref" href="#hmlargeS"><span class="id" title="abbreviation">hmlargeS</span></a>  := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlargeS"><span class="id" title="definition">mlargeS</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="mlarge" class="idref" href="#mlarge"><span class="id" title="definition">mlarge</span></a> <a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a> := <a class="idref" href="gaia_hydras.GLarge_Sets.html#hmlarge"><span class="id" title="abbreviation">hmlarge</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:1"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GLarge_Sets.html#s:2"><span class="id" title="variable">s</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="mlargeS" class="idref" href="#mlargeS"><span class="id" title="definition">mlargeS</span></a> <a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> <a id="s:4" class="idref" href="#s:4"><span class="id" title="binder">s</span></a> := <a class="idref" href="gaia_hydras.GLarge_Sets.html#hmlargeS"><span class="id" title="abbreviation">hmlargeS</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:3"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GLarge_Sets.html#s:4"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hlarge" class="idref" href="#hlarge"><span class="id" title="abbreviation">hlarge</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="hlargeS" class="idref" href="#hlargeS"><span class="id" title="abbreviation">hlargeS</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="large" class="idref" href="#large"><span class="id" title="definition">large</span></a> <a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="A:6" class="idref" href="#A:6"><span class="id" title="binder">A</span></a> := <a class="idref" href="gaia_hydras.GLarge_Sets.html#hlarge"><span class="id" title="abbreviation">hlarge</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:5"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GLarge_Sets.html#A:6"><span class="id" title="variable">A</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="largeS" class="idref" href="#largeS"><span class="id" title="definition">largeS</span></a> <a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="A:8" class="idref" href="#A:8"><span class="id" title="binder">A</span></a> := <a class="idref" href="gaia_hydras.GLarge_Sets.html#hlargeS"><span class="id" title="abbreviation">hlargeS</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:7"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GLarge_Sets.html#A:8"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hL_spec" class="idref" href="#hL_spec"><span class="id" title="abbreviation">hL_spec</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_spec" class="idref" href="#L_spec"><span class="id" title="definition">L_spec</span></a> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:9"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GLarge_Sets.html#f:10"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_unicity" class="idref" href="#mlarge_unicity"><span class="id" title="lemma">mlarge_unicity</span></a> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> <a id="k:12" class="idref" href="#k:12"><span class="id" title="binder">k</span></a> <a id="l:13" class="idref" href="#l:13"><span class="id" title="binder">l</span></a> <a id="l':14" class="idref" href="#l':14"><span class="id" title="binder">l'</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GLarge_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:11"><span class="id" title="variable">a</span></a> (<span class="id" title="definition">index_iota</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#k:12"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#l:13"><span class="id" title="variable">l</span></a><span class="id" title="notation">.+1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GLarge_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:11"><span class="id" title="variable">a</span></a> (<span class="id" title="definition">index_iota</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#k:12"><span class="id" title="variable">k</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#l':14"><span class="id" title="variable">l'</span></a><span class="id" title="notation">.+1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GLarge_Sets.html#l:13"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#l':14"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_fin_ok" class="idref" href="#L_fin_ok"><span class="id" title="lemma">L_fin_ok</span></a> <a id="i:15" class="idref" href="#i:15"><span class="id" title="binder">i</span></a> : <a class="idref" href="gaia_hydras.GLarge_Sets.html#L_spec"><span class="id" title="definition">L_spec</span></a> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#i:15"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#i:15"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Theorem_4_5" class="idref" href="#Theorem_4_5"><span class="id" title="lemma">Theorem_4_5</span></a> (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a>: <span class="id" title="inductive">T1</span>)(<a id="Ha:17" class="idref" href="#Ha:17"><span class="id" title="binder">Ha</span></a> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:16"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> <a id="B:19" class="idref" href="#B:19"><span class="id" title="binder">B</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HA:20" class="idref" href="#HA:20"><span class="id" title="binder">HA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted.Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#A:18"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HB:21" class="idref" href="#HB:21"><span class="id" title="binder">HB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted.Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#B:19"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HAB:22" class="idref" href="#HAB:22"><span class="id" title="binder">HAB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#incl"><span class="id" title="definition">List.incl</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#A:18"><span class="id" title="variable">A</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#B:19"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GLarge_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#A:18"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GLarge_Sets.html#B:19"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GPaths.html b/theories/html/gaia_hydras.GPaths.html
new file mode 100644
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+++ b/theories/html/gaia_hydras.GPaths.html
@@ -0,0 +1,250 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GPaths</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GPaths</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Gaia-compatible accessibility in epsilon0  
+
+<div class="paragraph"> </div>
+
+(imported from <span class="inlinecode"><span class="id" title="var">hydras.Epsilon0.Paths</span></span>) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">CantorOrdinal</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Importation of Ketonen-Solovay's  machinery into gaia's world
+    (work in progress) 
+ 
+<div class="paragraph"> </div>
+
+ small transition step associated with canonical sequences 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="htransition" class="idref" href="#htransition"><span class="id" title="abbreviation">htransition</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#transition"><span class="id" title="definition">Epsilon0.Paths.transition</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hbounded_transitionS" class="idref" href="#hbounded_transitionS"><span class="id" title="abbreviation">hbounded_transitionS</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#bounded_transitionS"><span class="id" title="definition">Paths.bounded_transitionS</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transition" class="idref" href="#transition"><span class="id" title="definition">transition</span></a> <a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a> (<a id="a:2" class="idref" href="#a:2"><span class="id" title="binder">a</span></a> <a id="b:3" class="idref" href="#b:3"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:1"><span class="id" title="variable">i</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:2"><span class="id" title="variable">a</span></a><span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:3"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:2"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:1"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bounded_transitionS" class="idref" href="#bounded_transitionS"><span class="id" title="definition">bounded_transitionS</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> (<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#i:7"><span class="id" title="variable">i</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.GPaths.html#n:4"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.GPaths.html#transition"><span class="id" title="definition">transition</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:7"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:6"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transitionb" class="idref" href="#transitionb"><span class="id" title="definition">transitionb</span></a> <a id="i:8" class="idref" href="#i:8"><span class="id" title="binder">i</span></a> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:8"><span class="id" title="variable">i</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:9"><span class="id" title="variable">a</span></a><span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:10"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:8"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="transitionP" class="idref" href="#transitionP"><span class="id" title="lemma">transitionP</span></a> <a id="i:11" class="idref" href="#i:11"><span class="id" title="binder">i</span></a> <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#transition"><span class="id" title="definition">transition</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:11"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:13"><span class="id" title="variable">b</span></a> )<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.GPaths.html#transitionb"><span class="id" title="definition">transitionb</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:11"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:13"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+TODO : define <span class="inlinecode"><span class="id" title="var">path_to</span></span> as a boolean function 
+</div>
+<div class="code">
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hpath_to" class="idref" href="#hpath_to"><span class="id" title="abbreviation">hpath_to</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hpath" class="idref" href="#hpath"><span class="id" title="abbreviation">hpath</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#path"><span class="id" title="definition">path</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hpathS" class="idref" href="#hpathS"><span class="id" title="abbreviation">hpathS</span></a> <span class="id" title="var">from</span> <span class="id" title="var">s</span> <span class="id" title="var">to</span> := (<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <span class="id" title="var">to</span> <span class="id" title="var">s</span> <span class="id" title="var">from</span>).<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hconst_pathS" class="idref" href="#hconst_pathS"><span class="id" title="abbreviation">hconst_pathS</span></a>  := <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hconst_path" class="idref" href="#hconst_path"><span class="id" title="abbreviation">hconst_path</span></a>  := <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hgnawS" class="idref" href="#hgnawS"><span class="id" title="abbreviation">hgnawS</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hgnaw" class="idref" href="#hgnaw"><span class="id" title="abbreviation">hgnaw</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hstandard_gnaw" class="idref" href="#hstandard_gnaw"><span class="id" title="abbreviation">hstandard_gnaw</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">path_to</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> : <span class="inlinecode"><span class="id" title="var">b</span></span> is accessible from <span class="inlinecode"><span class="id" title="var">alpha</span></span> with trace <span class="inlinecode"><span class="id" title="var">s</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="path_to" class="idref" href="#path_to"><span class="id" title="definition">path_to</span></a> (<a id="to:14" class="idref" href="#to:14"><span class="id" title="binder">to</span></a>: <span class="id" title="inductive">T1</span>)(<a id="s:15" class="idref" href="#s:15"><span class="id" title="binder">s</span></a>: <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="from:16" class="idref" href="#from:16"><span class="id" title="binder">from</span></a>:<span class="id" title="inductive">T1</span>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#hpath_to"><span class="id" title="abbreviation">hpath_to</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:14"><span class="id" title="variable">to</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#s:15"><span class="id" title="variable">s</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#from:16"><span class="id" title="variable">from</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="path" class="idref" href="#path"><span class="id" title="abbreviation">path</span></a> <span class="id" title="var">from</span> <span class="id" title="var">s</span> <span class="id" title="var">to</span> := (<a class="idref" href="gaia_hydras.GPaths.html#path_to"><span class="id" title="definition">path_to</span></a> <span class="id" title="var">to</span> <span class="id" title="var">s</span> <span class="id" title="var">from</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="acc_from" class="idref" href="#acc_from"><span class="id" title="definition">acc_from</span></a> <a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a> <a id="b:18" class="idref" href="#b:18"><span class="id" title="binder">b</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s:19" class="idref" href="#s:19"><span class="id" title="binder">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.GPaths.html#path"><span class="id" title="abbreviation">path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#s:19"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:18"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="const_path" class="idref" href="#const_path"><span class="id" title="definition">const_path</span></a> <a id="i:20" class="idref" href="#i:20"><span class="id" title="binder">i</span></a> <a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a> <a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a> :=  <a class="idref" href="gaia_hydras.GPaths.html#hconst_path"><span class="id" title="abbreviation">hconst_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:20"><span class="id" title="variable">i</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:21"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:22"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="standard_path" class="idref" href="#standard_path"><span class="id" title="definition">standard_path</span></a> <a id="i:23" class="idref" href="#i:23"><span class="id" title="binder">i</span></a> <a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="j:25" class="idref" href="#j:25"><span class="id" title="binder">j</span></a> <a id="b:26" class="idref" href="#b:26"><span class="id" title="binder">b</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">Paths.standard_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:23"><span class="id" title="variable">i</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:24"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#j:25"><span class="id" title="variable">j</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:26"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="standard_gnaw" class="idref" href="#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a id="i:27" class="idref" href="#i:27"><span class="id" title="binder">i</span></a> <a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#hstandard_gnaw"><span class="id" title="abbreviation">hstandard_gnaw</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:27"><span class="id" title="variable">i</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:28"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#l:29"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="gnaw" class="idref" href="#gnaw"><span class="id" title="definition">gnaw</span></a> <a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="s:31" class="idref" href="#s:31"><span class="id" title="binder">s</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#hgnaw"><span class="id" title="abbreviation">hgnaw</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:30"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#s:31"><span class="id" title="variable">s</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="gnawS" class="idref" href="#gnawS"><span class="id" title="definition">gnawS</span></a> <a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a> <a id="s:33" class="idref" href="#s:33"><span class="id" title="binder">s</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#hgnawS"><span class="id" title="abbreviation">hgnawS</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:32"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#s:33"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pathS" class="idref" href="#pathS"><span class="id" title="definition">pathS</span></a> (<a id="from:34" class="idref" href="#from:34"><span class="id" title="binder">from</span></a>: <span class="id" title="inductive">T1</span>)(<a id="s:35" class="idref" href="#s:35"><span class="id" title="binder">s</span></a>: <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="to:36" class="idref" href="#to:36"><span class="id" title="binder">to</span></a>: <span class="id" title="inductive">T1</span>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#hpathS"><span class="id" title="abbreviation">hpathS</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#from:34"><span class="id" title="variable">from</span></a>) <a class="idref" href="gaia_hydras.GPaths.html#s:35"><span class="id" title="variable">s</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:36"><span class="id" title="variable">to</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="path_tob" class="idref" href="#path_tob"><span class="id" title="definition">path_tob</span></a> (<a id="b:37" class="idref" href="#b:37"><span class="id" title="binder">b</span></a>: <span class="id" title="inductive">T1</span>) (<a id="s:38" class="idref" href="#s:38"><span class="id" title="binder">s</span></a>: <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:39" class="idref" href="#a:39"><span class="id" title="binder">a</span></a>:<span class="id" title="inductive">T1</span>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.GPaths.html#s:38"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <span class="id" title="notation">[::]</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="notation">[::</span> <span class="id" title="var">i</span><span class="id" title="notation">]</span> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.GPaths.html#transitionb"><span class="id" title="definition">transitionb</span></a> <span class="id" title="var">i</span> <a class="idref" href="gaia_hydras.GPaths.html#a:39"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:37"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">i</span> <span class="id" title="notation">::</span> <span class="id" title="var">s</span> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.GPaths.html#path_tob:40"><span class="id" title="definition">path_tob</span></a>  <a class="idref" href="gaia_hydras.GPaths.html#b:37"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GPaths.html#s:38"><span class="id" title="variable">s</span></a> (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:39"><span class="id" title="variable">a</span></a> <span class="id" title="var">i</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+SSreflect's iota was already defined in Prelude 
+<div class="paragraph"> </div>
+
+ To simplify 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_inv1" class="idref" href="#path_to_inv1"><span class="id" title="lemma">path_to_inv1</span></a> <a id="to:42" class="idref" href="#to:42"><span class="id" title="binder">to</span></a> <a id="i:43" class="idref" href="#i:43"><span class="id" title="binder">i</span></a> <a id="from:44" class="idref" href="#from:44"><span class="id" title="binder">from</span></a> : <a class="idref" href="gaia_hydras.GPaths.html#path_to"><span class="id" title="definition">path_to</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:42"><span class="id" title="variable">to</span></a> <span class="id" title="notation">[::</span> <a class="idref" href="gaia_hydras.GPaths.html#i:43"><span class="id" title="variable">i</span></a><span class="id" title="notation">]</span> <a class="idref" href="gaia_hydras.GPaths.html#from:44"><span class="id" title="variable">from</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#i:43"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.GPaths.html#transition"><span class="id" title="definition">transition</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:43"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#from:44"><span class="id" title="variable">from</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:42"><span class="id" title="variable">to</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_iff1" class="idref" href="#path_to_iff1"><span class="id" title="lemma">path_to_iff1</span></a> <a id="to:45" class="idref" href="#to:45"><span class="id" title="binder">to</span></a> <a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a> <a id="from:47" class="idref" href="#from:47"><span class="id" title="binder">from</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#from:47"><span class="id" title="variable">from</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:46"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#from:47"><span class="id" title="variable">from</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#path_to"><span class="id" title="definition">path_to</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:45"><span class="id" title="variable">to</span></a> <span class="id" title="notation">[::</span> <a class="idref" href="gaia_hydras.GPaths.html#i:46"><span class="id" title="variable">i</span></a><span class="id" title="notation">]</span> <a class="idref" href="gaia_hydras.GPaths.html#from:47"><span class="id" title="variable">from</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.GPaths.html#to:45"><span class="id" title="variable">to</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#from:47"><span class="id" title="variable">from</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:46"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#from:47"><span class="id" title="variable">from</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_adapt" class="idref" href="#iota_adapt"><span class="id" title="lemma">iota_adapt</span></a> <a id="i:48" class="idref" href="#i:48"><span class="id" title="binder">i</span></a> <a id="l:49" class="idref" href="#l:49"><span class="id" title="binder">l</span></a>: <span class="id" title="definition">iota</span> <a class="idref" href="gaia_hydras.GPaths.html#i:48"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#l:49"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">MoreLists.iota_from</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:48"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#l:49"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_iota_from" class="idref" href="#standard_gnaw_iota_from"><span class="id" title="lemma">standard_gnaw_iota_from</span></a> <a id="i:50" class="idref" href="#i:50"><span class="id" title="binder">i</span></a> <a id="a:51" class="idref" href="#a:51"><span class="id" title="binder">a</span></a> <a id="l:52" class="idref" href="#l:52"><span class="id" title="binder">l</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#i:50"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:50"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:51"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#l:52"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.GPaths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:51"><span class="id" title="variable">a</span></a> (<span class="id" title="definition">iota</span> <a class="idref" href="gaia_hydras.GPaths.html#i:50"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#l:52"><span class="id" title="variable">l</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_def" class="idref" href="#interval_def"><span class="id" title="lemma">interval_def</span></a> <a id="i:53" class="idref" href="#i:53"><span class="id" title="binder">i</span></a> <a id="j:54" class="idref" href="#j:54"><span class="id" title="binder">j</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">MoreLists.interval</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:53"><span class="id" title="variable">i</span></a> <a class="idref" href="gaia_hydras.GPaths.html#j:54"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">index_iota</span> <a class="idref" href="gaia_hydras.GPaths.html#i:53"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GPaths.html#j:54"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_LT" class="idref" href="#path_to_LT"><span class="id" title="lemma">path_to_LT</span></a> <a id="b:55" class="idref" href="#b:55"><span class="id" title="binder">b</span></a> <a id="s:56" class="idref" href="#s:56"><span class="id" title="binder">s</span></a> <a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#path_to"><span class="id" title="definition">path_to</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:55"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GPaths.html#s:56"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#b:55"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:55"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:57"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_path_to" class="idref" href="#LT_path_to"><span class="id" title="lemma">LT_path_to</span></a> (<a id="a:58" class="idref" href="#a:58"><span class="id" title="binder">a</span></a> <a id="b:59" class="idref" href="#b:59"><span class="id" title="binder">b</span></a> : <span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:58"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#b:59"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:59"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:58"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.GPaths.html#path_to"><span class="id" title="definition">path_to</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:59"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.GPaths.html#s:60"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:58"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="KS_thm_2_4" class="idref" href="#KS_thm_2_4"><span class="id" title="lemma">KS_thm_2_4</span></a> (<a id="lambda:61" class="idref" href="#lambda:61"><span class="id" title="binder">lambda</span></a> : <span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#lambda:61"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1limit</span> <a class="idref" href="gaia_hydras.GPaths.html#lambda:61"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:62" class="idref" href="#i:62"><span class="id" title="binder">i</span></a> <a id="j:63" class="idref" href="#j:63"><span class="id" title="binder">j</span></a>, (<a class="idref" href="gaia_hydras.GPaths.html#i:62"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#j:63"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#const_path"><span class="id" title="definition">const_path</span></a> 1 (<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#lambda:61"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GPaths.html#j:63"><span class="id" title="variable">j</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.GCanon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="gaia_hydras.GPaths.html#lambda:61"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:62"><span class="id" title="variable">i</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cor12" class="idref" href="#Cor12"><span class="id" title="lemma">Cor12</span></a> (<a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a>: <span class="id" title="inductive">T1</span>) : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:65" class="idref" href="#b:65"><span class="id" title="binder">b</span></a> <a id="i:66" class="idref" href="#i:66"><span class="id" title="binder">i</span></a> <a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a>, <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#b:65"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:65"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.GPaths.html#i:66"><span class="id" title="variable">i</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#n:67"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:66"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:65"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#n:67"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:65"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma2_6_1" class="idref" href="#Lemma2_6_1"><span class="id" title="lemma">Lemma2_6_1</span></a> (<a id="a:68" class="idref" href="#a:68"><span class="id" title="binder">a</span></a>:<span class="id" title="inductive">T1</span>) :<br/>
+<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:68"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:69" class="idref" href="#b:69"><span class="id" title="binder">b</span></a>, <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#b:69"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:69"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:68"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.GPaths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#n:70"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:68"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:69"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="constant_to_standard_path" class="idref" href="#constant_to_standard_path"><span class="id" title="lemma">constant_to_standard_path</span></a> (<a id="a:71" class="idref" href="#a:71"><span class="id" title="binder">a</span></a> <a id="b:72" class="idref" href="#b:72"><span class="id" title="binder">b</span></a> : <span class="id" title="inductive">T1</span>) (<a id="i:73" class="idref" href="#i:73"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:73"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:72"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="constructor">zero</span>  <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:74" class="idref" href="#j:74"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.GPaths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:73"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#j:74"><span class="id" title="variable">j</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:72"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span>  <a id="LT_to_standard_path" class="idref" href="#LT_to_standard_path"><span class="id" title="lemma">LT_to_standard_path</span></a> (<a id="a:75" class="idref" href="#a:75"><span class="id" title="binder">a</span></a> <a id="b:76" class="idref" href="#b:76"><span class="id" title="binder">b</span></a> : <span class="id" title="inductive">T1</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.GPaths.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:76"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.GPaths.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="n:77" class="idref" href="#n:77"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:78" class="idref" href="#j:78"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.GPaths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="gaia_hydras.GPaths.html#n:77"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.GPaths.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.GPaths.html#j:78"><span class="id" title="variable">j</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:76"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab469"></a><h1 class="section">Adaptation to E0</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hCanon_plus" class="idref" href="#hCanon_plus"><span class="id" title="abbreviation">hCanon_plus</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Canon_plus" class="idref" href="#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a id="i:79" class="idref" href="#i:79"><span class="id" title="binder">i</span></a> <a id="a:80" class="idref" href="#a:80"><span class="id" title="binder">a</span></a> <a id="b:81" class="idref" href="#b:81"><span class="id" title="binder">b</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GPaths.html#hCanon_plus"><span class="id" title="abbreviation">hCanon_plus</span></a> <a class="idref" href="gaia_hydras.GPaths.html#i:79"><span class="id" title="variable">i</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#a:80"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.GPaths.html#b:81"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab470"></a><h1 class="section">Examples</h1>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="ex_path1" class="idref" href="#ex_path1"><span class="id" title="definition">ex_path1</span></a> : <a class="idref" href="gaia_hydras.GPaths.html#path"><span class="id" title="abbreviation">path</span></a> (<span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">(</span><span class="id" title="notation">\</span><span class="id" title="notation">F</span> 2<span class="id" title="notation">)</span>) <span class="id" title="notation">[::</span> 2<span class="id" title="notation">;</span> 2<span class="id" title="notation">;</span> 2<span class="id" title="notation">]</span> <span class="id" title="definition">T1omega</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path2" class="idref" href="#ex_path2"><span class="id" title="definition">ex_path2</span></a>: <a class="idref" href="gaia_hydras.GPaths.html#path"><span class="id" title="abbreviation">path</span></a> (<span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 2) <span class="id" title="notation">[::</span> 3<span class="id" title="notation">;</span> 4<span class="id" title="notation">;</span> 5<span class="id" title="notation">;</span> 6<span class="id" title="notation">]</span> <span class="id" title="definition">T1omega</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path3" class="idref" href="#ex_path3"><span class="id" title="definition">ex_path3</span></a>: <a class="idref" href="gaia_hydras.GPaths.html#path"><span class="id" title="abbreviation">path</span></a> (<span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 2) (<span class="id" title="definition">index_iota</span> 3 15) <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path4" class="idref" href="#ex_path4"><span class="id" title="definition">ex_path4</span></a>: <a class="idref" href="gaia_hydras.GPaths.html#path"><span class="id" title="abbreviation">path</span></a> (<span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 2) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#repeat"><span class="id" title="definition">List.repeat</span></a> 3 8) <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GPrelude.html b/theories/html/gaia_hydras.GPrelude.html
new file mode 100644
index 00000000..27c58c88
--- /dev/null
+++ b/theories/html/gaia_hydras.GPrelude.html
@@ -0,0 +1,44 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GPrelude</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GPrelude</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Learning ssreflect:  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">Lemma</span> <a id="diffP" class="idref" href="#diffP"><span class="id" title="lemma">diffP</span></a> {<a id="T:1" class="idref" href="#T:1"><span class="id" title="binder">T</span></a>:<span class="id" title="abbreviation">eqType</span>}(<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a> <a id="j:3" class="idref" href="#j:3"><span class="id" title="binder">j</span></a>:<a class="idref" href="gaia_hydras.GPrelude.html#T:1"><span class="id" title="variable">T</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="gaia_hydras.GPrelude.html#i:2"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.GPrelude.html#j:3"><span class="id" title="variable">j</span></a>) (<a class="idref" href="gaia_hydras.GPrelude.html#i:2"><span class="id" title="variable">i</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.GPrelude.html#j:3"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.GaiaToHydra.html b/theories/html/gaia_hydras.GaiaToHydra.html
new file mode 100644
index 00000000..346da598
--- /dev/null
+++ b/theories/html/gaia_hydras.GaiaToHydra.html
@@ -0,0 +1,60 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.GaiaToHydra</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.GaiaToHydra</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> .<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="module">Epsilon0.T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hmultA" class="idref" href="#hmultA"><span class="id" title="lemma">hmultA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1mul</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.GaiaToHydra.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.GaiaToHydra.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hmult_dist" class="idref" href="#hmult_dist"><span class="id" title="lemma">hmult_dist</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1add</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.HydraGaia_Examples.html b/theories/html/gaia_hydras.HydraGaia_Examples.html
new file mode 100644
index 00000000..9b4fcf68
--- /dev/null
+++ b/theories/html/gaia_hydras.HydraGaia_Examples.html
@@ -0,0 +1,99 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.HydraGaia_Examples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.HydraGaia_Examples</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a> <a class="idref" href="gaia_hydras.GCanon.html#"><span class="id" title="library">GCanon</span></a> <a class="idref" href="gaia_hydras.GHessenberg.html#"><span class="id" title="library">GHessenberg</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.GLarge_Sets.html#"><span class="id" title="library">GLarge_Sets</span></a>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 42.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ab410a966ac148e9b78c65c6cdf301fd" class="idref" href="#ab410a966ac148e9b78c65c6cdf301fd"><span class="id" title="definition">α</span></a> : <span class="id" title="inductive">T1</span> :=<br/>
+&nbsp;<span class="id" title="definition">T1omega</span> <span class="id" title="notation">+</span>  <span class="id" title="definition">phi0</span> <span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 3 <span class="id" title="notation">+</span> <span class="id" title="definition">phi0</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 5) <span class="id" title="notation">×</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 4 <span class="id" title="notation">+</span> <span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="definition">T1omega</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="dc5233cb1d950ecad15b1e9b2514f665" class="idref" href="#dc5233cb1d950ecad15b1e9b2514f665"><span class="id" title="definition">β</span></a> : <span class="id" title="inductive">T1</span> :=  <span class="id" title="definition">phi0</span> (<span class="id" title="definition">phi0</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 2)).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Hessenberg's sum 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="gaia_hydras.GHessenberg.html#oplus"><span class="id" title="definition">oplus</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="definition">T1lt</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 42) <span class="id" title="definition">T1omega</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> 42 <span class="id" title="notation">&lt;</span> <span class="id" title="definition">T1omega</span>)%<span class="id" title="var">ca</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> 42 <span class="id" title="notation">&lt;</span> <span class="id" title="definition">T1omega</span>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1.T1nat</span></a> 42) <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1.T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> 42)%<span class="id" title="var">t1</span> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1.T1omega</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.ON_gfinite.html b/theories/html/gaia_hydras.ON_gfinite.html
new file mode 100644
index 00000000..3a4789e7
--- /dev/null
+++ b/theories/html/gaia_hydras.ON_gfinite.html
@@ -0,0 +1,86 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.ON_gfinite</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.ON_gfinite</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ordinals a la mathcomp 
+</div>
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="finord_lt" class="idref" href="#finord_lt"><span class="id" title="definition">finord_lt</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> <a id="beta:3" class="idref" href="#beta:3"><span class="id" title="binder">beta</span></a>: <span class="id" title="notation">'</span><span class="id" title="notation">I_n</span>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.ON_gfinite.html#alpha:2"><span class="id" title="variable">alpha</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.ON_gfinite.html#beta:3"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span>  <a id="finord_compare" class="idref" href="#finord_compare"><span class="id" title="instance">finord_compare</span></a> (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> (<span class="id" title="notation">'</span><span class="id" title="notation">I_n</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:5" class="idref" href="#alpha:5"><span class="id" title="binder">alpha</span></a> <a id="beta:6" class="idref" href="#beta:6"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#compare"><span class="id" title="definition">Nat.compare</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#alpha:5"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#beta:6"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finord_compare_correct" class="idref" href="#finord_compare_correct"><span class="id" title="lemma">finord_compare_correct</span></a> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> <a id="beta:9" class="idref" href="#beta:9"><span class="id" title="binder">beta</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">I_n</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_lt"><span class="id" title="definition">finord_lt</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:7"><span class="id" title="variable">n</span></a>) <a class="idref" href="gaia_hydras.ON_gfinite.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#beta:9"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#beta:9"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="finord_sto" class="idref" href="#finord_sto"><span class="id" title="instance">finord_sto</span></a> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_lt"><span class="id" title="definition">finord_lt</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:10"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="finord_comp" class="idref" href="#finord_comp"><span class="id" title="instance">finord_comp</span></a> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_lt"><span class="id" title="definition">finord_lt</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:11"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_compare"><span class="id" title="instance">finord_compare</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:11"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finord_lt_wf" class="idref" href="#finord_lt_wf"><span class="id" title="lemma">finord_lt_wf</span></a> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_lt"><span class="id" title="definition">finord_lt</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:12"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="finord_ON" class="idref" href="#finord_ON"><span class="id" title="instance">finord_ON</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_lt"><span class="id" title="definition">finord_lt</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:23"><span class="id" title="variable">n</span></a>) (@<a class="idref" href="gaia_hydras.ON_gfinite.html#finord_compare"><span class="id" title="instance">finord_compare</span></a> <a class="idref" href="gaia_hydras.ON_gfinite.html#n:23"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Examples 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Example</span> <a id="o_33_of_42" class="idref" href="#o_33_of_42"><span class="id" title="definition">o_33_of_42</span></a>: <span class="id" title="notation">'</span><span class="id" title="notation">I_42</span> := @<span class="id" title="constructor">Ordinal</span> 42 33 <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Example</span> <a id="o_36_of_42" class="idref" href="#o_36_of_42"><span class="id" title="definition">o_36_of_42</span></a>: <span class="id" title="notation">'</span><span class="id" title="notation">I_42</span> :=  @<span class="id" title="constructor">Ordinal</span> 42 36 <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,682 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.T1Bridge</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.T1Bridge</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab417"></a><h1 class="section">Bridge between Hydra-battle's and Gaia's <span class="inlinecode"><span class="id" title="var">T1</span></span>  (Experimental)</h1>
+
+
+<div class="paragraph"> </div>
+
+This library introduces tools for making  definitions and lemmas from
+ <a href="../ordinals/"> Hydra battles's ordinals</a>  compatible with 
+ <a href="https://github.com/coq-community/gaia/tree/master/theories/schutte">Gaia's ssete9 library</a>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The name <span class="inlinecode"><span class="id" title="var">T1</span></span> denotes Gaia's data type. We use <span class="inlinecode"><span class="id" title="var">T1.T1</span></span> or <span class="inlinecode"><span class="id" title="var">hT1</span></span> for Hydra battles ordinal terms.
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hT1" class="idref" href="#hT1"><span class="id" title="abbreviation">hT1</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1.T1</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="T1" class="idref" href="#T1"><span class="id" title="abbreviation">T1</span></a> := <span class="id" title="inductive">ssete9.CantorOrdinal.T1</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We rename Hydra battle's total order on <span class="inlinecode"><span class="id" title="var">hT1</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Restrictions to terms in normal form 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LT" class="idref" href="#LT"><span class="id" title="abbreviation">LT</span></a> := (<span class="id" title="definition">restrict</span> <span class="id" title="definition">T1nf</span> <span class="id" title="definition">T1lt</span>).<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LE" class="idref" href="#LE"><span class="id" title="abbreviation">LE</span></a> := (<span class="id" title="definition">restrict</span> <span class="id" title="definition">T1nf</span> <span class="id" title="definition">T1le</span>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Translation functions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="h2g" class="idref" href="#h2g"><span class="id" title="definition">h2g</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:1"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <span class="id" title="constructor">cons</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g:2"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:1"><span class="id" title="variable">a</span></a>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g:2"><span class="id" title="definition">h2g</span></a> <span class="id" title="var">b</span>) <span class="id" title="keyword">else</span> <span class="id" title="constructor">zero</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="g2h" class="idref" href="#g2h"><span class="id" title="definition">g2h</span></a> (<a id="a:4" class="idref" href="#a:4"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:4"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">is</span> <span class="id" title="constructor">cons</span> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h:5"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:4"><span class="id" title="variable">a</span></a>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h:5"><span class="id" title="definition">g2h</span></a> <span class="id" title="var">b</span>) <span class="id" title="keyword">else</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h2g_g2hK" class="idref" href="#h2g_g2hK"><span class="id" title="lemma">h2g_g2hK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_h2gK" class="idref" href="#g2h_h2gK"><span class="id" title="lemma">g2h_h2gK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_eqE" class="idref" href="#g2h_eqE"><span class="id" title="lemma">g2h_eqE</span></a> (<a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:8"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:8"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h2g_eqE" class="idref" href="#h2g_eqE"><span class="id" title="lemma">h2g_eqE</span></a> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> :<a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:10"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_diffE" class="idref" href="#g2h_diffE"><span class="id" title="lemma">g2h_diffE</span></a> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:12"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:12"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h2g_diffE" class="idref" href="#h2g_diffE"><span class="id" title="lemma">h2g_diffE</span></a> (<a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a> <a id="b:14" class="idref" href="#b:14"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:14"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:14"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Pretty printing 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="T1pp" class="idref" href="#T1pp"><span class="id" title="definition">T1pp</span></a> (<a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#pp"><span class="id" title="definition">pp</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:15"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab418"></a><h1 class="section">refinement of constants, functions, etc.</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="refines0" class="idref" href="#refines0"><span class="id" title="definition">refines0</span></a> (<a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>)(<a id="y:17" class="idref" href="#y:17"><span class="id" title="binder">y</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) :=  <a class="idref" href="gaia_hydras.T1Bridge.html#y:17"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:16"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="refines1" class="idref" href="#refines1"><span class="id" title="definition">refines1</span></a> (<a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) (<a id="f':19" class="idref" href="#f':19"><span class="id" title="binder">f'</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:20" class="idref" href="#x:20"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#f':19"><span class="id" title="variable">f'</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:20"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:20"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="refines2" class="idref" href="#refines2"><span class="id" title="definition">refines2</span></a> (<a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) (<a id="f':22" class="idref" href="#f':22"><span class="id" title="binder">f'</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> ) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> <a id="y:24" class="idref" href="#y:24"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#f':22"><span class="id" title="variable">f'</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:23"><span class="id" title="variable">x</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:24"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:24"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="refinesPred" class="idref" href="#refinesPred"><span class="id" title="definition">refinesPred</span></a> (<a id="hP:25" class="idref" href="#hP:25"><span class="id" title="binder">hP</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="gP:26" class="idref" href="#gP:26"><span class="id" title="binder">gP</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:27" class="idref" href="#x:27"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#hP:25"><span class="id" title="variable">hP</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:27"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#gP:26"><span class="id" title="variable">gP</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:27"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="refinesRel" class="idref" href="#refinesRel"><span class="id" title="definition">refinesRel</span></a> (<a id="hR:28" class="idref" href="#hR:28"><span class="id" title="binder">hR</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)<br/>
+&nbsp;&nbsp;(<a id="gR:29" class="idref" href="#gR:29"><span class="id" title="binder">gR</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:30" class="idref" href="#x:30"><span class="id" title="binder">x</span></a> <a id="y:31" class="idref" href="#y:31"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#hR:28"><span class="id" title="variable">hR</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:30"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:31"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#gR:29"><span class="id" title="variable">gR</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:30"><span class="id" title="variable">x</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:31"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="refines1_R" class="idref" href="#refines1_R"><span class="id" title="lemma">refines1_R</span></a> <a id="f:32" class="idref" href="#f:32"><span class="id" title="binder">f</span></a> <a id="f':33" class="idref" href="#f':33"><span class="id" title="binder">f'</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#refines1"><span class="id" title="definition">refines1</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f:32"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f':33"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <span class="id" title="keyword">∀</span> <a id="y:34" class="idref" href="#y:34"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#f:32"><span class="id" title="variable">f</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:34"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f':33"><span class="id" title="variable">f'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:34"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="refines2_R" class="idref" href="#refines2_R"><span class="id" title="lemma">refines2_R</span></a> <a id="f:35" class="idref" href="#f:35"><span class="id" title="binder">f</span></a> <a id="f':36" class="idref" href="#f':36"><span class="id" title="binder">f'</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#refines2"><span class="id" title="definition">refines2</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f:35"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f':36"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <span class="id" title="keyword">∀</span> <a id="y:37" class="idref" href="#y:37"><span class="id" title="binder">y</span></a> <a id="z:38" class="idref" href="#z:38"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#f:35"><span class="id" title="variable">f</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:37"><span class="id" title="variable">y</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#z:38"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f':36"><span class="id" title="variable">f'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:37"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#z:38"><span class="id" title="variable">z</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab419"></a><h2 class="section">Refinements of usual constants</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_ref" class="idref" href="#zero_ref"><span class="id" title="lemma">zero_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines0"><span class="id" title="definition">refines0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="one_ref" class="idref" href="#one_ref"><span class="id" title="lemma">one_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines0"><span class="id" title="definition">refines0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">T1.one</span></a> <span class="id" title="definition">one</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Finite_ref" class="idref" href="#Finite_ref"><span class="id" title="lemma">Finite_ref</span></a> (<a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#refines0"><span class="id" title="definition">refines0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1.T1nat</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:39"><span class="id" title="variable">n</span></a>) (<span class="id" title="notation">\</span><span class="id" title="notation">F</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:39"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_ref" class="idref" href="#omega_ref"><span class="id" title="lemma">omega_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines0"><span class="id" title="definition">refines0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1.T1omega</span></a> <span class="id" title="definition">T1omega</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab420"></a><h2 class="section">unary functions and predicates</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_ref" class="idref" href="#succ_ref"><span class="id" title="lemma">succ_ref</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#refines1"><span class="id" title="definition">refines1</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <span class="id" title="definition">T1succ</span>.<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_ref" class="idref" href="#phi0_ref"><span class="id" title="lemma">phi0_ref</span></a> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#refines0"><span class="id" title="definition">refines0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:40"><span class="id" title="variable">x</span></a>) (<span class="id" title="definition">phi0</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:40"><span class="id" title="variable">x</span></a>)).  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_phi0" class="idref" href="#g2h_phi0"><span class="id" title="lemma">g2h_phi0</span></a> <a id="a:41" class="idref" href="#a:41"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<span class="id" title="definition">phi0</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:41"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:41"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ap_ref" class="idref" href="#ap_ref"><span class="id" title="lemma">ap_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refinesPred"><span class="id" title="definition">refinesPred</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ap"><span class="id" title="inductive">Epsilon0.T1.ap</span></a> <span class="id" title="definition">T1ap</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab421"></a><h2 class="section">Equality and comparison</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1eq_refl" class="idref" href="#T1eq_refl"><span class="id" title="lemma">T1eq_refl</span></a> (<a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <span class="id" title="definition">T1eq</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:42"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1eqE" class="idref" href="#T1eqE"><span class="id" title="lemma">T1eqE</span></a> <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a>: <span class="id" title="definition">T1eq</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:44"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1eq_h2g" class="idref" href="#T1eq_h2g"><span class="id" title="lemma">T1eq_h2g</span></a> (<a id="a:45" class="idref" href="#a:45"><span class="id" title="binder">a</span></a> <a id="b:46" class="idref" href="#b:46"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) : <span class="id" title="definition">T1eq</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:45"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:46"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:45"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:46"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_ref" class="idref" href="#compare_ref"><span class="id" title="lemma">compare_ref</span></a> (<a id="x:47" class="idref" href="#x:47"><span class="id" title="binder">x</span></a> <a id="y:48" class="idref" href="#y:48"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#compare_T1"><span class="id" title="instance">T1.compare_T1</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:47"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:48"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <span class="id" title="definition">T1lt</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:47"><span class="id" title="variable">x</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:48"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Datatypes.Eq</span></a> ⇒ <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:47"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:48"><span class="id" title="variable">y</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Datatypes.Gt</span></a> ⇒ <span class="id" title="definition">T1lt</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:48"><span class="id" title="variable">y</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:47"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decide_hltE" class="idref" href="#decide_hltE"><span class="id" title="lemma">decide_hltE</span></a> (<a id="a:49" class="idref" href="#a:49"><span class="id" title="binder">a</span></a> <a id="b:50" class="idref" href="#b:50"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:50"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:49"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:50"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_ref" class="idref" href="#lt_ref"><span class="id" title="lemma">lt_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refinesRel"><span class="id" title="definition">refinesRel</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <span class="id" title="definition">T1lt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_ref" class="idref" href="#le_ref"><span class="id" title="lemma">le_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refinesRel"><span class="id" title="definition">refinesRel</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#le"><span class="id" title="abbreviation">T1.le</span></a> <span class="id" title="definition">T1le</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab422"></a><h2 class="section">Ordinal addition</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_ref" class="idref" href="#plus_ref"><span class="id" title="lemma">plus_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines2"><span class="id" title="definition">refines2</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1.T1add</span></a> <span class="id" title="definition">T1add</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab423"></a><h2 class="section">Ordinal multiplication</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_mult_ref" class="idref" href="#Proof_of_mult_ref"><span class="id" title="section">Proof_of_mult_ref</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="T1mul_a0E" class="idref" href="#T1mul_a0E"><span class="id" title="lemma">T1mul_a0E</span></a> (<a id="c:51" class="idref" href="#c:51"><span class="id" title="binder">c</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#c:51"><span class="id" title="variable">c</span></a> <span class="id" title="notation">×</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="T1mul_cons_cons_E" class="idref" href="#T1mul_cons_cons_E"><span class="id" title="lemma">T1mul_cons_cons_E</span></a> <a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> <a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a> <a id="n':54" class="idref" href="#n':54"><span class="id" title="binder">n'</span></a> <a id="b':55" class="idref" href="#b':55"><span class="id" title="binder">b'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">cons</span> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:53"><span class="id" title="variable">b</span></a> <span class="id" title="notation">×</span> <span class="id" title="constructor">cons</span> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':54"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b':55"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">cons</span> <span class="id" title="constructor">zero</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#n:52"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':54"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:52"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':54"><span class="id" title="variable">n'</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#b':55"><span class="id" title="variable">b'</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="T1mulE4" class="idref" href="#T1mulE4"><span class="id" title="lemma">T1mulE4</span></a> <a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a> <a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> <a id="b:58" class="idref" href="#b:58"><span class="id" title="binder">b</span></a> <a id="n':59" class="idref" href="#n':59"><span class="id" title="binder">n'</span></a> <a id="b':60" class="idref" href="#b':60"><span class="id" title="binder">b'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a:56"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="notation">(</span><span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:58"><span class="id" title="variable">b</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">×</span> <span class="id" title="notation">(</span><span class="id" title="constructor">cons</span> <span class="id" title="constructor">zero</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':59"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b':60"><span class="id" title="variable">b'</span></a><span class="id" title="notation">)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:56"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#n:57"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':59"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:57"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':59"><span class="id" title="variable">n'</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#b:58"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="multE4" class="idref" href="#multE4"><span class="id" title="lemma">multE4</span></a> <a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> <a id="b:63" class="idref" href="#b:63"><span class="id" title="binder">b</span></a> <a id="n':64" class="idref" href="#n':64"><span class="id" title="binder">n'</span></a> <a id="b':65" class="idref" href="#b':65"><span class="id" title="binder">b'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">Epsilon0.T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">Epsilon0.T1.T1mul</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:63"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">Epsilon0.T1.zero</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n':64"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b':65"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:61"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#n:62"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':64"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:62"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n':64"><span class="id" title="variable">n'</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#b:63"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="T1mulE5" class="idref" href="#T1mulE5"><span class="id" title="lemma">T1mulE5</span></a> <a id="a:66" class="idref" href="#a:66"><span class="id" title="binder">a</span></a> <a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a> <a id="b:68" class="idref" href="#b:68"><span class="id" title="binder">b</span></a> <a id="a':69" class="idref" href="#a':69"><span class="id" title="binder">a'</span></a> <a id="n':70" class="idref" href="#n':70"><span class="id" title="binder">n'</span></a> <a id="b':71" class="idref" href="#b':71"><span class="id" title="binder">b'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a':69"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">!=</span> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:68"><span class="id" title="variable">b</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">×</span> <span class="id" title="notation">(</span><span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a':69"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n':70"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b':71"><span class="id" title="variable">b'</span></a><span class="id" title="notation">)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">cons</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#a:66"><span class="id" title="variable">a</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a':69"><span class="id" title="variable">a'</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#n':70"><span class="id" title="variable">n'</span></a> (<span class="id" title="definition">T1mul</span> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:68"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#b':71"><span class="id" title="variable">b'</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="multE5" class="idref" href="#multE5"><span class="id" title="lemma">multE5</span></a> <a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> <a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> <a id="b:74" class="idref" href="#b:74"><span class="id" title="binder">b</span></a> <a id="a':75" class="idref" href="#a':75"><span class="id" title="binder">a'</span></a> <a id="n':76" class="idref" href="#n':76"><span class="id" title="binder">n'</span></a> <a id="b':77" class="idref" href="#b':77"><span class="id" title="binder">b'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a':75"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">Epsilon0.T1.T1mul</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:74"><span class="id" title="variable">b</span></a>)  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a':75"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n':76"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b':77"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1.T1add</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:72"><span class="id" title="variable">a</span></a>  <a class="idref" href="gaia_hydras.T1Bridge.html#a':75"><span class="id" title="variable">a'</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#n':76"><span class="id" title="variable">n'</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1.T1mul</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:74"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#b':77"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="h2g_cons" class="idref" href="#h2g_cons"><span class="id" title="lemma">h2g_cons</span></a> <a id="a:78" class="idref" href="#a:78"><span class="id" title="binder">a</span></a> <a id="n:79" class="idref" href="#n:79"><span class="id" title="binder">n</span></a> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:79"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:80"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">cons</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:78"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#n:79"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:80"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="g2h_cons" class="idref" href="#g2h_cons"><span class="id" title="lemma">g2h_cons</span></a> <a id="a:81" class="idref" href="#a:81"><span class="id" title="binder">a</span></a> <a id="n:82" class="idref" href="#n:82"><span class="id" title="binder">n</span></a> <a id="b:83" class="idref" href="#b:83"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<span class="id" title="constructor">cons</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:81"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:82"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:83"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:81"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#n:82"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:83"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="h2g_zero" class="idref" href="#h2g_zero"><span class="id" title="lemma">h2g_zero</span></a>  : <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="constructor">zero</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="g2h_zero" class="idref" href="#g2h_zero"><span class="id" title="lemma">g2h_zero</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <span class="id" title="constructor">zero</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="mult_ref0" class="idref" href="#mult_ref0"><span class="id" title="lemma">mult_ref0</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines2"><span class="id" title="definition">refines2</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1.T1mul</span></a> <span class="id" title="definition">T1mul</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.T1Bridge.html#Proof_of_mult_ref"><span class="id" title="section">Proof_of_mult_ref</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_ref" class="idref" href="#mult_ref"><span class="id" title="lemma">mult_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refines2"><span class="id" title="definition">refines2</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1.T1mul</span></a> <span class="id" title="definition">T1mul</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_multE" class="idref" href="#g2h_multE"><span class="id" title="lemma">g2h_multE</span></a> (<a id="a:84" class="idref" href="#a:84"><span class="id" title="binder">a</span></a> <a id="b:85" class="idref" href="#b:85"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#a:84"><span class="id" title="variable">a</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.T1Bridge.html#b:85"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1.T1mul</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:84"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:85"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_plusE" class="idref" href="#g2h_plusE"><span class="id" title="lemma">g2h_plusE</span></a> (<a id="a:86" class="idref" href="#a:86"><span class="id" title="binder">a</span></a> <a id="b:87" class="idref" href="#b:87"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#a:86"><span class="id" title="variable">a</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.T1Bridge.html#b:87"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1.T1add</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:86"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:87"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab424"></a><h1 class="section">Ordinal terms in normal form</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_ref" class="idref" href="#nf_ref"><span class="id" title="lemma">nf_ref</span></a> (<a id="a:88" class="idref" href="#a:88"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>)  : <a class="idref" href="hydras.Epsilon0.T1.html#nf_b"><span class="id" title="definition">T1.nf_b</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">T1nf</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:88"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_ref" class="idref" href="#LT_ref"><span class="id" title="lemma">LT_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refinesRel"><span class="id" title="definition">refinesRel</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">T1.LT</span></a>  <a class="idref" href="gaia_hydras.T1Bridge.html#LT"><span class="id" title="abbreviation">LT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_ref" class="idref" href="#LE_ref"><span class="id" title="lemma">LE_ref</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#refinesRel"><span class="id" title="definition">refinesRel</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LE"><span class="id" title="definition">T1.LE</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#LE"><span class="id" title="abbreviation">LE</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Limits, successors, etc 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_ref" class="idref" href="#T1limit_ref"><span class="id" title="lemma">T1limit_ref</span></a> (<a id="a:89" class="idref" href="#a:89"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">Epsilon0.T1.T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1.T1limit</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:89"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">T1limit</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:89"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1is_succ_ref" class="idref" href="#T1is_succ_ref"><span class="id" title="lemma">T1is_succ_ref</span></a> (<a id="a:90" class="idref" href="#a:90"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">Epsilon0.T1.T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1.T1is_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">T1is_succ</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:90"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hnf_g2h" class="idref" href="#hnf_g2h"><span class="id" title="lemma">hnf_g2h</span></a> <a id="a:91" class="idref" href="#a:91"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:91"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:91"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_succ" class="idref" href="#g2h_succ"><span class="id" title="lemma">g2h_succ</span></a> <a id="a:92" class="idref" href="#a:92"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<span class="id" title="definition">T1succ</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:92"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:92"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hlt_iff" class="idref" href="#hlt_iff"><span class="id" title="lemma">hlt_iff</span></a> <a id="a:93" class="idref" href="#a:93"><span class="id" title="binder">a</span></a> <a id="b:94" class="idref" href="#b:94"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:93"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:94"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:93"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:94"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1lt_iff" class="idref" href="#T1lt_iff"><span class="id" title="lemma">T1lt_iff</span></a> <a id="a:95" class="idref" href="#a:95"><span class="id" title="binder">a</span></a> <a id="b:96" class="idref" href="#b:96"><span class="id" title="binder">b</span></a>: <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:95"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.T1Bridge.html#b:96"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a:95"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#b:96"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:95"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:96"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1le_iff" class="idref" href="#T1le_iff"><span class="id" title="lemma">T1le_iff</span></a> (<a id="a:97" class="idref" href="#a:97"><span class="id" title="binder">a</span></a> <a id="b:98" class="idref" href="#b:98"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T1Bridge.html#a:97"><span class="id" title="variable">a</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.T1Bridge.html#b:98"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#le"><span class="id" title="abbreviation">T1.le</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:97"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:98"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab425"></a><h1 class="section">Well formed ordinals as a data type</h1>
+
+</div>
+<div class="code">
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0" class="idref" href="#hE0"><span class="id" title="abbreviation">hE0</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0.E0</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hcnf" class="idref" href="#hcnf"><span class="id" title="abbreviation">hcnf</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">E0.cnf</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0lt" class="idref" href="#hE0lt"><span class="id" title="abbreviation">hE0lt</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0.E0lt</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0le" class="idref" href="#hE0le"><span class="id" title="abbreviation">hE0le</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0le"><span class="id" title="definition">E0.E0le</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0zero" class="idref" href="#hE0zero"><span class="id" title="abbreviation">hE0zero</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0.E0zero</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0omega" class="idref" href="#hE0omega"><span class="id" title="abbreviation">hE0omega</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0.E0_omega</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0phi0" class="idref" href="#hE0phi0"><span class="id" title="abbreviation">hE0phi0</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0.E0_phi0</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0fin" class="idref" href="#hE0fin"><span class="id" title="abbreviation">hE0fin</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0.E0fin</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0limit" class="idref" href="#hE0limit"><span class="id" title="abbreviation">hE0limit</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0.E0limit</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hE0is_succ" class="idref" href="#hE0is_succ"><span class="id" title="abbreviation">hE0is_succ</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0.E0is_succ</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="E0" class="idref" href="#E0"><span class="id" title="record">E0</span></a> := <a id="mkE0" class="idref" href="#mkE0"><span class="id" title="constructor">mkE0</span></a> { <a id="cnf" class="idref" href="#cnf"><span class="id" title="projection">cnf</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a> ; <span class="id" title="var">_</span> : <span class="id" title="definition">T1nf</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf:100"><span class="id" title="method">cnf</span></a> <span class="id" title="notation">==</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>}.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a><a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">:</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">E0</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="e0Sub" class="idref" href="#e0Sub"><span class="id" title="definition">e0Sub</span></a> := <span class="id" title="notation">[</span><span class="id" title="notation">subType</span> <span class="id" title="notation">for</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a><span class="id" title="notation">]</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="keyword">fun</span> (<a id="x:101" class="idref" href="#x:101"><span class="id" title="binder">x</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) ⇒ <span class="id" title="projection">val</span> <a class="idref" href="gaia_hydras.T1Bridge.html#x:101"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="omeganf" class="idref" href="#omeganf"><span class="id" title="lemma">omeganf</span></a> : <span class="id" title="definition">T1nf</span> <span class="id" title="definition">T1omega</span> <span class="id" title="notation">==</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="projection">Sub</span> <span class="id" title="definition">T1omega</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="projection">Sub</span> <span class="id" title="definition">T1omega</span> <a class="idref" href="gaia_hydras.T1Bridge.html#omeganf"><span class="id" title="lemma">omeganf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#e0Sub"><span class="id" title="definition">e0Sub</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="projection">Sub</span> <span class="id" title="definition">T1omega</span> <a class="idref" href="gaia_hydras.T1Bridge.html#omeganf"><span class="id" title="lemma">omeganf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0eqb" class="idref" href="#E0eqb"><span class="id" title="definition">E0eqb</span></a> (<a id="a:102" class="idref" href="#a:102"><span class="id" title="binder">a</span></a> <a id="b:103" class="idref" href="#b:103"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>):= <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:102"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:103"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gE0_eq_intro" class="idref" href="#gE0_eq_intro"><span class="id" title="lemma">gE0_eq_intro</span></a> <a id="a:104" class="idref" href="#a:104"><span class="id" title="binder">a</span></a> <a id="b:105" class="idref" href="#b:105"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:104"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:105"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:104"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:105"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0_eq_mixin" class="idref" href="#E0_eq_mixin"><span class="id" title="definition">E0_eq_mixin</span></a> :  <span class="id" title="record">Equality.mixin_of</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0_eqtype" class="idref" href="#E0_eqtype"><span class="id" title="definition">E0_eqtype</span></a> :=  <span class="id" title="constructor">Equality.Pack</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_eq_mixin"><span class="id" title="definition">E0_eq_mixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="keyword">Structure</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_eqtype"><span class="id" title="definition">E0_eqtype</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ppE0" class="idref" href="#ppE0"><span class="id" title="definition">ppE0</span></a> (<a id="a:106" class="idref" href="#a:106"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) := <a class="idref" href="gaia_hydras.T1Bridge.html#T1pp"><span class="id" title="definition">T1pp</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:106"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0lt" class="idref" href="#E0lt"><span class="id" title="definition">E0lt</span></a> (<a id="a:107" class="idref" href="#a:107"><span class="id" title="binder">a</span></a> <a id="b:108" class="idref" href="#b:108"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) := <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:107"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:108"><span class="id" title="variable">b</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0le" class="idref" href="#E0le"><span class="id" title="definition">E0le</span></a>  (<a id="a:109" class="idref" href="#a:109"><span class="id" title="binder">a</span></a> <a id="b:110" class="idref" href="#b:110"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) := <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:109"><span class="id" title="variable">a</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:110"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0zero" class="idref" href="#E0zero"><span class="id" title="definition">E0zero</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> := @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> <span class="id" title="constructor">zero</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>]<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="E0_succ" class="idref" href="#E0_succ"><span class="id" title="definition">E0_succ</span></a> (<a id="a:111" class="idref" href="#a:111"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=  @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<span class="id" title="definition">T1succ</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:111"><span class="id" title="variable">a</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>]<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="E0_pred" class="idref" href="#E0_pred"><span class="id" title="definition">E0_pred</span></a> (<a id="a:112" class="idref" href="#a:112"><span class="id" title="binder">a</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>:=  @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<span class="id" title="definition">T1pred</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:112"><span class="id" title="variable">a</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="E0fin" class="idref" href="#E0fin"><span class="id" title="definition">E0fin</span></a> (<a id="n:113" class="idref" href="#n:113"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:113"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">p</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin:114"><span class="id" title="definition">E0fin</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0_omega" class="idref" href="#E0_omega"><span class="id" title="definition">E0_omega</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> := @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> <span class="id" title="definition">T1omega</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>]  <span class="id" title="keyword">Definition</span> <a id="E0_phi0" class="idref" href="#E0_phi0"><span class="id" title="definition">E0_phi0</span></a> (<a id="a:116" class="idref" href="#a:116"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> := @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<span class="id" title="definition">phi0</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:116"><span class="id" title="variable">a</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0plus" class="idref" href="#E0plus"><span class="id" title="definition">E0plus</span></a> (<a id="a:117" class="idref" href="#a:117"><span class="id" title="binder">a</span></a> <a id="beta:118" class="idref" href="#beta:118"><span class="id" title="binder">beta</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<span class="id" title="definition">T1add</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:117"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:118"><span class="id" title="variable">beta</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0mul" class="idref" href="#E0mul"><span class="id" title="definition">E0mul</span></a> (<a id="a:119" class="idref" href="#a:119"><span class="id" title="binder">a</span></a> <a id="beta:120" class="idref" href="#beta:120"><span class="id" title="binder">beta</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<span class="id" title="definition">T1mul</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:119"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:120"><span class="id" title="variable">beta</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0fin_cnf" class="idref" href="#E0fin_cnf"><span class="id" title="lemma">E0fin_cnf</span></a> (<a id="n:121" class="idref" href="#n:121"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:121"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">\</span><span class="id" title="notation">F</span> <a class="idref" href="gaia_hydras.T1Bridge.html#n:121"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0_h2g" class="idref" href="#E0_h2g"><span class="id" title="definition">E0_h2g</span></a> (<a id="a:122" class="idref" href="#a:122"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hE0"><span class="id" title="abbreviation">hE0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>:= @<a class="idref" href="gaia_hydras.T1Bridge.html#mkE0"><span class="id" title="constructor">mkE0</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">E0.cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:122"><span class="id" title="variable">a</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="E0_g2h" class="idref" href="#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a id="a:123" class="idref" href="#a:123"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#hE0"><span class="id" title="abbreviation">hE0</span></a> := @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">E0.mkord</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:123"><span class="id" title="variable">a</span></a>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0limit" class="idref" href="#E0limit"><span class="id" title="definition">E0limit</span></a> <a id="a:124" class="idref" href="#a:124"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#hE0limit"><span class="id" title="abbreviation">hE0limit</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:124"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0is_succ" class="idref" href="#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a id="a:125" class="idref" href="#a:125"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#hE0is_succ"><span class="id" title="abbreviation">hE0is_succ</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:125"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_h2g_nf" class="idref" href="#E0_h2g_nf"><span class="id" title="lemma">E0_h2g_nf</span></a> (<a id="a:126" class="idref" href="#a:126"><span class="id" title="binder">a</span></a>:<a class="idref" href="gaia_hydras.T1Bridge.html#hE0"><span class="id" title="abbreviation">hE0</span></a>) : <span class="id" title="definition">T1nf</span> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_h2g"><span class="id" title="definition">E0_h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:126"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gE0lt_iff" class="idref" href="#gE0lt_iff"><span class="id" title="lemma">gE0lt_iff</span></a> <a id="a:127" class="idref" href="#a:127"><span class="id" title="binder">a</span></a> <a id="beta:128" class="idref" href="#beta:128"><span class="id" title="binder">beta</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:127"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:128"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:127"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:128"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gE0le_iff" class="idref" href="#gE0le_iff"><span class="id" title="lemma">gE0le_iff</span></a> <a id="a:129" class="idref" href="#a:129"><span class="id" title="binder">a</span></a> <a id="beta:130" class="idref" href="#beta:130"><span class="id" title="binder">beta</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0le"><span class="id" title="definition">E0le</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:129"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:130"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:129"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:130"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_h2g_g2hK" class="idref" href="#E0_h2g_g2hK"><span class="id" title="lemma">E0_h2g_g2hK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_h2g"><span class="id" title="definition">E0_h2g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_g2h_h2gK" class="idref" href="#E0_g2h_h2gK"><span class="id" title="lemma">E0_g2h_h2gK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_h2g"><span class="id" title="definition">E0_h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_E0_succ" class="idref" href="#g2h_E0_succ"><span class="id" title="lemma">g2h_E0_succ</span></a> <a id="a:131" class="idref" href="#a:131"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:131"><span class="id" title="variable">a</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0.E0_succ</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:131"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="E0is_succ_succ" class="idref" href="#E0is_succ_succ"><span class="id" title="lemma">E0is_succ_succ</span></a> <a id="a:132" class="idref" href="#a:132"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a>  (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:132"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0is_succE" class="idref" href="#E0is_succE"><span class="id" title="lemma">E0is_succE</span></a> <a id="a:133" class="idref" href="#a:133"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:133"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:134" class="idref" href="#beta:134"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:133"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:134"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_eqE" class="idref" href="#E0_eqE"><span class="id" title="lemma">E0_eqE</span></a> (<a id="x:135" class="idref" href="#x:135"><span class="id" title="binder">x</span></a> <a id="y:136" class="idref" href="#y:136"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#x:135"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:136"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:135"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:136"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_diffE" class="idref" href="#E0_diffE"><span class="id" title="lemma">E0_diffE</span></a> (<a id="x:137" class="idref" href="#x:137"><span class="id" title="binder">x</span></a> <a id="y:138" class="idref" href="#y:138"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) : <a class="idref" href="gaia_hydras.T1Bridge.html#x:137"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:138"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:137"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#y:138"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_pred_succK" class="idref" href="#E0_pred_succK"><span class="id" title="lemma">E0_pred_succK</span></a> <a id="x:139" class="idref" href="#x:139"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_succ"><span class="id" title="definition">E0_succ</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:139"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#x:139"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_E0zero" class="idref" href="#g2h_E0zero"><span class="id" title="lemma">g2h_E0zero</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0zero"><span class="id" title="definition">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0.E0zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0g2h_Fin" class="idref" href="#E0g2h_Fin"><span class="id" title="lemma">E0g2h_Fin</span></a> <a id="i:140" class="idref" href="#i:140"><span class="id" title="binder">i</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0fin"><span class="id" title="definition">E0fin</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#i:140"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0.E0fin</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#i:140"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0g2h_phi0" class="idref" href="#E0g2h_phi0"><span class="id" title="lemma">E0g2h_phi0</span></a> <a id="a:141" class="idref" href="#a:141"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_phi0"><span class="id" title="definition">E0_phi0</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:141"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0.E0_phi0</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:141"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0g2h_mulE" class="idref" href="#E0g2h_mulE"><span class="id" title="lemma">E0g2h_mulE</span></a> (<a id="a:142" class="idref" href="#a:142"><span class="id" title="binder">a</span></a> <a id="b:143" class="idref" href="#b:143"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0mul"><span class="id" title="definition">E0mul</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:142"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:143"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0mul"><span class="id" title="instance">E0.E0mul</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:142"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:143"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0g2h_plusE" class="idref" href="#E0g2h_plusE"><span class="id" title="lemma">E0g2h_plusE</span></a> (<a id="a:144" class="idref" href="#a:144"><span class="id" title="binder">a</span></a> <a id="b:145" class="idref" href="#b:145"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>): <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0plus"><span class="id" title="definition">E0plus</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:144"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:145"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0add"><span class="id" title="instance">E0.E0add</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:144"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:145"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0g2h_omegaE" class="idref" href="#E0g2h_omegaE"><span class="id" title="lemma">E0g2h_omegaE</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0_g2h"><span class="id" title="definition">E0_g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0_omega"><span class="id" title="definition">E0_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hE0omega"><span class="id" title="abbreviation">hE0omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Basics.html#"><span class="id" title="library">Basics</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+TODO: simplify this proof !!! 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gE0lt_wf" class="idref" href="#gE0lt_wf"><span class="id" title="lemma">gE0lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L1'" class="idref" href="#L1'"><span class="id" title="lemma">L1'</span></a> (<a id="a:150" class="idref" href="#a:150"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) : <span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.T1Bridge.html#a:150"><span class="id" title="variable">a</span></a> <span class="id" title="notation">×</span> <span class="id" title="definition">T1omega</span><span class="id" title="notation">)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">T1omega</span> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:150"><span class="id" title="variable">a</span></a> <span class="id" title="notation">×</span> <span class="id" title="definition">T1omega</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Sequences and limits 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="g2h_seq" class="idref" href="#g2h_seq"><span class="id" title="definition">g2h_seq</span></a> (<a id="s:151" class="idref" href="#s:151"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) <a id="n:152" class="idref" href="#n:152"><span class="id" title="binder">n</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#s:151"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:152"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="h2g_seq" class="idref" href="#h2g_seq"><span class="id" title="definition">h2g_seq</span></a> (<a id="s:153" class="idref" href="#s:153"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) <a id="n:154" class="idref" href="#n:154"><span class="id" title="binder">n</span></a> := <a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#s:153"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:154"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="gstrict_lub" class="idref" href="#gstrict_lub"><span class="id" title="definition">gstrict_lub</span></a> (<a id="s:155" class="idref" href="#s:155"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) (<a id="lambda:156" class="idref" href="#lambda:156"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:157" class="idref" href="#i:157"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#LT"><span class="id" title="abbreviation">LT</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#i:157"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#lambda:156"><span class="id" title="variable">lambda</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:158" class="idref" href="#a:158"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:159" class="idref" href="#i:159"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.T1Bridge.html#LE"><span class="id" title="abbreviation">LE</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#i:159"><span class="id" title="variable">i</span></a>) <a class="idref" href="gaia_hydras.T1Bridge.html#a:158"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#LE"><span class="id" title="abbreviation">LE</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#lambda:156"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="gaia_hydras.T1Bridge.html#a:158"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="strict_lub_ref" class="idref" href="#strict_lub_ref"><span class="id" title="lemma">strict_lub_ref</span></a> (<a id="s:160" class="idref" href="#s:160"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) (<a id="lambda:161" class="idref" href="#lambda:161"><span class="id" title="binder">lambda</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#s:160"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#lambda:161"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#gstrict_lub"><span class="id" title="definition">gstrict_lub</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g_seq"><span class="id" title="definition">h2g_seq</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#s:160"><span class="id" title="variable">s</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#lambda:161"><span class="id" title="variable">lambda</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Search</span> ( <span class="id" title="var">_</span> <span class="id" title="notation">×</span> ?<span class="id" title="keyword">beta</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="keyword">beta</span>)%<span class="id" title="var">ca</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> ( <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> ?<span class="id" title="keyword">beta</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="keyword">beta</span>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span>  <a id="T1compare" class="idref" href="#T1compare"><span class="id" title="instance">T1compare</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="a:162" class="idref" href="#a:162"><span class="id" title="binder">a</span></a> <a id="beta:163" class="idref" href="#beta:163"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:162"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:163"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_g2h" class="idref" href="#compare_g2h"><span class="id" title="lemma">compare_g2h</span></a> (<a id="a:164" class="idref" href="#a:164"><span class="id" title="binder">a</span></a> <a id="beta:165" class="idref" href="#beta:165"><span class="id" title="binder">beta</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:164"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:165"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:164"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:165"><span class="id" title="variable">beta</span></a> .<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_h2g" class="idref" href="#compare_h2g"><span class="id" title="lemma">compare_h2g</span></a> (<a id="a:166" class="idref" href="#a:166"><span class="id" title="binder">a</span></a> <a id="beta:167" class="idref" href="#beta:167"><span class="id" title="binder">beta</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#hT1"><span class="id" title="abbreviation">hT1</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:166"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:167"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:166"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:167"><span class="id" title="variable">beta</span></a> .<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab426"></a><h1 class="section">Make E0 an ordinal notation</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1compare_correct" class="idref" href="#T1compare_correct"><span class="id" title="lemma">T1compare_correct</span></a> (<a id="a:168" class="idref" href="#a:168"><span class="id" title="binder">a</span></a> <a id="b:169" class="idref" href="#b:169"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.T1Bridge.html#T1"><span class="id" title="abbreviation">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Bridge.html#a:168"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:169"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#a:168"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#b:169"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="E0compare" class="idref" href="#E0compare"><span class="id" title="instance">E0compare</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:170" class="idref" href="#alpha:170"><span class="id" title="binder">alpha</span></a> <a id="beta:171" class="idref" href="#beta:171"><span class="id" title="binder">beta</span></a>: <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) ⇒ <a class="idref" href="gaia_hydras.T1Bridge.html#T1compare"><span class="id" title="instance">T1compare</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#alpha:170"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="gaia_hydras.T1Bridge.html#cnf"><span class="id" title="projection">cnf</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:171"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0compare_correct" class="idref" href="#E0compare_correct"><span class="id" title="lemma">E0compare_correct</span></a> (<a id="alpha:172" class="idref" href="#alpha:172"><span class="id" title="binder">alpha</span></a> <a id="beta:173" class="idref" href="#beta:173"><span class="id" title="binder">beta</span></a> : <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#alpha:172"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:173"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#alpha:172"><span class="id" title="variable">alpha</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#beta:173"><span class="id" title="variable">beta</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="E0_sto" class="idref" href="#E0_sto"><span class="id" title="instance">E0_sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="E0_comp" class="idref" href="#E0_comp"><span class="id" title="instance">E0_comp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Epsilon0" class="idref" href="#Epsilon0"><span class="id" title="instance">Epsilon0</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">T1omega</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab427"></a><h1 class="section">Abstract properties of arithmetic functions (with SSreflect inequalities)</h1>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strict_mono" class="idref" href="#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a id="f:174" class="idref" href="#f:174"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:175" class="idref" href="#n:175"><span class="id" title="binder">n</span></a> <a id="p:176" class="idref" href="#p:176"><span class="id" title="binder">p</span></a>, (<a class="idref" href="gaia_hydras.T1Bridge.html#n:175"><span class="id" title="variable">n</span></a><span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#p:176"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f:174"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:175"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#f:174"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#p:176"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates_from" class="idref" href="#dominates_from"><span class="id" title="definition">dominates_from</span></a> (<a id="n:177" class="idref" href="#n:177"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:178" class="idref" href="#g:178"><span class="id" title="binder">g</span></a> <a id="f:179" class="idref" href="#f:179"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:180" class="idref" href="#p:180"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="gaia_hydras.T1Bridge.html#n:177"><span class="id" title="variable">n</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.T1Bridge.html#p:180"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.T1Bridge.html#f:179"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#p:180"><span class="id" title="variable">p</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.T1Bridge.html#g:178"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#p:180"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates" class="idref" href="#dominates"><span class="id" title="definition">dominates</span></a> <a id="g:181" class="idref" href="#g:181"><span class="id" title="binder">g</span></a> <a id="f:182" class="idref" href="#f:182"><span class="id" title="binder">f</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:183" class="idref" href="#n:183"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:183"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g:181"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f:182"><span class="id" title="variable">f</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates_strong" class="idref" href="#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> <a id="g:184" class="idref" href="#g:184"><span class="id" title="binder">g</span></a> <a id="f:185" class="idref" href="#f:185"><span class="id" title="binder">f</span></a>  := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:186" class="idref" href="#n:186"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:186"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#g:184"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#f:185"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fun_le" class="idref" href="#fun_le"><span class="id" title="definition">fun_le</span></a>  <a id="f:187" class="idref" href="#f:187"><span class="id" title="binder">f</span></a> <a id="g:188" class="idref" href="#g:188"><span class="id" title="binder">g</span></a> := <span class="id" title="keyword">∀</span> <a id="n:189" class="idref" href="#n:189"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="gaia_hydras.T1Bridge.html#f:187"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:189"><span class="id" title="variable">n</span></a> <span class="id" title="notation">≤</span>  <a class="idref" href="gaia_hydras.T1Bridge.html#g:188"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.T1Bridge.html#n:189"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+++ b/theories/html/gaia_hydras.T1Choice.html
@@ -0,0 +1,216 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.T1Choice</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.T1Choice</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Experimental !!!!  This file is a draft !!! 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Type <span class="inlinecode"><span class="id" title="var">T1</span></span> vs generic trees 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T12Tree" class="idref" href="#T12Tree"><span class="id" title="definition">T12Tree</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a>: <span class="id" title="inductive">T1</span>): <span class="id" title="inductive">GenTree.tree</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.T1Choice.html#a:1"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">is</span> <span class="id" title="constructor">cons</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">then</span> <span class="id" title="constructor">GenTree.Node</span> <span class="id" title="var">n</span> <span class="id" title="notation">[::</span> <a class="idref" href="gaia_hydras.T1Choice.html#T12Tree:2"><span class="id" title="definition">T12Tree</span></a> <span class="id" title="var">b</span><span class="id" title="notation">;</span> <a class="idref" href="gaia_hydras.T1Choice.html#T12Tree:2"><span class="id" title="definition">T12Tree</span></a> <span class="id" title="var">c</span><span class="id" title="notation">]</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="constructor">GenTree.Leaf</span> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Tree2T1" class="idref" href="#Tree2T1"><span class="id" title="definition">Tree2T1</span></a> (<a id="t:4" class="idref" href="#t:4"><span class="id" title="binder">t</span></a>: <span class="id" title="inductive">GenTree.tree</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <span class="id" title="inductive">T1</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.T1Choice.html#t:4"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <span class="id" title="constructor">GenTree.Leaf</span> 0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="constructor">zero</span><br/>
+&nbsp;&nbsp;| <span class="id" title="constructor">GenTree.Node</span> <span class="id" title="var">n</span> <span class="id" title="notation">[::</span> <span class="id" title="var">t1</span><span class="id" title="notation">;</span> <span class="id" title="var">t2</span><span class="id" title="notation">]</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.T1Choice.html#Tree2T1:5"><span class="id" title="definition">Tree2T1</span></a> <span class="id" title="var">t1</span>, <a class="idref" href="gaia_hydras.T1Choice.html#Tree2T1:5"><span class="id" title="definition">Tree2T1</span></a> <span class="id" title="var">t2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">b</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">c</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<span class="id" title="constructor">cons</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="TreeT1K" class="idref" href="#TreeT1K"><span class="id" title="lemma">TreeT1K</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T12Tree"><span class="id" title="definition">T12Tree</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#Tree2T1"><span class="id" title="definition">Tree2T1</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+to remove (useless) 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span>  <a id="T12Tree_inj" class="idref" href="#T12Tree_inj"><span class="id" title="lemma">T12Tree_inj</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T12Tree"><span class="id" title="definition">T12Tree</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1mixin" class="idref" href="#T1mixin"><span class="id" title="definition">T1mixin</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="record">Countable.mixin_of</span> <span class="id" title="inductive">T1</span> := <span class="id" title="definition">PcanCountMixin</span> <a class="idref" href="gaia_hydras.T1Choice.html#TreeT1K"><span class="id" title="lemma">TreeT1K</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="T1Choice" class="idref" href="#T1Choice"><span class="id" title="definition">T1Choice</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <span class="id" title="abbreviation">ChoiceType</span> <span class="id" title="inductive">T1</span> (<span class="id" title="abbreviation">CountChoiceMixin</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1mixin"><span class="id" title="definition">T1mixin</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_pos" class="idref" href="#ex_pos"><span class="id" title="definition">ex_pos</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a>: <span class="id" title="inductive">T1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <span class="id" title="constructor">zero</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.T1Choice.html#alpha:7"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="some_pos" class="idref" href="#some_pos"><span class="id" title="definition">some_pos</span></a>: <span class="id" title="inductive">T1</span> := <span class="id" title="definition">xchoose</span> <a class="idref" href="gaia_hydras.T1Choice.html#ex_pos"><span class="id" title="definition">ex_pos</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="some_pos'" class="idref" href="#some_pos'"><span class="id" title="definition">some_pos'</span></a> : <span class="id" title="inductive">T1</span> := <span class="id" title="definition">choose</span> (<span class="id" title="keyword">fun</span> <a id="p:8" class="idref" href="#p:8"><span class="id" title="binder">p</span></a> : <span class="id" title="inductive">T1</span> ⇒ <span class="id" title="constructor">zero</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.T1Choice.html#p:8"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">T1omega</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">zero</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">:</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1Choice</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">!=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">some_pos'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+   = EqType T1 (EqMixin (@T1eqP))
+     : eqType 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+   = Choice.Pack
+         {|
+           Choice.base := EqMixin (@T1eqP);
+           Choice.mixin := PcanChoiceMixin (pcan_pickleK TreeT1K)
+         |}
+     : choiceType
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1_le_Mixin" class="idref" href="#T1_le_Mixin"><span class="id" title="definition">T1_le_Mixin</span></a> := <span class="id" title="abbreviation">leOrderMixin</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1Choice"><span class="id" title="definition">T1Choice</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1min" class="idref" href="#T1min"><span class="id" title="definition">T1min</span></a> <a id="a:10" class="idref" href="#a:10"><span class="id" title="binder">a</span></a> <a id="b:11" class="idref" href="#b:11"><span class="id" title="binder">b</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Choice.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#b:11"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#b:11"><span class="id" title="variable">b</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1max" class="idref" href="#T1max"><span class="id" title="definition">T1max</span></a> <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Choice.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#a:12"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1ltE" class="idref" href="#T1ltE"><span class="id" title="lemma">T1ltE</span></a> <a id="x:14" class="idref" href="#x:14"><span class="id" title="binder">x</span></a> <a id="y:15" class="idref" href="#y:15"><span class="id" title="binder">y</span></a> : <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Choice.html#x:14"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:15"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.T1Choice.html#y:15"><span class="id" title="variable">y</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.T1Choice.html#x:14"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <span class="id" title="definition">T1le</span> <a class="idref" href="gaia_hydras.T1Choice.html#x:14"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:15"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1minE" class="idref" href="#T1minE"><span class="id" title="lemma">T1minE</span></a> <a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a> <a id="y:17" class="idref" href="#y:17"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.T1Choice.html#T1min"><span class="id" title="definition">T1min</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:17"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Choice.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:17"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:17"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1maxE" class="idref" href="#T1maxE"><span class="id" title="lemma">T1maxE</span></a> <a id="x:18" class="idref" href="#x:18"><span class="id" title="binder">x</span></a> <a id="y:19" class="idref" href="#y:19"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.T1Choice.html#T1max"><span class="id" title="definition">T1max</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#x:18"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:19"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="definition">T1lt</span> <a class="idref" href="gaia_hydras.T1Choice.html#x:18"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:19"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#y:19"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#x:18"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1le_asym" class="idref" href="#T1le_asym"><span class="id" title="lemma">T1le_asym</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">ssrbool.antisymmetric</span></a> <span class="id" title="definition">T1le</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1leOrderMixin" class="idref" href="#T1leOrderMixin"><span class="id" title="definition">T1leOrderMixin</span></a> : <span class="id" title="abbreviation">leOrderMixin</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1Choice"><span class="id" title="definition">T1Choice</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="abbreviation">LeOrderMixin</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1ltE"><span class="id" title="lemma">T1ltE</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1minE"><span class="id" title="lemma">T1minE</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1maxE"><span class="id" title="lemma">T1maxE</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1le_asym"><span class="id" title="lemma">T1le_asym</span></a> <span class="id" title="lemma">T1le_trans</span> <span class="id" title="lemma">T1le_total</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="T1orderType" class="idref" href="#T1orderType"><span class="id" title="definition">T1orderType</span></a> :=<br/>
+&nbsp;&nbsp;@<span class="id" title="definition">OrderOfChoiceType</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1Choice"><span class="id" title="definition">T1Choice</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1leOrderMixin"><span class="id" title="definition">T1leOrderMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">@</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Order.le</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">tt</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">T1orderType</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <span class="id" title="definition">T1omega</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.T1Choice.html#T1orderType"><span class="id" title="definition">T1orderType</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">((</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">:</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">T1orderType</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">:</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">T1orderType</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">))%</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">O</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="b8382ebd4d1ecdc4b5385bb249fc75b4" class="idref" href="#b8382ebd4d1ecdc4b5385bb249fc75b4"><span class="id" title="notation">&quot;</span></a>x &lt;= y" := (@<span class="id" title="definition">Order.le</span> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1orderType"><span class="id" title="definition">T1orderType</span></a>  <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":::x_'&lt;'_x" class="idref" href="#:::x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt; y" := (@<span class="id" title="definition">Order.lt</span> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.T1Choice.html#T1orderType"><span class="id" title="definition">T1orderType</span></a>  <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <span class="id" title="definition">Order.le</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">@</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">Order.le</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">tt</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">T1orderType</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">T1omega</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">Order.POrderTheory</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm3" class="idref" href="#Unnamed_thm3"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm3" class="idref" href="#Unnamed_thm3"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm3" class="idref" href="#Unnamed_thm3"><span class="id" title="definition">T1omega</span></a>.<br/>
+<span class="id" title="keyword">Set Printing All</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm4" class="idref" href="#Unnamed_thm4"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm4" class="idref" href="#Unnamed_thm4"><span class="id" title="definition">&lt;</span></a> <a id="Unnamed_thm4" class="idref" href="#Unnamed_thm4"><span class="id" title="definition">T1succ</span></a> <a id="Unnamed_thm4" class="idref" href="#Unnamed_thm4"><span class="id" title="definition">T1omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span>  <a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">~~</span></a> <a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">(&lt;%</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">O</span></a> <a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">(</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">:</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">T1orderType</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">(</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">:</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">T1orderType</span></a><a id="Unnamed_thm5" class="idref" href="#Unnamed_thm5"><span class="id" title="definition">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm6" class="idref" href="#Unnamed_thm6"><span class="id" title="definition">~~(</span></a><a id="Unnamed_thm6" class="idref" href="#Unnamed_thm6"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm6" class="idref" href="#Unnamed_thm6"><span class="id" title="definition">&lt;</span></a> <a id="Unnamed_thm6" class="idref" href="#Unnamed_thm6"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm6" class="idref" href="#Unnamed_thm6"><span class="id" title="definition">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">~~</span></a> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">@</span></a><a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">Order.lt</span></a> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">tt</span></a> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">(</span></a><a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">T1orderType</span></a><a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm7" class="idref" href="#Unnamed_thm7"><span class="id" title="definition">T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm8" class="idref" href="#Unnamed_thm8"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm8" class="idref" href="#Unnamed_thm8"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm8" class="idref" href="#Unnamed_thm8"><span class="id" title="definition">T1omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="gaia_hydras.T1Bridge.html#E0"><span class="id" title="record">E0</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.T2Bridge.html b/theories/html/gaia_hydras.T2Bridge.html
new file mode 100644
index 00000000..4442c65b
--- /dev/null
+++ b/theories/html/gaia_hydras.T2Bridge.html
@@ -0,0 +1,114 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.T2Bridge</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.T2Bridge</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab471"></a><h1 class="section">Bridge between Hydra-battle's and Gaia's <span class="inlinecode"><span class="id" title="var">T1</span></span>  (Experimental)</h1>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#"><span class="id" title="library">T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#"><span class="id" title="library">Gamma0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">ssete9</span>.<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="module">Gamma0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Hydra-Battles' type for ordinal terms below <span class="inlinecode"><span class="id" title="var">Gamma00</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">Gamma0.T2.T2</span></a>.<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">Gamma0.T2.T2</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hT2" class="idref" href="#hT2"><span class="id" title="abbreviation">hT2</span></a> := <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">hydras.Gamma0.T2.T2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Gaia's type for ordinal terms below <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> 
+</div>
+<div class="code">
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="T2" class="idref" href="#T2"><span class="id" title="abbreviation">T2</span></a> := <span class="id" title="inductive">ssete9.Gamma0.T2</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hcons" class="idref" href="#hcons"><span class="id" title="abbreviation">hcons</span></a> := <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="hzero" class="idref" href="#hzero"><span class="id" title="abbreviation">hzero</span></a> := <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">hydras.Gamma0.T2.zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="h2g" class="idref" href="#h2g"><span class="id" title="definition">h2g</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T2Bridge.html#hT2"><span class="id" title="abbreviation">hT2</span></a>) : <a class="idref" href="gaia_hydras.T2Bridge.html#T2"><span class="id" title="abbreviation">T2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.T2Bridge.html#alpha:1"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.T2Bridge.html#hzero"><span class="id" title="abbreviation">hzero</span></a> ⇒ <span class="id" title="constructor">zero</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.T2Bridge.html#hcons"><span class="id" title="abbreviation">hcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <span class="id" title="constructor">cons</span> (<a class="idref" href="gaia_hydras.T2Bridge.html#h2g:2"><span class="id" title="definition">h2g</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="gaia_hydras.T2Bridge.html#h2g:2"><span class="id" title="definition">h2g</span></a> <span class="id" title="var">b</span>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.T2Bridge.html#h2g:2"><span class="id" title="definition">h2g</span></a> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="g2h" class="idref" href="#g2h"><span class="id" title="definition">g2h</span></a> (<a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="gaia_hydras.T2Bridge.html#T2"><span class="id" title="abbreviation">T2</span></a>) : <a class="idref" href="gaia_hydras.T2Bridge.html#hT2"><span class="id" title="abbreviation">hT2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.T2Bridge.html#alpha:4"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="constructor">zero</span> ⇒ <a class="idref" href="gaia_hydras.T2Bridge.html#hzero"><span class="id" title="abbreviation">hzero</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="constructor">cons</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="gaia_hydras.T2Bridge.html#hcons"><span class="id" title="abbreviation">hcons</span></a> (<a class="idref" href="gaia_hydras.T2Bridge.html#g2h:5"><span class="id" title="definition">g2h</span></a> <span class="id" title="var">a</span>)(<a class="idref" href="gaia_hydras.T2Bridge.html#g2h:5"><span class="id" title="definition">g2h</span></a> <span class="id" title="var">b</span>)  <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.T2Bridge.html#g2h:5"><span class="id" title="definition">g2h</span></a> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h2g_g2hK" class="idref" href="#h2g_g2hK"><span class="id" title="lemma">h2g_g2hK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#h2g"><span class="id" title="definition">h2g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_h2gK" class="idref" href="#g2h_h2gK"><span class="id" title="lemma">g2h_h2gK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#g2h"><span class="id" title="definition">g2h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g2h_eqE" class="idref" href="#g2h_eqE"><span class="id" title="lemma">g2h_eqE</span></a> (<a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.T2Bridge.html#T2"><span class="id" title="abbreviation">T2</span></a>) :  <a class="idref" href="gaia_hydras.T2Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#g2h"><span class="id" title="definition">g2h</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#b:8"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#b:8"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h2g_eqE" class="idref" href="#h2g_eqE"><span class="id" title="lemma">h2g_eqE</span></a> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> :<a class="idref" href="gaia_hydras.T2Bridge.html#hT2"><span class="id" title="abbreviation">hT2</span></a>) :  <a class="idref" href="gaia_hydras.T2Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#h2g"><span class="id" title="definition">h2g</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#b:10"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="keyword">fun</span> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.T2Bridge.html#hT2"><span class="id" title="abbreviation">hT2</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.T2Bridge.html#b:12"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <span class="id" title="definition">T2lt</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.nfwfgaia.html b/theories/html/gaia_hydras.nfwfgaia.html
new file mode 100644
index 00000000..99c626d2
--- /dev/null
+++ b/theories/html/gaia_hydras.nfwfgaia.html
@@ -0,0 +1,3901 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.nfwfgaia</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.nfwfgaia</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Copy of gaia's ssete9.v (for experimenting Alectryon documentation)  Changes from original proof script are signaled as additions/changes or inside Alectryon snippets 
+ 
+<div class="paragraph"> </div>
+
+<a id="lab428"></a><h1 class="section">Ordinals in Pure Coq</h1>
+
+  Copyright INRIA (2013-2013) Marelle Team (José Grimm).
+  After a work of Castéran
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssreflect</span> <span class="id" title="library">ssrfun</span> <span class="id" title="library">ssrbool</span> <span class="id" title="library">eqtype</span> <span class="id" title="library">ssrnat</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">fintype</span> <span class="id" title="library">bigop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab429"></a><h1 class="section">prelude</h1>
+
+<div class="paragraph"> </div>
+
+ Useful lemmas missing in ssreflect 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="if_simpl" class="idref" href="#if_simpl"><span class="id" title="lemma">if_simpl</span></a> (<a id="p:1" class="idref" href="#p:1"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltn_simpl1" class="idref" href="#ltn_simpl1"><span class="id" title="lemma">ltn_simpl1</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> <a id="n':3" class="idref" href="#n':3"><span class="id" title="binder">n'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#n':3"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqn_simpl1" class="idref" href="#eqn_simpl1"><span class="id" title="lemma">eqn_simpl1</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> <a id="n':5" class="idref" href="#n':5"><span class="id" title="binder">n'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#n':5"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:4"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:4"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltn_simpl2" class="idref" href="#ltn_simpl2"><span class="id" title="lemma">ltn_simpl2</span></a> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> <a id="n':7" class="idref" href="#n':7"><span class="id" title="binder">n'</span></a> <a id="n'':8" class="idref" href="#n'':8"><span class="id" title="binder">n''</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':7"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':7"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':8"><span class="id" title="variable">n''</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':8"><span class="id" title="variable">n''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n':7"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':8"><span class="id" title="variable">n''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqn_simpl2" class="idref" href="#eqn_simpl2"><span class="id" title="lemma">eqn_simpl2</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="n':10" class="idref" href="#n':10"><span class="id" title="binder">n'</span></a> <a id="n'':11" class="idref" href="#n'':11"><span class="id" title="binder">n''</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':10"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':10"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':11"><span class="id" title="variable">n''</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':11"><span class="id" title="variable">n''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n':10"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n'':11"><span class="id" title="variable">n''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltn_add_le" class="idref" href="#ltn_add_le"><span class="id" title="lemma">ltn_add_le</span></a> <a id="m1:12" class="idref" href="#m1:12"><span class="id" title="binder">m1</span></a> <a id="m2:13" class="idref" href="#m2:13"><span class="id" title="binder">m2</span></a> <a id="n1:14" class="idref" href="#n1:14"><span class="id" title="binder">n1</span></a> <a id="n2:15" class="idref" href="#n2:15"><span class="id" title="binder">n2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:12"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:14"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:13"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:15"><span class="id" title="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:12"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:13"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:14"><span class="id" title="variable">n1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:15"><span class="id" title="variable">n2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltn_add_el" class="idref" href="#ltn_add_el"><span class="id" title="lemma">ltn_add_el</span></a> <a id="m1:16" class="idref" href="#m1:16"><span class="id" title="binder">m1</span></a> <a id="m2:17" class="idref" href="#m2:17"><span class="id" title="binder">m2</span></a> <a id="n1:18" class="idref" href="#n1:18"><span class="id" title="binder">n1</span></a> <a id="n2:19" class="idref" href="#n2:19"><span class="id" title="binder">n2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:16"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:18"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:17"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:19"><span class="id" title="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:16"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:17"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:18"><span class="id" title="variable">n1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:19"><span class="id" title="variable">n2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltn_add_ll" class="idref" href="#ltn_add_ll"><span class="id" title="lemma">ltn_add_ll</span></a> <a id="m1:20" class="idref" href="#m1:20"><span class="id" title="binder">m1</span></a> <a id="m2:21" class="idref" href="#m2:21"><span class="id" title="binder">m2</span></a> <a id="n1:22" class="idref" href="#n1:22"><span class="id" title="binder">n1</span></a> <a id="n2:23" class="idref" href="#n2:23"><span class="id" title="binder">n2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:20"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:22"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:21"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:23"><span class="id" title="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:20"><span class="id" title="variable">m1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:21"><span class="id" title="variable">m2</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:22"><span class="id" title="variable">n1</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:23"><span class="id" title="variable">n2</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+The ssreflect comparison on nat is WF 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_wf" class="idref" href="#lt_wf"><span class="id" title="lemma">lt_wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<span class="id" title="keyword">fun</span> (<a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#a:24"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:25"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab430"></a><h2 class="section">Example 1</h2>
+
+<div class="paragraph"> </div>
+
+ An example of function defined by transfinite induction using <span class="inlinecode"><span class="id" title="keyword">Fix</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Wf_ex" class="idref" href="#Wf_ex"><span class="id" title="module">Wf_ex</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex.f_spec" class="idref" href="#Wf_ex.f_spec"><span class="id" title="definition">f_spec</span></a> <a id="f:26" class="idref" href="#f:26"><span class="id" title="binder">f</span></a> <a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:27"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">m</span><span class="id" title="notation">.+2</span> <span class="id" title="keyword">then</span>  <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#f:26"><span class="id" title="variable">f</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#f:26"><span class="id" title="variable">f</span></a> <span class="id" title="var">m</span><span class="id" title="notation">.+1).+1</span> <span class="id" title="notation">).+1</span> <span class="id" title="keyword">else</span> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex.f_spec_simp" class="idref" href="#Wf_ex.f_spec_simp"><span class="id" title="lemma">f_spec_simp</span></a> <a id="f:29" class="idref" href="#f:29"><span class="id" title="binder">f</span></a> <a id="n:30" class="idref" href="#n:30"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:29"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:29"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:31"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:29"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:30"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex.f0" class="idref" href="#Wf_ex.f0"><span class="id" title="lemma">f0</span></a> <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> <a id="p:33" class="idref" href="#p:33"><span class="id" title="binder">p</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#p:33"><span class="id" title="variable">p</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:32"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:33"><span class="id" title="variable">p</span></a><span class="id" title="notation">.+2</span> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:32"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex.f1" class="idref" href="#Wf_ex.f1"><span class="id" title="definition">f1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="z:35" class="idref" href="#z:35"><span class="id" title="binder">z</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#z:35"><span class="id" title="variable">z</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:36" class="idref" href="#y:36"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:36"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:35"><span class="id" title="variable">z</span></a><span class="id" title="notation">.-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:37" class="idref" href="#y:37"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:37"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:34"><span class="id" title="variable">x</span></a><span class="id" title="notation">.-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex.f2" class="idref" href="#Wf_ex.f2"><span class="id" title="definition">f2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Fix"><span class="id" title="definition">Fix</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#lt_wf"><span class="id" title="lemma">lt_wf</span></a> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f1"><span class="id" title="definition">f1</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex.f" class="idref" href="#Wf_ex.f"><span class="id" title="definition">f</span></a> (<a id="x:38" class="idref" href="#x:38"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <span class="id" title="abbreviation">sval</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:38"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex.f_eqn" class="idref" href="#Wf_ex.f_eqn"><span class="id" title="lemma">f_eqn</span></a> <a id="x:39" class="idref" href="#x:39"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:39"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f1"><span class="id" title="definition">f1</span></a>  (<span class="id" title="keyword">fun</span> <a id="y:40" class="idref" href="#y:40"><span class="id" title="binder">y</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:40"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex.f_correct" class="idref" href="#Wf_ex.f_correct"><span class="id" title="lemma">f_correct</span></a> <a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f"><span class="id" title="definition">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:41"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex.f"><span class="id" title="definition">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:41"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex"><span class="id" title="module">Wf_ex</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab431"></a><h2 class="section">Example 2</h2>
+
+<div class="paragraph"> </div>
+
+ Second example,  
+  f(n) = 1 + \sum<sub>(i &lt; n)</sub> f(i) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Wfsum" class="idref" href="#Wfsum"><span class="id" title="module">Wfsum</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.psum" class="idref" href="#Wfsum.psum"><span class="id" title="definition">psum</span></a> (<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a> := <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a><span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:43"><span class="id" title="variable">n</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#i:44"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.f_spec" class="idref" href="#Wfsum.f_spec"><span class="id" title="definition">f_spec</span></a> <a id="f:47" class="idref" href="#f:47"><span class="id" title="binder">f</span></a>:= <span class="id" title="keyword">∀</span> <a id="n:48" class="idref" href="#n:48"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:47"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:48"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.psum"><span class="id" title="definition">psum</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:47"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:48"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wfsum.f_spec_simp" class="idref" href="#Wfsum.f_spec_simp"><span class="id" title="lemma">f_spec_simp</span></a> <a id="f:49" class="idref" href="#f:49"><span class="id" title="binder">f</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:50"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2 <span class="id" title="notation">^</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:50"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wfsum.psum_exten" class="idref" href="#Wfsum.psum_exten"><span class="id" title="lemma">psum_exten</span></a> <a id="n:51" class="idref" href="#n:51"><span class="id" title="binder">n</span></a> <a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a> <a id="g:53" class="idref" href="#g:53"><span class="id" title="binder">g</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="k:54" class="idref" href="#k:54"><span class="id" title="binder">k</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#k:54"><span class="id" title="variable">k</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:51"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:54"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:53"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:54"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.psum"><span class="id" title="definition">psum</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:51"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.psum"><span class="id" title="definition">psum</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:53"><span class="id" title="variable">g</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:51"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wfsum.lt_dec" class="idref" href="#Wfsum.lt_dec"><span class="id" title="lemma">lt_dec</span></a> <a id="n:55" class="idref" href="#n:55"><span class="id" title="binder">n</span></a> <a id="m:56" class="idref" href="#m:56"><span class="id" title="binder">m</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:55"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:56"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:55"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:56"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.extension" class="idref" href="#Wfsum.extension"><span class="id" title="definition">extension</span></a>  (<a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:59" class="idref" href="#p:59"><span class="id" title="binder">p</span></a> : <span class="id" title="keyword">∀</span> <a id="k:58" class="idref" href="#k:58"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#k:58"><span class="id" title="variable">k</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="k:60" class="idref" href="#k:60"><span class="id" title="binder">k</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.lt_dec"><span class="id" title="lemma">lt_dec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:60"><span class="id" title="variable">k</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:57"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">x</span> ⇒  <a class="idref" href="gaia_hydras.nfwfgaia.html#p:59"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:60"><span class="id" title="variable">k</span></a> <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ 0 <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.f1" class="idref" href="#Wfsum.f1"><span class="id" title="definition">f1</span></a> (<a id="n:61" class="idref" href="#n:61"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="h:63" class="idref" href="#h:63"><span class="id" title="binder">h</span></a> : <span class="id" title="keyword">∀</span> <a id="z:62" class="idref" href="#z:62"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#z:62"><span class="id" title="variable">z</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:61"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.psum"><span class="id" title="definition">psum</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.extension"><span class="id" title="definition">extension</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#h:63"><span class="id" title="variable">h</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n:61"><span class="id" title="variable">n</span></a><span class="id" title="notation">).+1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.f2" class="idref" href="#Wfsum.f2"><span class="id" title="definition">f2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Fix"><span class="id" title="definition">Fix</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#lt_wf"><span class="id" title="lemma">lt_wf</span></a> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f1"><span class="id" title="definition">f1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wfsum.f_eqn" class="idref" href="#Wfsum.f_eqn"><span class="id" title="lemma">f_eqn</span></a> <a id="x:64" class="idref" href="#x:64"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:64"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f1"><span class="id" title="definition">f1</span></a> (<span class="id" title="var">n</span>:=<a class="idref" href="gaia_hydras.nfwfgaia.html#x:64"><span class="id" title="variable">x</span></a>) (<span class="id" title="keyword">fun</span> <a id="y:65" class="idref" href="#y:65"><span class="id" title="binder">y</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:65"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wfsum.f" class="idref" href="#Wfsum.f"><span class="id" title="definition">f</span></a> (<a id="x:66" class="idref" href="#x:66"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:66"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wfsum.f_correct" class="idref" href="#Wfsum.f_correct"><span class="id" title="lemma">f_correct</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum.f"><span class="id" title="definition">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wfsum"><span class="id" title="module">Wfsum</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab432"></a><h2 class="section">Example 3</h2>
+
+<div class="paragraph"> </div>
+
+ We consider here only even numbers, show that comparison is WF, define a
+ function by transfinite induction and show it is correct. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Wf_ex3" class="idref" href="#Wf_ex3"><span class="id" title="module">Wf_ex3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex3.lte" class="idref" href="#Wf_ex3.lte"><span class="id" title="definition">lte</span></a> <a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a> <a id="m:68" class="idref" href="#m:68"><span class="id" title="binder">m</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:67"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:68"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:67"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:68"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.lte_wf" class="idref" href="#Wf_ex3.lte_wf"><span class="id" title="lemma">lte_wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte"><span class="id" title="definition">lte</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex3.f_spec" class="idref" href="#Wf_ex3.f_spec"><span class="id" title="definition">f_spec</span></a> <a id="f:70" class="idref" href="#f:70"><span class="id" title="binder">f</span></a> <a id="n:71" class="idref" href="#n:71"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:71"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">m</span><span class="id" title="notation">.+4</span> <span class="id" title="keyword">then</span>  <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#f:70"><span class="id" title="variable">f</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#f:70"><span class="id" title="variable">f</span></a> (<span class="id" title="var">m</span><span class="id" title="notation">.+2</span>)<span class="id" title="notation">).*2</span><span class="id" title="notation">.+2</span> <span class="id" title="notation">).+1</span> <span class="id" title="keyword">else</span> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f_spec_simp" class="idref" href="#Wf_ex3.f_spec_simp"><span class="id" title="lemma">f_spec_simp</span></a> <a id="f:73" class="idref" href="#f:73"><span class="id" title="binder">f</span></a> <a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:75" class="idref" href="#n:75"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a><span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:75"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:75"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:75"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#n:74"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-1</span><span class="id" title="notation">)./2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f_spec_simp1" class="idref" href="#Wf_ex3.f_spec_simp1"><span class="id" title="lemma">f_spec_simp1</span></a> <a id="f:77" class="idref" href="#f:77"><span class="id" title="binder">f</span></a> <a id="n:78" class="idref" href="#n:78"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:79" class="idref" href="#n:79"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a><span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:79"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:79"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:79"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:77"><span class="id" title="variable">f</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:78"><span class="id" title="variable">n</span></a><span class="id" title="notation">.*2</span><span class="id" title="notation">.+2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:78"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f_spec_simp2" class="idref" href="#Wf_ex3.f_spec_simp2"><span class="id" title="lemma">f_spec_simp2</span></a> <a id="f:80" class="idref" href="#f:80"><span class="id" title="binder">f</span></a> <a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:82" class="idref" href="#n:82"><span class="id" title="binder">n</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#f:80"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:82"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:80"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:82"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#f:80"><span class="id" title="variable">f</span></a>(<a class="idref" href="gaia_hydras.nfwfgaia.html#n:81"><span class="id" title="variable">n</span></a><span class="id" title="notation">.*2</span><span class="id" title="notation">.+3</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:81"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f0a" class="idref" href="#Wf_ex3.f0a"><span class="id" title="lemma">f0a</span></a> <a id="y:84" class="idref" href="#y:84"><span class="id" title="binder">y</span></a> <a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a>: <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:85"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:85"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:84"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#n:85"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span><span class="id" title="notation">)./2</span><span class="id" title="notation">.-1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#y:84"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:85"><span class="id" title="variable">n</span></a><span class="id" title="notation">./2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte"><span class="id" title="definition">lte</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#y:84"><span class="id" title="variable">y</span></a><span class="id" title="notation">.*2</span><span class="id" title="notation">).+2</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:85"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+4</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f0b" class="idref" href="#Wf_ex3.f0b"><span class="id" title="lemma">f0b</span></a> <a id="a:86" class="idref" href="#a:86"><span class="id" title="binder">a</span></a> <a id="b:87" class="idref" href="#b:87"><span class="id" title="binder">b</span></a>: <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:86"><span class="id" title="variable">a</span></a><span class="id" title="notation">.*2</span><span class="id" title="notation">.+2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:87"><span class="id" title="variable">b</span></a> <span class="id" title="notation">≤</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a:86"><span class="id" title="variable">a</span></a><span class="id" title="notation">.*2</span><span class="id" title="notation">.+2</span><span class="id" title="notation">)./2</span><span class="id" title="notation">.-1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:87"><span class="id" title="variable">b</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:86"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f0c" class="idref" href="#Wf_ex3.f0c"><span class="id" title="lemma">f0c</span></a> <a id="n:88" class="idref" href="#n:88"><span class="id" title="binder">n</span></a>:  <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:88"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte"><span class="id" title="definition">lte</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:88"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:88"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+4</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.odd_dec" class="idref" href="#Wf_ex3.odd_dec"><span class="id" title="lemma">odd_dec</span></a> <a id="n:89" class="idref" href="#n:89"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:89"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:89"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex3.f1" class="idref" href="#Wf_ex3.f1"><span class="id" title="definition">f1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:90" class="idref" href="#x:90"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="z:91" class="idref" href="#z:91"><span class="id" title="binder">z</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte"><span class="id" title="definition">lte</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:91"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:90"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:92" class="idref" href="#y:92"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:91"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:92"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#z:91"><span class="id" title="variable">z</span></a><span class="id" title="notation">./2</span><span class="id" title="notation">).-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:93" class="idref" href="#y:93"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:90"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:93"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#x:90"><span class="id" title="variable">x</span></a><span class="id" title="notation">./2</span><span class="id" title="notation">).-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex3.f2" class="idref" href="#Wf_ex3.f2"><span class="id" title="definition">f2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Fix"><span class="id" title="definition">Fix</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte_wf"><span class="id" title="lemma">lte_wf</span></a> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f1"><span class="id" title="definition">f1</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Wf_ex3.f" class="idref" href="#Wf_ex3.f"><span class="id" title="definition">f</span></a> (<a id="x:94" class="idref" href="#x:94"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <span class="id" title="abbreviation">sval</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:94"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f_eqn" class="idref" href="#Wf_ex3.f_eqn"><span class="id" title="lemma">f_eqn</span></a> <a id="x:95" class="idref" href="#x:95"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:95"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f1"><span class="id" title="definition">f1</span></a>  (<span class="id" title="keyword">fun</span> <a id="y:96" class="idref" href="#y:96"><span class="id" title="binder">y</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f2"><span class="id" title="definition">f2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:96"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Wf_ex3.f_correct" class="idref" href="#Wf_ex3.f_correct"><span class="id" title="lemma">f_correct</span></a> <a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a><span class="id" title="definition">odd</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f"><span class="id" title="definition">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec"><span class="id" title="definition">f_spec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3.f"><span class="id" title="definition">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:97"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Wf_ex3"><span class="id" title="module">Wf_ex3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab433"></a><h2 class="section">More on accessiblity</h2>
+
+<div class="paragraph"> </div>
+
+ We show that there is no striclty decreasing function with domain <span class="inlinecode"><span class="id" title="var">nat</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a id="Sequences" class="idref" href="#Sequences"><span class="id" title="section">Sequences</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Sequences.A" class="idref" href="#Sequences.A"><span class="id" title="variable">A</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Sequences.R" class="idref" href="#Sequences.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="acc_rec" class="idref" href="#acc_rec"><span class="id" title="lemma">acc_rec</span></a> <a id="a:100" class="idref" href="#a:100"><span class="id" title="binder">a</span></a> <a id="b:101" class="idref" href="#b:101"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.R"><span class="id" title="variable">R</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:100"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:101"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.R"><span class="id" title="variable">R</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:101"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.R"><span class="id" title="variable">R</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:100"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Sequences.W" class="idref" href="#Sequences.W"><span class="id" title="variable">W</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="not_decreasing" class="idref" href="#not_decreasing"><span class="id" title="lemma">not_decreasing</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:103" class="idref" href="#f:103"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:104" class="idref" href="#i:104"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:103"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#i:104"><span class="id" title="variable">i</span></a><span class="id" title="notation">.+1</span>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:103"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#i:104"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Sequences"><span class="id" title="section">Sequences</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We show here an induction principle;  we could use it for ordinals in NF 
+form.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="restricted_recursion" class="idref" href="#restricted_recursion"><span class="id" title="section">restricted_recursion</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a id="restricted_recursion.A" class="idref" href="#restricted_recursion.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="restricted_recursion.P" class="idref" href="#restricted_recursion.P"><span class="id" title="variable">P</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#A:108"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>)(<a id="restricted_recursion.R" class="idref" href="#restricted_recursion.R"><span class="id" title="variable">R</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#A:108"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#A:108"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="restrict" class="idref" href="#restrict"><span class="id" title="definition">restrict</span></a> <a id="a:111" class="idref" href="#a:111"><span class="id" title="binder">a</span></a> <a id="b:112" class="idref" href="#b:112"><span class="id" title="binder">b</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:111"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.R"><span class="id" title="variable">R</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:111"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:112"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:112"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="well_founded_P" class="idref" href="#well_founded_P"><span class="id" title="definition">well_founded_P</span></a> := <span class="id" title="keyword">∀</span> <a id="a:113" class="idref" href="#a:113"><span class="id" title="binder">a</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:113"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:113"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="P_well_founded_induction_type" class="idref" href="#P_well_founded_induction_type"><span class="id" title="lemma">P_well_founded_induction_type</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#well_founded_P"><span class="id" title="definition">well_founded_P</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="Q:114" class="idref" href="#Q:114"><span class="id" title="binder">Q</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:115" class="idref" href="#x:115"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:115"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:116" class="idref" href="#y:116"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:116"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.R"><span class="id" title="variable">R</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:116"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:115"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:114"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:116"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:114"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:115"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:117" class="idref" href="#a:117"><span class="id" title="binder">a</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:117"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:114"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:117"><span class="id" title="variable">a</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#restricted_recursion"><span class="id" title="section">restricted_recursion</span></a>.<br/>
+<span class="id" title="keyword">Module</span> <a id="CantorOrdinal" class="idref" href="#CantorOrdinal"><span class="id" title="module">CantorOrdinal</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab434"></a><h1 class="section">The type T1</h1>
+
+<div class="paragraph"> </div>
+
+ This type represents all ordinals less that  
+  &epsilon; <sub>0</sub> ,
+ via the 
+Cantor Normal Form. More exactly <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> represents 
+  &omega; <sup> A</sup> * (n.+1) + B
+ if <span class="inlinecode"><span class="id" title="var">a</span></span> represents A and  <span class="inlinecode"><span class="id" title="var">b</span></span> represents B. 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="CantorOrdinal.T1" class="idref" href="#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="CantorOrdinal.zero" class="idref" href="#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1:120"><span class="id" title="inductive">T1</span></a><br/>
+| <a id="CantorOrdinal.cons" class="idref" href="#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1:120"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1:120"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1:120"><span class="id" title="inductive">T1</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab435"></a><h2 class="section">Equality</h2>
+
+<div class="paragraph"> </div>
+
+ we define a boolean equality, the use the mechanism of canonical
+ structures provided by ssreflect 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1eq" class="idref" href="#CantorOrdinal.T1eq"><span class="id" title="definition">T1eq</span></a> <a id="x:122" class="idref" href="#x:122"><span class="id" title="binder">x</span></a> <a id="y:123" class="idref" href="#y:123"><span class="id" title="binder">y</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">x</span>} :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:122"><span class="id" title="variable">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#y:123"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1eq:124"><span class="id" title="definition">T1eq</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><span class="id" title="notation">==</span> <span class="id" title="var">n'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1eq:124"><span class="id" title="definition">T1eq</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1eqP" class="idref" href="#CantorOrdinal.T1eqP"><span class="id" title="lemma">T1eqP</span></a> : <span class="id" title="definition">Equality.axiom</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq"><span class="id" title="definition">T1eq</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="CantorOrdinal.T1_eqMixin" class="idref" href="#CantorOrdinal.T1_eqMixin"><span class="id" title="definition">T1_eqMixin</span></a> := <span class="id" title="abbreviation">EqMixin</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eqP"><span class="id" title="lemma">T1eqP</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <a id="CantorOrdinal.T1_eqType" class="idref" href="#CantorOrdinal.T1_eqType"><span class="id" title="definition">T1_eqType</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <span class="id" title="abbreviation">EqType</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1_eqMixin"><span class="id" title="definition">T1_eqMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eqP"><span class="id" title="lemma">T1eqP</span></a> {<span class="id" title="var">x</span> <span class="id" title="var">y</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1eqE" class="idref" href="#CantorOrdinal.T1eqE"><span class="id" title="lemma">T1eqE</span></a> <a id="a:127" class="idref" href="#a:127"><span class="id" title="binder">a</span></a> <a id="n:128" class="idref" href="#n:128"><span class="id" title="binder">n</span></a> <a id="b:129" class="idref" href="#b:129"><span class="id" title="binder">b</span></a> <a id="a':130" class="idref" href="#a':130"><span class="id" title="binder">a'</span></a> <a id="n':131" class="idref" href="#n':131"><span class="id" title="binder">n'</span></a> <a id="b':132" class="idref" href="#b':132"><span class="id" title="binder">b'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:127"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:128"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:129"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':130"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':131"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':132"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:127"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':130"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:128"><span class="id" title="variable">n</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':131"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:129"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':132"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">cantor_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">cantor_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ca</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Some definitions
+<ul class="doclist">
+<li>  &phi;<sub>0</sub>(x) is <span class="inlinecode"></span> <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">0</span> <span class="inlinecode"><span class="id" title="var">zero</span></span> <span class="inlinecode"></span>, 
+    it represents   &omega; <sup>x</sup>
+
+</li>
+<li> one is   &phi;<sub>0</sub>(0)
+
+</li>
+<li> omega is   &phi;<sub>0</sub>(1)
+
+</li>
+<li> bad is an example of an ordinal not in normal form
+
+</li>
+<li> <span class="inlinecode"></span> <span class="inlinecode"><span class="id" title="keyword">fun</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">:=</span> <span class="inlinecode">\<span class="id" title="var">F</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is the canonical injection of <span class="inlinecode"><span class="id" title="var">nat</span></span> into <span class="inlinecode"><span class="id" title="var">T1</span></span>
+
+</li>
+<li> the log of <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> is <span class="inlinecode"><span class="id" title="var">a</span></span>
+
+</li>
+<li> an ordinal is AP if it is in the image of  &phi;<sub>0</sub>.
+
+</li>
+</ul>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.phi0" class="idref" href="#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a id="a:133" class="idref" href="#a:133"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:133"><span class="id" title="variable">a</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.one" class="idref" href="#CantorOrdinal.one"><span class="id" title="definition">one</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1omega" class="idref" href="#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1bad" class="idref" href="#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1nat" class="idref" href="#CantorOrdinal.T1nat"><span class="id" title="definition">T1nat</span></a> (<a id="n:134" class="idref" href="#n:134"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:134"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">p</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">p</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1log" class="idref" href="#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a id="a:136" class="idref" href="#a:136"><span class="id" title="binder">a</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:136"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:136"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1ap" class="idref" href="#CantorOrdinal.T1ap"><span class="id" title="definition">T1ap</span></a> <a id="x:138" class="idref" href="#x:138"><span class="id" title="binder">x</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:138"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="511f6a25e6d530048c45997becda2220" class="idref" href="#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">&quot;</span></a>\F n" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nat"><span class="id" title="definition">T1nat</span></a> <span class="id" title="var">n</span>)(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 29) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1F_inj" class="idref" href="#CantorOrdinal.T1F_inj"><span class="id" title="lemma">T1F_inj</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nat"><span class="id" title="definition">T1nat</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1phi0_zero" class="idref" href="#CantorOrdinal.T1phi0_zero"><span class="id" title="lemma">T1phi0_zero</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> 1.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1phi0_zero'" class="idref" href="#CantorOrdinal.T1phi0_zero'"><span class="id" title="lemma">T1phi0_zero'</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1log_phi0" class="idref" href="#CantorOrdinal.T1log_phi0"><span class="id" title="lemma">T1log_phi0</span></a> <a id="x:140" class="idref" href="#x:140"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:140"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:140"><span class="id" title="variable">x</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ap_phi0" class="idref" href="#CantorOrdinal.T1ap_phi0"><span class="id" title="lemma">T1ap_phi0</span></a> <a id="x:141" class="idref" href="#x:141"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap"><span class="id" title="definition">T1ap</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:141"><span class="id" title="variable">x</span></a>).  <br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab436"></a><h2 class="section">Order on T1</h2>
+
+<div class="paragraph"> </div>
+
+ We give here a recursion definition of comparison.
+Essentially,  &phi;<sub>0</sub>(x) is strictly increasing, 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1lt" class="idref" href="#CantorOrdinal.T1lt"><span class="id" title="definition">T1lt</span></a> <a id="x:142" class="idref" href="#x:142"><span class="id" title="binder">x</span></a> <a id="y:143" class="idref" href="#y:143"><span class="id" title="binder">y</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">x</span>} :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:142"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:143"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <span class="id" title="var">a</span> <span class="id" title="notation">==</span> <span class="id" title="var">a'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span> <span class="id" title="notation">&lt;</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <span class="id" title="notation">==</span> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">b</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">b'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:143"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+<span class="id" title="keyword">where</span>  <a id="CantorOrdinal.::cantor_scope:x_'&lt;'_x" class="idref" href="#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt; y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1lt:144"><span class="id" title="definition">T1lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1le" class="idref" href="#CantorOrdinal.T1le"><span class="id" title="definition">T1le</span></a> (<a id="x:148" class="idref" href="#x:148"><span class="id" title="binder">x</span></a> <a id="y:149" class="idref" href="#y:149"><span class="id" title="binder">y</span></a> :<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:148"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:149"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:148"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:149"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="0ccb34c09c66b2db1577192ef152bbf9" class="idref" href="#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">&quot;</span></a>x &lt;= y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le"><span class="id" title="definition">T1le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="81970e39e8d1381cef754a4956da95e7" class="idref" href="#81970e39e8d1381cef754a4956da95e7"><span class="id" title="notation">&quot;</span></a>x &gt;= y" := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="CantorOrdinal.::cantor_scope:x_'&gt;'_x" class="idref" href="#CantorOrdinal.::cantor_scope:x_'&gt;'_x"><span class="id" title="notation">&quot;</span></a>x &gt; y"  := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>)  : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lenn" class="idref" href="#CantorOrdinal.T1lenn"><span class="id" title="lemma">T1lenn</span></a> <a id="x:150" class="idref" href="#x:150"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:150"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:150"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T1lenn</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltnn" class="idref" href="#CantorOrdinal.T1ltnn"><span class="id" title="lemma">T1ltnn</span></a> <a id="x:151" class="idref" href="#x:151"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:151"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:151"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_ne" class="idref" href="#CantorOrdinal.T1lt_ne"><span class="id" title="lemma">T1lt_ne</span></a> <a id="a:152" class="idref" href="#a:152"><span class="id" title="binder">a</span></a> <a id="b:153" class="idref" href="#b:153"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:152"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:153"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:152"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:153"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_ne'" class="idref" href="#CantorOrdinal.T1lt_ne'"><span class="id" title="lemma">T1lt_ne'</span></a> <a id="a:154" class="idref" href="#a:154"><span class="id" title="binder">a</span></a> <a id="b:155" class="idref" href="#b:155"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:154"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:155"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:155"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:154"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltW" class="idref" href="#CantorOrdinal.T1ltW"><span class="id" title="lemma">T1ltW</span></a> <a id="a:156" class="idref" href="#a:156"><span class="id" title="binder">a</span></a> <a id="b:157" class="idref" href="#b:157"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:156"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:157"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:156"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:157"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_eqVlt" class="idref" href="#CantorOrdinal.T1le_eqVlt"><span class="id" title="lemma">T1le_eqVlt</span></a> <a id="a:158" class="idref" href="#a:158"><span class="id" title="binder">a</span></a> <a id="b:159" class="idref" href="#b:159"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:158"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:159"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:158"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:159"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:158"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:159"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_neAle" class="idref" href="#CantorOrdinal.T1lt_neAle"><span class="id" title="lemma">T1lt_neAle</span></a> <a id="a:160" class="idref" href="#a:160"><span class="id" title="binder">a</span></a> <a id="b:161" class="idref" href="#b:161"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:160"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:161"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:160"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:161"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:160"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:161"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltn0" class="idref" href="#CantorOrdinal.T1ltn0"><span class="id" title="lemma">T1ltn0</span></a> <a id="x:162" class="idref" href="#x:162"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:162"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le0n" class="idref" href="#CantorOrdinal.T1le0n"><span class="id" title="lemma">T1le0n</span></a> <a id="x:163" class="idref" href="#x:163"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:163"><span class="id" title="variable">x</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1len0" class="idref" href="#CantorOrdinal.T1len0"><span class="id" title="lemma">T1len0</span></a> <a id="x:164" class="idref" href="#x:164"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:164"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:164"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt0n" class="idref" href="#CantorOrdinal.T1lt0n"><span class="id" title="lemma">T1lt0n</span></a> <a id="x:165" class="idref" href="#x:165"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:165"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:165"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ge1" class="idref" href="#CantorOrdinal.T1ge1"><span class="id" title="lemma">T1ge1</span></a> <a id="x:166" class="idref" href="#x:166"><span class="id" title="binder">x</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:166"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:166"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt1" class="idref" href="#CantorOrdinal.T1lt1"><span class="id" title="lemma">T1lt1</span></a> <a id="x:167" class="idref" href="#x:167"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:167"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:167"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nat_inc" class="idref" href="#CantorOrdinal.T1nat_inc"><span class="id" title="lemma">T1nat_inc</span></a> <a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a> <a id="p:169" class="idref" href="#p:169"><span class="id" title="binder">p</span></a> : (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:168"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:169"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:169"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+This is an alternative version of less-or-equal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_consE" class="idref" href="#CantorOrdinal.T1le_consE"><span class="id" title="lemma">T1le_consE</span></a>  <a id="a:170" class="idref" href="#a:170"><span class="id" title="binder">a</span></a> <a id="n:171" class="idref" href="#n:171"><span class="id" title="binder">n</span></a> <a id="b:172" class="idref" href="#b:172"><span class="id" title="binder">b</span></a> <a id="a':173" class="idref" href="#a':173"><span class="id" title="binder">a'</span></a> <a id="n':174" class="idref" href="#n':174"><span class="id" title="binder">n'</span></a> <a id="b':175" class="idref" href="#b':175"><span class="id" title="binder">b'</span></a>:<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:170"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:171"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:172"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':173"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':174"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':175"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:170"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':173"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:170"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':173"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:171"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':174"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:171"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':174"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:172"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':175"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We have exactly one of: <span class="inlinecode"><span class="id" title="var">a</span></span> is less than, greater than, or equal to <span class="inlinecode"><span class="id" title="var">b</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_trichotomy" class="idref" href="#CantorOrdinal.T1lt_trichotomy"><span class="id" title="lemma">T1lt_trichotomy</span></a> <a id="a:176" class="idref" href="#a:176"><span class="id" title="binder">a</span></a> <a id="b:177" class="idref" href="#b:177"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:176"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:177"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:176"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:177"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:177"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:176"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_anti" class="idref" href="#CantorOrdinal.T1lt_anti"><span class="id" title="lemma">T1lt_anti</span></a> <a id="b:178" class="idref" href="#b:178"><span class="id" title="binder">b</span></a> <a id="a:179" class="idref" href="#a:179"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:179"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:178"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:178"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:179"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1leNgt" class="idref" href="#CantorOrdinal.T1leNgt"><span class="id" title="lemma">T1leNgt</span></a> <a id="a:180" class="idref" href="#a:180"><span class="id" title="binder">a</span></a> <a id="b:181" class="idref" href="#b:181"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:180"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:181"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:181"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:180"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltNge" class="idref" href="#CantorOrdinal.T1ltNge"><span class="id" title="lemma">T1ltNge</span></a> <a id="a:182" class="idref" href="#a:182"><span class="id" title="binder">a</span></a> <a id="b:183" class="idref" href="#b:183"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:182"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:183"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:183"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:182"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1eq_le" class="idref" href="#CantorOrdinal.T1eq_le"><span class="id" title="lemma">T1eq_le</span></a> <a id="m:184" class="idref" href="#m:184"><span class="id" title="binder">m</span></a> <a id="n:185" class="idref" href="#n:185"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:184"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:185"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:184"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:185"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:184"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_total" class="idref" href="#CantorOrdinal.T1le_total"><span class="id" title="lemma">T1le_total</span></a> <a id="m:186" class="idref" href="#m:186"><span class="id" title="binder">m</span></a> <a id="n:187" class="idref" href="#n:187"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:186"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:187"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:187"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:186"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+The next three definitions are similar to to those defined in ssrnat.
+we shall use <span class="inlinecode"><span class="id" title="var">T1ltgtP</span></span> a lot.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="CantorOrdinal.T1ltn_xor_geq" class="idref" href="#CantorOrdinal.T1ltn_xor_geq"><span class="id" title="inductive">T1ltn_xor_geq</span></a> <a id="m:188" class="idref" href="#m:188"><span class="id" title="binder">m</span></a> <a id="n:189" class="idref" href="#n:189"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.T1LtnNotGeq" class="idref" href="#CantorOrdinal.T1LtnNotGeq"><span class="id" title="constructor">T1LtnNotGeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:188"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:189"><span class="id" title="variable">n</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1ltn_xor_geq:190"><span class="id" title="inductive">T1ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:188"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:189"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.T1GeqNotLtn" class="idref" href="#CantorOrdinal.T1GeqNotLtn"><span class="id" title="constructor">T1GeqNotLtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:189"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:188"><span class="id" title="variable">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1ltn_xor_geq:190"><span class="id" title="inductive">T1ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:188"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:189"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="CantorOrdinal.T1leq_xor_gtn" class="idref" href="#CantorOrdinal.T1leq_xor_gtn"><span class="id" title="inductive">T1leq_xor_gtn</span></a> <a id="m:192" class="idref" href="#m:192"><span class="id" title="binder">m</span></a> <a id="n:193" class="idref" href="#n:193"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.T1GeqNotGtn" class="idref" href="#CantorOrdinal.T1GeqNotGtn"><span class="id" title="constructor">T1GeqNotGtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:192"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:193"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1leq_xor_gtn:194"><span class="id" title="inductive">T1leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:192"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:193"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.T1GtnNotLeq" class="idref" href="#CantorOrdinal.T1GtnNotLeq"><span class="id" title="constructor">T1GtnNotLeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:193"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:192"><span class="id" title="variable">m</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T1leq_xor_gtn:194"><span class="id" title="inductive">T1leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:192"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:193"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="CantorOrdinal.compare_T1" class="idref" href="#CantorOrdinal.compare_T1"><span class="id" title="inductive">compare_T1</span></a> <a id="m:196" class="idref" href="#m:196"><span class="id" title="binder">m</span></a> <a id="n:197" class="idref" href="#n:197"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.CompareT1Lt" class="idref" href="#CantorOrdinal.CompareT1Lt"><span class="id" title="constructor">CompareT1Lt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T1:198"><span class="id" title="inductive">compare_T1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.CompareT1Gt" class="idref" href="#CantorOrdinal.CompareT1Gt"><span class="id" title="constructor">CompareT1Gt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T1:198"><span class="id" title="inductive">compare_T1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="CantorOrdinal.CompareT1Eq" class="idref" href="#CantorOrdinal.CompareT1Eq"><span class="id" title="constructor">CompareT1Eq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T1:198"><span class="id" title="inductive">compare_T1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:196"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:197"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1leP" class="idref" href="#CantorOrdinal.T1leP"><span class="id" title="lemma">T1leP</span></a> <a id="x:200" class="idref" href="#x:200"><span class="id" title="binder">x</span></a> <a id="y:201" class="idref" href="#y:201"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1leq_xor_gtn"><span class="id" title="inductive">T1leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:200"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:201"><span class="id" title="variable">y</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:200"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:201"><span class="id" title="variable">y</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#y:201"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:200"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltP" class="idref" href="#CantorOrdinal.T1ltP"><span class="id" title="lemma">T1ltP</span></a> <a id="m:202" class="idref" href="#m:202"><span class="id" title="binder">m</span></a> <a id="n:203" class="idref" href="#n:203"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltn_xor_geq"><span class="id" title="inductive">T1ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:202"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:203"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:203"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:202"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:202"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:203"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1ltgtP" class="idref" href="#CantorOrdinal.T1ltgtP"><span class="id" title="lemma">T1ltgtP</span></a> <a id="m:204" class="idref" href="#m:204"><span class="id" title="binder">m</span></a> <a id="n:205" class="idref" href="#n:205"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.compare_T1"><span class="id" title="inductive">compare_T1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:204"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:205"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:204"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:205"><span class="id" title="variable">n</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:205"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:204"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:204"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:205"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We show here transitivity of comparison, using <span class="inlinecode"><span class="id" title="var">T1ltgtP</span></span> <span class="inlinecode"></span>.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_trans" class="idref" href="#CantorOrdinal.T1lt_trans"><span class="id" title="lemma">T1lt_trans</span></a> <a id="b:206" class="idref" href="#b:206"><span class="id" title="binder">b</span></a> <a id="a:207" class="idref" href="#a:207"><span class="id" title="binder">a</span></a> <a id="c:208" class="idref" href="#c:208"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:207"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:206"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:206"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:208"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:207"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:208"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_le_trans" class="idref" href="#CantorOrdinal.T1lt_le_trans"><span class="id" title="lemma">T1lt_le_trans</span></a> <a id="b:209" class="idref" href="#b:209"><span class="id" title="binder">b</span></a> <a id="a:210" class="idref" href="#a:210"><span class="id" title="binder">a</span></a> <a id="c:211" class="idref" href="#c:211"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:210"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:209"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:209"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:211"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:210"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:211"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_lt_trans" class="idref" href="#CantorOrdinal.T1le_lt_trans"><span class="id" title="lemma">T1le_lt_trans</span></a> <a id="b:212" class="idref" href="#b:212"><span class="id" title="binder">b</span></a> <a id="a:213" class="idref" href="#a:213"><span class="id" title="binder">a</span></a> <a id="c:214" class="idref" href="#c:214"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:213"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:212"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:212"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:214"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:213"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:214"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_trans" class="idref" href="#CantorOrdinal.T1le_trans"><span class="id" title="lemma">T1le_trans</span></a> <a id="b:215" class="idref" href="#b:215"><span class="id" title="binder">b</span></a> <a id="a:216" class="idref" href="#a:216"><span class="id" title="binder">a</span></a> <a id="c:217" class="idref" href="#c:217"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:216"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:215"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:215"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:217"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:216"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:217"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The following lemma implies 
+x &lt; &omega; <sup> x</sup>,  so all ordinals are less than 
+ &epsilon; <sub>0</sub>  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.head_lt_cons" class="idref" href="#CantorOrdinal.head_lt_cons"><span class="id" title="lemma">head_lt_cons</span></a> <a id="a:218" class="idref" href="#a:218"><span class="id" title="binder">a</span></a> <a id="n:219" class="idref" href="#n:219"><span class="id" title="binder">n</span></a> <a id="b:220" class="idref" href="#b:220"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:218"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:218"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:219"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:220"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_cons_le" class="idref" href="#CantorOrdinal.T1lt_cons_le"><span class="id" title="lemma">T1lt_cons_le</span></a> <a id="a:221" class="idref" href="#a:221"><span class="id" title="binder">a</span></a> <a id="n:222" class="idref" href="#n:222"><span class="id" title="binder">n</span></a> <a id="b:223" class="idref" href="#b:223"><span class="id" title="binder">b</span></a> <a id="a':224" class="idref" href="#a':224"><span class="id" title="binder">a'</span></a> <a id="n':225" class="idref" href="#n':225"><span class="id" title="binder">n'</span></a> <a id="b':226" class="idref" href="#b':226"><span class="id" title="binder">b'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:221"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:222"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:223"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':224"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':225"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':226"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:221"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':224"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_cons_le" class="idref" href="#CantorOrdinal.T1le_cons_le"><span class="id" title="lemma">T1le_cons_le</span></a> <a id="a:227" class="idref" href="#a:227"><span class="id" title="binder">a</span></a> <a id="n:228" class="idref" href="#n:228"><span class="id" title="binder">n</span></a> <a id="b:229" class="idref" href="#b:229"><span class="id" title="binder">b</span></a> <a id="a':230" class="idref" href="#a':230"><span class="id" title="binder">a'</span></a> <a id="n':231" class="idref" href="#n':231"><span class="id" title="binder">n'</span></a> <a id="b':232" class="idref" href="#b':232"><span class="id" title="binder">b'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:227"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:228"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:229"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':230"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':231"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':232"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:227"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':230"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.phi0_lt" class="idref" href="#CantorOrdinal.phi0_lt"><span class="id" title="lemma">phi0_lt</span></a> <a id="a:233" class="idref" href="#a:233"><span class="id" title="binder">a</span></a> <a id="b:234" class="idref" href="#b:234"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:233"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:234"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:233"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:234"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.phi0_le" class="idref" href="#CantorOrdinal.phi0_le"><span class="id" title="lemma">phi0_le</span></a> <a id="a:235" class="idref" href="#a:235"><span class="id" title="binder">a</span></a> <a id="b:236" class="idref" href="#b:236"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:235"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:236"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:235"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:236"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.phi0_lt1" class="idref" href="#CantorOrdinal.phi0_lt1"><span class="id" title="lemma">phi0_lt1</span></a> <a id="a:237" class="idref" href="#a:237"><span class="id" title="binder">a</span></a> <a id="n:238" class="idref" href="#n:238"><span class="id" title="binder">n</span></a> <a id="b:239" class="idref" href="#b:239"><span class="id" title="binder">b</span></a> <a id="a':240" class="idref" href="#a':240"><span class="id" title="binder">a'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:237"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:238"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:239"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':240"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:237"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':240"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab437"></a><h2 class="section">Normal form</h2>
+
+<div class="paragraph"> </div>
+
+ There exists a strictly infinite  decreasing sequence of ordinals, 
+so the order is not well founded 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="CantorOrdinal.lt_not_wf" class="idref" href="#CantorOrdinal.lt_not_wf"><span class="id" title="lemma">lt_not_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt"><span class="id" title="definition">T1lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+We say that <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> is NF if 
+)b &lt;&phi;<sub>0</sub>(a).
+If <span class="inlinecode"><span class="id" title="var">b</span></span> is <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a'</span></span> <span class="inlinecode"><span class="id" title="var">n'</span></span> <span class="inlinecode"><span class="id" title="var">b'</span></span>, this says that <span class="inlinecode"><span class="id" title="var">b</span></span> is less than <span class="inlinecode"><span class="id" title="var">b'</span></span>. 
+If <span class="inlinecode"><span class="id" title="var">a</span></span> is zero,  this says that <span class="inlinecode"><span class="id" title="var">b</span>=0</span>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1nf" class="idref" href="#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a id="x:250" class="idref" href="#x:250"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:250"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">_</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1nf:251"><span class="id" title="definition">T1nf</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1nf:251"><span class="id" title="definition">T1nf</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <span class="id" title="var">b</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_cons0" class="idref" href="#CantorOrdinal.T1nf_cons0"><span class="id" title="lemma">T1nf_cons0</span></a> <a id="a:253" class="idref" href="#a:253"><span class="id" title="binder">a</span></a> <a id="n:254" class="idref" href="#n:254"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:253"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:253"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:254"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_cons_cons" class="idref" href="#CantorOrdinal.T1nf_cons_cons"><span class="id" title="lemma">T1nf_cons_cons</span></a> <a id="a:255" class="idref" href="#a:255"><span class="id" title="binder">a</span></a> <a id="n:256" class="idref" href="#n:256"><span class="id" title="binder">n</span></a> <a id="a':257" class="idref" href="#a':257"><span class="id" title="binder">a'</span></a> <a id="n':258" class="idref" href="#n':258"><span class="id" title="binder">n'</span></a> <a id="b':259" class="idref" href="#b':259"><span class="id" title="binder">b'</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:255"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:256"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':257"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':258"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':259"><span class="id" title="variable">b'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':257"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:255"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_consa" class="idref" href="#CantorOrdinal.T1nf_consa"><span class="id" title="lemma">T1nf_consa</span></a> <a id="a:260" class="idref" href="#a:260"><span class="id" title="binder">a</span></a> <a id="n:261" class="idref" href="#n:261"><span class="id" title="binder">n</span></a> <a id="b:262" class="idref" href="#b:262"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:260"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:261"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:262"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:260"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_consb" class="idref" href="#CantorOrdinal.T1nf_consb"><span class="id" title="lemma">T1nf_consb</span></a> <a id="a:263" class="idref" href="#a:263"><span class="id" title="binder">a</span></a> <a id="n:264" class="idref" href="#n:264"><span class="id" title="binder">n</span></a> <a id="b:265" class="idref" href="#b:265"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:263"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:264"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:265"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:265"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_finite1" class="idref" href="#CantorOrdinal.T1nf_finite1"><span class="id" title="lemma">T1nf_finite1</span></a> <a id="n:266" class="idref" href="#n:266"><span class="id" title="binder">n</span></a> <a id="b:267" class="idref" href="#b:267"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:266"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:267"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:267"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_finite" class="idref" href="#CantorOrdinal.T1nf_finite"><span class="id" title="lemma">T1nf_finite</span></a> <a id="n:268" class="idref" href="#n:268"><span class="id" title="binder">n</span></a> <a id="b:269" class="idref" href="#b:269"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:268"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:269"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:269"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nfCE" class="idref" href="#CantorOrdinal.T1nfCE"><span class="id" title="lemma">T1nfCE</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.  <br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We show here that the restriction of <span class="inlinecode"><span class="id" title="var">T1lt</span></span> to NF ordinals is well-founded,
+  then prove two induction principles. Note that <span class="inlinecode"><span class="id" title="var">nf_Wf'</span></span> says every 
+ NF <span class="inlinecode"><span class="id" title="var">x</span></span> is accessible by the relation: <span class="inlinecode"><span class="id" title="var">u</span>&lt;<span class="id" title="var">v</span></span>, <span class="inlinecode"><span class="id" title="var">u</span></span> and <span class="inlinecode"><span class="id" title="var">v</span></span> NF.
+ If <span class="inlinecode"><span class="id" title="var">x</span></span> is not NF it is trivially accessible. The proof is a bit tricky 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="CantorOrdinal.AddLocalNotation" class="idref" href="#CantorOrdinal.AddLocalNotation"><span class="id" title="section">AddLocalNotation</span></a>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>]  <span class="id" title="keyword">Notation</span> <a id="CantorOrdinal.LT" class="idref" href="#CantorOrdinal.LT"><span class="id" title="abbreviation">LT</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(@<a class="idref" href="gaia_hydras.nfwfgaia.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> (<span class="id" title="keyword">fun</span> <a id="x:270" class="idref" href="#x:270"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:270"><span class="id" title="variable">x</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:271" class="idref" href="#x:271"><span class="id" title="binder">x</span></a> <a id="y:272" class="idref" href="#y:272"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:271"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:272"><span class="id" title="variable">y</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_Wf" class="idref" href="#CantorOrdinal.nf_Wf"><span class="id" title="lemma">nf_Wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt"><span class="id" title="definition">T1lt</span></a>).   
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_Wf'" class="idref" href="#CantorOrdinal.nf_Wf'"><span class="id" title="lemma">nf_Wf'</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#well_founded_P"><span class="id" title="definition">well_founded_P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt"><span class="id" title="definition">T1lt</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1transfinite_induction" class="idref" href="#CantorOrdinal.T1transfinite_induction"><span class="id" title="lemma">T1transfinite_induction</span></a> <a id="P:275" class="idref" href="#P:275"><span class="id" title="binder">P</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:276" class="idref" href="#x:276"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:276"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:277" class="idref" href="#y:277"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:277"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#y:277"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:276"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:275"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:277"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:275"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:276"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:278" class="idref" href="#a:278"><span class="id" title="binder">a</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:278"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:275"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:278"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1transfinite_induction_Q" class="idref" href="#CantorOrdinal.T1transfinite_induction_Q"><span class="id" title="lemma">T1transfinite_induction_Q</span></a> (<a id="P:279" class="idref" href="#P:279"><span class="id" title="binder">P</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) (<a id="Q:280" class="idref" href="#Q:280"><span class="id" title="binder">Q</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:281" class="idref" href="#x:281"><span class="id" title="binder">x</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:280"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:281"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:281"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:282" class="idref" href="#y:282"><span class="id" title="binder">y</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:280"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:282"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:282"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#y:282"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:281"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:279"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:282"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:279"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:281"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:283" class="idref" href="#a:283"><span class="id" title="binder">a</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:283"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:280"><span class="id" title="variable">Q</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:283"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:279"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:283"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1nf_rect" class="idref" href="#CantorOrdinal.T1nf_rect"><span class="id" title="lemma">T1nf_rect</span></a> (<a id="P:289" class="idref" href="#P:289"><span class="id" title="binder">P</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:290" class="idref" href="#n:290"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:290"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:291" class="idref" href="#a:291"><span class="id" title="binder">a</span></a> <a id="n:292" class="idref" href="#n:292"><span class="id" title="binder">n</span></a> <a id="b:293" class="idref" href="#b:293"><span class="id" title="binder">b</span></a> <a id="n':294" class="idref" href="#n':294"><span class="id" title="binder">n'</span></a> <a id="b':295" class="idref" href="#b':295"><span class="id" title="binder">b'</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:291"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:292"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:293"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:291"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:292"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:293"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#b':295"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:291"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:292"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:293"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':295"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':295"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:291"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:292"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:293"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n':294"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':295"><span class="id" title="variable">b'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:296" class="idref" href="#a:296"><span class="id" title="binder">a</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:296"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:289"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:296"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.AddLocalNotation"><span class="id" title="section">AddLocalNotation</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab438"></a><h2 class="section">Successor</h2>
+
+<div class="paragraph"> </div>
+
+ We say that <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> is
+<ul class="doclist">
+<li> limit if <span class="inlinecode"><span class="id" title="var">a</span></span> is non-zero, <span class="inlinecode"><span class="id" title="var">b</span></span> is limit or zero
+
+</li>
+<li> finite if <span class="inlinecode"><span class="id" title="var">a</span></span> is zero
+
+</li>
+<li> a successor if <span class="inlinecode"><span class="id" title="var">a</span></span> is zero or <span class="inlinecode"><span class="id" title="var">b</span></span> is a successor
+
+</li>
+</ul>
+   and define its 
+<ul class="doclist">
+<li> successor as <span class="inlinecode">\<span class="id" title="var">F</span></span> <span class="inlinecode">(<span class="id" title="var">n</span>+2)</span> or <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">(<span class="id" title="var">succ</span></span> <span class="inlinecode"><span class="id" title="var">b</span>)</span>
+
+</li>
+<li> predecessor as <span class="inlinecode">\<span class="id" title="var">F</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> or <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">(<span class="id" title="var">pred</span></span> <span class="inlinecode"><span class="id" title="var">b</span>)</span>
+
+</li>
+<li> split <span class="inlinecode"><span class="id" title="var">u</span>,<span class="id" title="var">v</span></span> as <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">x</span>,</span> <span class="inlinecode"><span class="id" title="var">y</span></span> if <span class="inlinecode"><span class="id" title="var">b</span></span> split as <span class="inlinecode"><span class="id" title="var">x</span>,<span class="id" title="var">y</span></span> and <span class="inlinecode"><span class="id" title="var">a</span></span> is non-zero;
+      and as <span class="inlinecode">0,<span class="id" title="var">n</span>+1</span> if <span class="inlinecode"><span class="id" title="var">a</span></span> is zero
+
+</li>
+</ul>
+Note that if <span class="inlinecode"><span class="id" title="var">a</span>=0</span>, the quantity <span class="inlinecode"><span class="id" title="var">b</span></span> is ignored; but when <span class="inlinecode"><span class="id" title="var">x</span></span> is NF, 
+then <span class="inlinecode"><span class="id" title="var">b</span></span> is zero. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1limit" class="idref" href="#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a id="x:297" class="idref" href="#x:297"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:297"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1limit:298"><span class="id" title="definition">T1limit</span></a> <span class="id" title="var">b</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1finite" class="idref" href="#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a id="x:300" class="idref" href="#x:300"><span class="id" title="binder">x</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:300"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <span class="id" title="var">a</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1is_succ" class="idref" href="#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a id="x:302" class="idref" href="#x:302"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:302"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T1is_succ:303"><span class="id" title="definition">T1is_succ</span></a> <span class="id" title="var">b</span>  <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1succ" class="idref" href="#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a id="c:305" class="idref" href="#c:305"><span class="id" title="binder">c</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:305"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1succ:306"><span class="id" title="definition">T1succ</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1pred" class="idref" href="#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> (<a id="c:308" class="idref" href="#c:308"><span class="id" title="binder">c</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:308"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1pred:309"><span class="id" title="definition">T1pred</span></a> <span class="id" title="var">b</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1split" class="idref" href="#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a id="x:311" class="idref" href="#x:311"><span class="id" title="binder">x</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:311"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#T1split:312"><span class="id" title="definition">T1split</span></a> <span class="id" title="var">b</span> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:311"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_limit" class="idref" href="#CantorOrdinal.split_limit"><span class="id" title="lemma">split_limit</span></a> <a id="x:314" class="idref" href="#x:314"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:314"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:314"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:314"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_is_succ" class="idref" href="#CantorOrdinal.split_is_succ"><span class="id" title="lemma">split_is_succ</span></a> <a id="x:315" class="idref" href="#x:315"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:315"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:315"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_finite" class="idref" href="#CantorOrdinal.split_finite"><span class="id" title="lemma">split_finite</span></a> <a id="x:316" class="idref" href="#x:316"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:316"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:316"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_succ" class="idref" href="#CantorOrdinal.split_succ"><span class="id" title="lemma">split_succ</span></a> <a id="x:317" class="idref" href="#x:317"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:317"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:317"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_pred" class="idref" href="#CantorOrdinal.split_pred"><span class="id" title="lemma">split_pred</span></a> <a id="x:318" class="idref" href="#x:318"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:318"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:318"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_le" class="idref" href="#CantorOrdinal.split_le"><span class="id" title="lemma">split_le</span></a> <a id="x:319" class="idref" href="#x:319"><span class="id" title="binder">x</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:319"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:319"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_split" class="idref" href="#CantorOrdinal.nf_split"><span class="id" title="lemma">nf_split</span></a> <a id="x:320" class="idref" href="#x:320"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:320"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:320"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1finite1" class="idref" href="#CantorOrdinal.T1finite1"><span class="id" title="lemma">T1finite1</span></a> <a id="n:321" class="idref" href="#n:321"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:321"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1finite2" class="idref" href="#CantorOrdinal.T1finite2"><span class="id" title="lemma">T1finite2</span></a> <a id="x:322" class="idref" href="#x:322"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:322"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:322"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:322"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:322"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1finite2CE" class="idref" href="#CantorOrdinal.T1finite2CE"><span class="id" title="lemma">T1finite2CE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="n:323" class="idref" href="#n:323"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:323"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1finite_succ" class="idref" href="#CantorOrdinal.T1finite_succ"><span class="id" title="lemma">T1finite_succ</span></a> <a id="x:324" class="idref" href="#x:324"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:324"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:324"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1succ_nat" class="idref" href="#CantorOrdinal.T1succ_nat"><span class="id" title="lemma">T1succ_nat</span></a> <a id="n:325" class="idref" href="#n:325"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:325"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:325"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_omega" class="idref" href="#CantorOrdinal.nf_omega"><span class="id" title="lemma">nf_omega</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_finite" class="idref" href="#CantorOrdinal.nf_finite"><span class="id" title="lemma">nf_finite</span></a> <a id="n:326" class="idref" href="#n:326"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:326"><span class="id" title="variable">n</span></a>).  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_phi0" class="idref" href="#CantorOrdinal.nf_phi0"><span class="id" title="lemma">nf_phi0</span></a> <a id="a:327" class="idref" href="#a:327"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:327"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:327"><span class="id" title="variable">a</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_log" class="idref" href="#CantorOrdinal.nf_log"><span class="id" title="lemma">nf_log</span></a> <a id="a:328" class="idref" href="#a:328"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:328"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:328"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+An ordinal is zero, limit or a successor, exclusively. When we split <span class="inlinecode"><span class="id" title="var">x</span></span>
+the first component is zero or limit, the second is a natural number  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_pr1" class="idref" href="#CantorOrdinal.limit_pr1"><span class="id" title="lemma">limit_pr1</span></a> <a id="x:329" class="idref" href="#x:329"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:329"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:329"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:329"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_limit1" class="idref" href="#CantorOrdinal.split_limit1"><span class="id" title="lemma">split_limit1</span></a> <a id="x:330" class="idref" href="#x:330"><span class="id" title="binder">x</span></a> (<a id="y:331" class="idref" href="#y:331"><span class="id" title="binder">y</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:330"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:331"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:331"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If <span class="inlinecode"><span class="id" title="var">x</span></span> is limit, if <span class="inlinecode"><span class="id" title="var">y</span></span> is less than <span class="inlinecode"><span class="id" title="var">x</span></span>, so is the successor of <span class="inlinecode"><span class="id" title="var">y</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_pr" class="idref" href="#CantorOrdinal.limit_pr"><span class="id" title="lemma">limit_pr</span></a> <a id="x:332" class="idref" href="#x:332"><span class="id" title="binder">x</span></a> <a id="y:333" class="idref" href="#y:333"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:332"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:333"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:332"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:333"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:332"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pred_le" class="idref" href="#CantorOrdinal.pred_le"><span class="id" title="lemma">pred_le</span></a> <a id="a:334" class="idref" href="#a:334"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:334"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:334"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pred_lt" class="idref" href="#CantorOrdinal.pred_lt"><span class="id" title="lemma">pred_lt</span></a> <a id="a:335" class="idref" href="#a:335"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:335"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:335"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:335"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_lt" class="idref" href="#CantorOrdinal.succ_lt"><span class="id" title="lemma">succ_lt</span></a> <a id="a:336" class="idref" href="#a:336"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:336"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:336"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_succ" class="idref" href="#CantorOrdinal.nf_succ"><span class="id" title="lemma">nf_succ</span></a> <a id="a:337" class="idref" href="#a:337"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:337"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:337"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_pred" class="idref" href="#CantorOrdinal.nf_pred"><span class="id" title="lemma">nf_pred</span></a> <a id="a:338" class="idref" href="#a:338"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:338"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:338"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_pred" class="idref" href="#CantorOrdinal.succ_pred"><span class="id" title="lemma">succ_pred</span></a> <a id="x:339" class="idref" href="#x:339"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:339"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:339"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:339"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:339"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_predCE" class="idref" href="#CantorOrdinal.succ_predCE"><span class="id" title="lemma">succ_predCE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="y:340" class="idref" href="#y:340"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:340"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_p1" class="idref" href="#CantorOrdinal.succ_p1"><span class="id" title="lemma">succ_p1</span></a> <a id="x:341" class="idref" href="#x:341"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:341"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pred_succ" class="idref" href="#CantorOrdinal.pred_succ"><span class="id" title="lemma">pred_succ</span></a> <a id="x:342" class="idref" href="#x:342"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:342"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:342"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:342"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pred_succ_CE" class="idref" href="#CantorOrdinal.pred_succ_CE"><span class="id" title="lemma">pred_succ_CE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_inj" class="idref" href="#CantorOrdinal.succ_inj"><span class="id" title="lemma">succ_inj</span></a> <a id="x:343" class="idref" href="#x:343"><span class="id" title="binder">x</span></a> <a id="y:344" class="idref" href="#y:344"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:343"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:344"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:343"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:344"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:343"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#y:344"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_injCE" class="idref" href="#CantorOrdinal.succ_injCE"><span class="id" title="lemma">succ_injCE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_succ" class="idref" href="#CantorOrdinal.lt_succ_succ"><span class="id" title="lemma">lt_succ_succ</span></a> <a id="x:345" class="idref" href="#x:345"><span class="id" title="binder">x</span></a> <a id="y:346" class="idref" href="#y:346"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:345"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:346"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:345"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:346"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.le_succ_succ" class="idref" href="#CantorOrdinal.le_succ_succ"><span class="id" title="lemma">le_succ_succ</span></a> <a id="x:347" class="idref" href="#x:347"><span class="id" title="binder">x</span></a> <a id="y:348" class="idref" href="#y:348"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:347"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:348"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:347"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:348"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ2CE" class="idref" href="#CantorOrdinal.lt_succ2CE"><span class="id" title="lemma">lt_succ2CE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_succE" class="idref" href="#CantorOrdinal.lt_succ_succE"><span class="id" title="lemma">lt_succ_succE</span></a> <a id="x:349" class="idref" href="#x:349"><span class="id" title="binder">x</span></a> <a id="y:350" class="idref" href="#y:350"><span class="id" title="binder">y</span></a>:  <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:349"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:350"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:349"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:350"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:349"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:350"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Some properties of comparison and successor 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.le_succ_succE" class="idref" href="#CantorOrdinal.le_succ_succE"><span class="id" title="lemma">le_succ_succE</span></a> <a id="x:351" class="idref" href="#x:351"><span class="id" title="binder">x</span></a> <a id="y:352" class="idref" href="#y:352"><span class="id" title="binder">y</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:351"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:352"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:351"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:352"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:351"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:352"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_le_1" class="idref" href="#CantorOrdinal.lt_succ_le_1"><span class="id" title="lemma">lt_succ_le_1</span></a> <a id="a:353" class="idref" href="#a:353"><span class="id" title="binder">a</span></a> <a id="b:354" class="idref" href="#b:354"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:353"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:354"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:353"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:354"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_le_2" class="idref" href="#CantorOrdinal.lt_succ_le_2"><span class="id" title="lemma">lt_succ_le_2</span></a> <a id="a:355" class="idref" href="#a:355"><span class="id" title="binder">a</span></a> <a id="b:356" class="idref" href="#b:356"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:356"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:356"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_le_3" class="idref" href="#CantorOrdinal.lt_succ_le_3"><span class="id" title="lemma">lt_succ_le_3</span></a> <a id="a:357" class="idref" href="#a:357"><span class="id" title="binder">a</span></a> <a id="b:358" class="idref" href="#b:358"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:357"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:357"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:358"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:357"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:358"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.lt_succ_le_4" class="idref" href="#CantorOrdinal.lt_succ_le_4"><span class="id" title="lemma">lt_succ_le_4</span></a> <a id="a:359" class="idref" href="#a:359"><span class="id" title="binder">a</span></a> <a id="b:360" class="idref" href="#b:360"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:360"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:359"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:360"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:359"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:360"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.phi0_log" class="idref" href="#CantorOrdinal.phi0_log"><span class="id" title="lemma">phi0_log</span></a> <a id="a:361" class="idref" href="#a:361"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:361"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:361"><span class="id" title="variable">a</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.tail_lt_cons" class="idref" href="#CantorOrdinal.tail_lt_cons"><span class="id" title="lemma">tail_lt_cons</span></a> <a id="a:362" class="idref" href="#a:362"><span class="id" title="binder">a</span></a> <a id="n:363" class="idref" href="#n:363"><span class="id" title="binder">n</span></a> <a id="b:364" class="idref" href="#b:364"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:364"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:362"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:364"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:362"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:363"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:364"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab439"></a><h2 class="section">Addition</h2>
+
+<div class="paragraph"> </div>
+
+ The definition of addition and subtraction given here are straightforward
+given our interpretation of <span class="inlinecode"><span class="id" title="var">cons</span></span>
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1add" class="idref" href="#CantorOrdinal.T1add"><span class="id" title="definition">T1add</span></a> <a id="x:365" class="idref" href="#x:365"><span class="id" title="binder">x</span></a> <a id="y:366" class="idref" href="#y:366"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:365"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:366"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a'</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> (<span class="id" title="var">b</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">+</span><span class="id" title="var">n'</span><span class="id" title="notation">).+1</span> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:365"><span class="id" title="variable">x</span></a>      <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span>  <a class="idref" href="gaia_hydras.nfwfgaia.html#y:366"><span class="id" title="variable">y</span></a><br/>
+<span class="id" title="keyword">where</span> <a id="33ed9a5b2d2229e33bbdf1ef87454c0d" class="idref" href="#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">&quot;</span></a>a + b" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1add:367"><span class="id" title="definition">T1add</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1sub" class="idref" href="#CantorOrdinal.T1sub"><span class="id" title="definition">T1sub</span></a> <a id="x:370" class="idref" href="#x:370"><span class="id" title="binder">x</span></a> <a id="y:371" class="idref" href="#y:371"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:370"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:371"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:370"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:371"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">a'</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span> <span class="id" title="notation">&lt;</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <span class="id" title="notation">==</span><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> (<span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">-</span><span class="id" title="var">n'</span><span class="id" title="notation">).-1</span>) <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <span class="id" title="notation">==</span> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">b</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">b'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">-</span><span class="id" title="var">n'</span><span class="id" title="notation">).-1</span> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:370"><span class="id" title="variable">x</span></a>      <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><br/>
+<span class="id" title="keyword">where</span>  <a id="CantorOrdinal.::cantor_scope:x_'-'_x" class="idref" href="#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">&quot;</span></a>x - y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1sub:372"><span class="id" title="definition">T1sub</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>):<span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Easy properties 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.succ_is_add_one" class="idref" href="#CantorOrdinal.succ_is_add_one"><span class="id" title="lemma">succ_is_add_one</span></a> <a id="a:375" class="idref" href="#a:375"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:375"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:375"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add1Nfin" class="idref" href="#CantorOrdinal.add1Nfin"><span class="id" title="lemma">add1Nfin</span></a> <a id="a:376" class="idref" href="#a:376"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:376"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:376"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:376"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub1Nfin" class="idref" href="#CantorOrdinal.sub1Nfin"><span class="id" title="lemma">sub1Nfin</span></a> <a id="a:377" class="idref" href="#a:377"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:377"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:377"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:377"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub1a" class="idref" href="#CantorOrdinal.sub1a"><span class="id" title="lemma">sub1a</span></a> <a id="x:378" class="idref" href="#x:378"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:378"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:378"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:378"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:378"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub1b" class="idref" href="#CantorOrdinal.sub1b"><span class="id" title="lemma">sub1b</span></a> <a id="x:379" class="idref" href="#x:379"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:379"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_1aCE" class="idref" href="#CantorOrdinal.sub_1aCE"><span class="id" title="lemma">sub_1aCE</span></a> (<a id="a:380" class="idref" href="#a:380"><span class="id" title="binder">a</span></a>:=  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:380"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:380"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_1bCE" class="idref" href="#CantorOrdinal.sub_1bCE"><span class="id" title="lemma">sub_1bCE</span></a> (<a id="a:381" class="idref" href="#a:381"><span class="id" title="binder">a</span></a>:=  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) : <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:381"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:381"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1add0n" class="idref" href="#CantorOrdinal.T1add0n"><span class="id" title="lemma">T1add0n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add"><span class="id" title="definition">T1add</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1addn0" class="idref" href="#CantorOrdinal.T1addn0"><span class="id" title="lemma">T1addn0</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add"><span class="id" title="definition">T1add</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1subn0" class="idref" href="#CantorOrdinal.T1subn0"><span class="id" title="lemma">T1subn0</span></a> <a id="x:382" class="idref" href="#x:382"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:382"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:382"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1subnn" class="idref" href="#CantorOrdinal.T1subnn"><span class="id" title="lemma">T1subnn</span></a> <a id="x:383" class="idref" href="#x:383"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:383"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:383"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_int" class="idref" href="#CantorOrdinal.add_int"><span class="id" title="lemma">add_int</span></a> <a id="n:384" class="idref" href="#n:384"><span class="id" title="binder">n</span></a> <a id="m:385" class="idref" href="#m:385"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:384"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:385"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:384"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:385"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_int" class="idref" href="#CantorOrdinal.sub_int"><span class="id" title="lemma">sub_int</span></a> <a id="n:386" class="idref" href="#n:386"><span class="id" title="binder">n</span></a> <a id="m:387" class="idref" href="#m:387"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:386"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:387"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:386"><span class="id" title="variable">n</span></a> <span class="id" title="notation">-</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:387"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_fin_omega" class="idref" href="#CantorOrdinal.add_fin_omega"><span class="id" title="lemma">add_fin_omega</span></a> <a id="n:388" class="idref" href="#n:388"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:388"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.fooCE" class="idref" href="#CantorOrdinal.fooCE"><span class="id" title="lemma">fooCE</span></a> (<a id="x:389" class="idref" href="#x:389"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:389"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\(</span></a><span class="id" title="keyword">∀</span> <a id="u:390" class="idref" href="#u:390"><span class="id" title="binder">u</span></a> <a id="v:391" class="idref" href="#v:391"><span class="id" title="binder">v</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:390"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:389"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:390"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:391"><span class="id" title="variable">v</span></a><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.split_add" class="idref" href="#CantorOrdinal.split_add"><span class="id" title="lemma">split_add</span></a> <a id="x:392" class="idref" href="#x:392"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> :=<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:392"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:392"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:392"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_to_cons" class="idref" href="#CantorOrdinal.add_to_cons"><span class="id" title="lemma">add_to_cons</span></a> <a id="a:393" class="idref" href="#a:393"><span class="id" title="binder">a</span></a> <a id="n:394" class="idref" href="#n:394"><span class="id" title="binder">n</span></a> <a id="b:395" class="idref" href="#b:395"><span class="id" title="binder">b</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#b:395"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:393"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:393"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:395"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:393"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:395"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.addC_CE" class="idref" href="#CantorOrdinal.addC_CE"><span class="id" title="lemma">addC_CE</span></a> (<a id="a:396" class="idref" href="#a:396"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) (<a id="b:397" class="idref" href="#b:397"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:396"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:397"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:396"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:397"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:397"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:396"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+We say that <span class="inlinecode"><span class="id" title="var">x</span></span> is AP is the sum of two ordinals less than <span class="inlinecode"><span class="id" title="var">x</span></span> is 
+less than <span class="inlinecode"><span class="id" title="var">x</span></span>. This conditionq holds if <span class="inlinecode"><span class="id" title="var">x</span></span> has the form 
+)&phi;<sub>0</sub>(a); the converse is true when <span class="inlinecode"><span class="id" title="var">x</span></span> is non-zero.
+We may also assume everything NF.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr0" class="idref" href="#CantorOrdinal.ap_pr0"><span class="id" title="lemma">ap_pr0</span></a> <a id="a:398" class="idref" href="#a:398"><span class="id" title="binder">a</span></a> (<a id="x:399" class="idref" href="#x:399"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:398"><span class="id" title="variable">a</span></a>) <a id="b:400" class="idref" href="#b:400"><span class="id" title="binder">b</span></a> <a id="c:401" class="idref" href="#c:401"><span class="id" title="binder">c</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#b:400"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:399"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:401"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:399"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:400"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:401"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:399"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr1" class="idref" href="#CantorOrdinal.ap_pr1"><span class="id" title="lemma">ap_pr1</span></a> <a id="c:407" class="idref" href="#c:407"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:408" class="idref" href="#a:408"><span class="id" title="binder">a</span></a> <a id="b:409" class="idref" href="#b:409"><span class="id" title="binder">b</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:408"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:407"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:409"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:407"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:408"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:409"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:407"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:407"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap"><span class="id" title="definition">T1ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:407"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr2" class="idref" href="#CantorOrdinal.ap_pr2"><span class="id" title="lemma">ap_pr2</span></a> <a id="c:410" class="idref" href="#c:410"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:411" class="idref" href="#a:411"><span class="id" title="binder">a</span></a> <a id="b:412" class="idref" href="#b:412"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:411"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:412"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:411"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:412"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:411"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:412"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap"><span class="id" title="definition">T1ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:410"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr2CE" class="idref" href="#CantorOrdinal.ap_pr2CE"><span class="id" title="lemma">ap_pr2CE</span></a> (<a id="c:414" class="idref" href="#c:414"><span class="id" title="binder">c</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> 1 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>):<br/>
+&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="a:415" class="idref" href="#a:415"><span class="id" title="binder">a</span></a> <a id="b:416" class="idref" href="#b:416"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:415"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:416"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:415"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:414"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:416"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:414"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:415"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:416"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:414"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Alternate definition of an AP: if <span class="inlinecode"><span class="id" title="var">a</span>&lt;<span class="id" title="var">x</span></span> then <span class="inlinecode"><span class="id" title="var">a</span>+<span class="id" title="var">x</span>=<span class="id" title="var">x</span></span>. 
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_simpl1" class="idref" href="#CantorOrdinal.add_simpl1"><span class="id" title="lemma">add_simpl1</span></a> <a id="a:422" class="idref" href="#a:422"><span class="id" title="binder">a</span></a> <a id="n:423" class="idref" href="#n:423"><span class="id" title="binder">n</span></a> <a id="b:424" class="idref" href="#b:424"><span class="id" title="binder">b</span></a> <a id="n':425" class="idref" href="#n':425"><span class="id" title="binder">n'</span></a> <a id="b':426" class="idref" href="#b':426"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:422"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:422"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:423"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:424"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':425"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':426"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:422"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:423"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#b:424"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':425"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':426"><span class="id" title="variable">b'</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_simpl2" class="idref" href="#CantorOrdinal.add_simpl2"><span class="id" title="lemma">add_simpl2</span></a>  <a id="n:427" class="idref" href="#n:427"><span class="id" title="binder">n</span></a> <a id="b:428" class="idref" href="#b:428"><span class="id" title="binder">b</span></a> <a id="a':429" class="idref" href="#a':429"><span class="id" title="binder">a'</span></a> <a id="n':430" class="idref" href="#n':430"><span class="id" title="binder">n'</span></a> <a id="b':431" class="idref" href="#b':431"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a':429"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:427"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:428"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':429"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':430"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':431"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':429"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':430"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':431"><span class="id" title="variable">b'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr3" class="idref" href="#CantorOrdinal.ap_pr3"><span class="id" title="lemma">ap_pr3</span></a> <a id="a:432" class="idref" href="#a:432"><span class="id" title="binder">a</span></a> <a id="b:433" class="idref" href="#b:433"><span class="id" title="binder">b</span></a> (<a id="x:434" class="idref" href="#x:434"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:432"><span class="id" title="variable">a</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#b:433"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:433"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:434"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ap_pr4" class="idref" href="#CantorOrdinal.ap_pr4"><span class="id" title="lemma">ap_pr4</span></a> <a id="x:435" class="idref" href="#x:435"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:436" class="idref" href="#b:436"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#b:436"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:435"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:436"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:435"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:435"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:435"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap"><span class="id" title="definition">T1ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:435"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+It follows tthat the sum of two NF ordinals is NF 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_add" class="idref" href="#CantorOrdinal.nf_add"><span class="id" title="lemma">nf_add</span></a> <a id="a:437" class="idref" href="#a:437"><span class="id" title="binder">a</span></a> <a id="b:438" class="idref" href="#b:438"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:437"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:438"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:437"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:438"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Results anbout addition subtraction comparison 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1add_eq0" class="idref" href="#CantorOrdinal.T1add_eq0"><span class="id" title="lemma">T1add_eq0</span></a> <a id="m:439" class="idref" href="#m:439"><span class="id" title="binder">m</span></a> <a id="n:440" class="idref" href="#n:440"><span class="id" title="binder">n</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:439"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:440"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:439"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:440"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_le1" class="idref" href="#CantorOrdinal.add_le1"><span class="id" title="lemma">add_le1</span></a> <a id="a:441" class="idref" href="#a:441"><span class="id" title="binder">a</span></a> <a id="b:442" class="idref" href="#b:442"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:441"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:441"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:442"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_le2" class="idref" href="#CantorOrdinal.add_le2"><span class="id" title="lemma">add_le2</span></a> <a id="a:443" class="idref" href="#a:443"><span class="id" title="binder">a</span></a> <a id="b:444" class="idref" href="#b:444"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:444"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:443"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:444"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.minus_lt" class="idref" href="#CantorOrdinal.minus_lt"><span class="id" title="lemma">minus_lt</span></a> <a id="a:445" class="idref" href="#a:445"><span class="id" title="binder">a</span></a> <a id="b:446" class="idref" href="#b:446"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:445"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:446"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:445"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:446"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.minus_le" class="idref" href="#CantorOrdinal.minus_le"><span class="id" title="lemma">minus_le</span></a> <a id="a:447" class="idref" href="#a:447"><span class="id" title="binder">a</span></a> <a id="b:448" class="idref" href="#b:448"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:447"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:448"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:447"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:448"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1sub0" class="idref" href="#CantorOrdinal.T1sub0"><span class="id" title="lemma">T1sub0</span></a> <a id="a:449" class="idref" href="#a:449"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:449"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:449"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_sub" class="idref" href="#CantorOrdinal.nf_sub"><span class="id" title="lemma">nf_sub</span></a> <a id="a:450" class="idref" href="#a:450"><span class="id" title="binder">a</span></a> <a id="b:451" class="idref" href="#b:451"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:450"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:451"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:450"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:451"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_le1" class="idref" href="#CantorOrdinal.sub_le1"><span class="id" title="lemma">sub_le1</span></a> <a id="a:452" class="idref" href="#a:452"><span class="id" title="binder">a</span></a> <a id="b:453" class="idref" href="#b:453"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:452"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:452"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:453"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:452"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_pr" class="idref" href="#CantorOrdinal.sub_pr"><span class="id" title="lemma">sub_pr</span></a> <a id="a:457" class="idref" href="#a:457"><span class="id" title="binder">a</span></a> <a id="b:458" class="idref" href="#b:458"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:458"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:457"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:458"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:457"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:458"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_inj" class="idref" href="#CantorOrdinal.add_inj"><span class="id" title="lemma">add_inj</span></a> <a id="a:459" class="idref" href="#a:459"><span class="id" title="binder">a</span></a> <a id="b:460" class="idref" href="#b:460"><span class="id" title="binder">b</span></a> <a id="c:461" class="idref" href="#c:461"><span class="id" title="binder">c</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:460"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:461"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:459"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:460"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:459"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:461"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:460"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:461"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_pr1" class="idref" href="#CantorOrdinal.sub_pr1"><span class="id" title="lemma">sub_pr1</span></a> <a id="a:462" class="idref" href="#a:462"><span class="id" title="binder">a</span></a> <a id="b:463" class="idref" href="#b:463"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:463"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:462"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:463"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:463"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:462"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:463"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:462"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_pr1CE" class="idref" href="#CantorOrdinal.sub_pr1CE"><span class="id" title="lemma">sub_pr1CE</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_pr1r" class="idref" href="#CantorOrdinal.sub_pr1r"><span class="id" title="lemma">sub_pr1r</span></a> <a id="a:464" class="idref" href="#a:464"><span class="id" title="binder">a</span></a> <a id="b:465" class="idref" href="#b:465"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:464"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:464"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:465"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:464"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:465"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.omega_minus_one" class="idref" href="#CantorOrdinal.omega_minus_one"><span class="id" title="lemma">omega_minus_one</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_nz" class="idref" href="#CantorOrdinal.sub_nz"><span class="id" title="lemma">sub_nz</span></a> <a id="a:466" class="idref" href="#a:466"><span class="id" title="binder">a</span></a> <a id="b:467" class="idref" href="#b:467"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:467"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:466"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:467"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:467"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:466"><span class="id" title="variable">a</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_nzCE" class="idref" href="#CantorOrdinal.sub_nzCE"><span class="id" title="lemma">sub_nzCE</span></a> (<a id="a:468" class="idref" href="#a:468"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) (<a id="b:469" class="idref" href="#b:469"><span class="id" title="binder">b</span></a> := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>)):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:468"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:469"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:469"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:468"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Associativity of addition 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1addA" class="idref" href="#CantorOrdinal.T1addA"><span class="id" title="lemma">T1addA</span></a> <a id="c1:470" class="idref" href="#c1:470"><span class="id" title="binder">c1</span></a> <a id="c2:471" class="idref" href="#c2:471"><span class="id" title="binder">c2</span></a> <a id="c3:472" class="idref" href="#c3:472"><span class="id" title="binder">c3</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c1:470"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c2:471"><span class="id" title="variable">c2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:472"><span class="id" title="variable">c3</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c1:470"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c2:471"><span class="id" title="variable">c2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:472"><span class="id" title="variable">c3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1addS" class="idref" href="#CantorOrdinal.T1addS"><span class="id" title="lemma">T1addS</span></a> <a id="a:473" class="idref" href="#a:473"><span class="id" title="binder">a</span></a> <a id="b:474" class="idref" href="#b:474"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:473"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:474"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:473"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:474"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_add2l" class="idref" href="#CantorOrdinal.T1le_add2l"><span class="id" title="lemma">T1le_add2l</span></a>  <a id="p:475" class="idref" href="#p:475"><span class="id" title="binder">p</span></a> <a id="m:476" class="idref" href="#m:476"><span class="id" title="binder">m</span></a> <a id="n:477" class="idref" href="#n:477"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:475"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:476"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:475"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:477"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:476"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:477"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_add2l" class="idref" href="#CantorOrdinal.T1lt_add2l"><span class="id" title="lemma">T1lt_add2l</span></a>  <a id="p:478" class="idref" href="#p:478"><span class="id" title="binder">p</span></a> <a id="m:479" class="idref" href="#m:479"><span class="id" title="binder">m</span></a> <a id="n:480" class="idref" href="#n:480"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:478"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:479"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:478"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:480"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:479"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:480"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_add2r" class="idref" href="#CantorOrdinal.T1lt_add2r"><span class="id" title="lemma">T1lt_add2r</span></a>  <a id="p:481" class="idref" href="#p:481"><span class="id" title="binder">p</span></a> <a id="m:482" class="idref" href="#m:482"><span class="id" title="binder">m</span></a> <a id="n:483" class="idref" href="#n:483"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:481"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:483"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:481"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:483"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_add2r" class="idref" href="#CantorOrdinal.T1le_add2r"><span class="id" title="lemma">T1le_add2r</span></a>  <a id="p:484" class="idref" href="#p:484"><span class="id" title="binder">p</span></a> <a id="m:485" class="idref" href="#m:485"><span class="id" title="binder">m</span></a> <a id="n:486" class="idref" href="#n:486"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:485"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:486"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:485"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:484"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:486"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:484"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1eq_add2l" class="idref" href="#CantorOrdinal.T1eq_add2l"><span class="id" title="lemma">T1eq_add2l</span></a>  <a id="p:487" class="idref" href="#p:487"><span class="id" title="binder">p</span></a> <a id="m:488" class="idref" href="#m:488"><span class="id" title="binder">m</span></a> <a id="n:489" class="idref" href="#n:489"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:487"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:488"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:487"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:489"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:488"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:489"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_le3" class="idref" href="#CantorOrdinal.add_le3"><span class="id" title="lemma">add_le3</span></a> <a id="a:490" class="idref" href="#a:490"><span class="id" title="binder">a</span></a> <a id="b:491" class="idref" href="#b:491"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:490"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:490"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:491"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:491"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_le4" class="idref" href="#CantorOrdinal.add_le4"><span class="id" title="lemma">add_le4</span></a> <a id="a:492" class="idref" href="#a:492"><span class="id" title="binder">a</span></a> <a id="b:493" class="idref" href="#b:493"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:493"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:492"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:492"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:493"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sub_pr1rCE" class="idref" href="#CantorOrdinal.sub_pr1rCE"><span class="id" title="lemma">sub_pr1rCE</span></a> (<a id="a:494" class="idref" href="#a:494"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) (<a id="b:495" class="idref" href="#b:495"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:494"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:495"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:495"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:494"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab440"></a><h2 class="section">Limits</h2>
+
+<div class="paragraph"> </div>
+
+ A limit ordinal is the supremum of a sequence of ordinals. We first show 
+  that some sequences are unbounded. We then show that, if the sequence is 
+  bounded, there is a least upper bound, more preciselly, if a property is 
+  satisfied for some NF ordinal, it is satisfied for a least NF aordinal.
+  This requires teh excluded middel principle.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.omega_tower" class="idref" href="#CantorOrdinal.omega_tower"><span class="id" title="definition">omega_tower</span></a> (<a id="n:496" class="idref" href="#n:496"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:496"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">p</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#omega_tower:497"><span class="id" title="definition">omega_tower</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.omega_tower_nf" class="idref" href="#CantorOrdinal.omega_tower_nf"><span class="id" title="lemma">omega_tower_nf</span></a> <a id="n:499" class="idref" href="#n:499"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_tower"><span class="id" title="definition">omega_tower</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:499"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.omega_tower_unbounded" class="idref" href="#CantorOrdinal.omega_tower_unbounded"><span class="id" title="lemma">omega_tower_unbounded</span></a> <a id="x:500" class="idref" href="#x:500"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:501" class="idref" href="#n:501"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_tower"><span class="id" title="definition">omega_tower</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:501"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:500"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.ex_middle" class="idref" href="#CantorOrdinal.ex_middle"><span class="id" title="definition">ex_middle</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:502" class="idref" href="#P:502"><span class="id" title="binder">P</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),  <span class="id" title="keyword">let</span> <a id="Q:504" class="idref" href="#Q:504"><span class="id" title="binder">Q</span></a> :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:503" class="idref" href="#x:503"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:503"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:502"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:503"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Q:504"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Q:504"><span class="id" title="variable">Q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.ex_middle_pick" class="idref" href="#CantorOrdinal.ex_middle_pick"><span class="id" title="lemma">ex_middle_pick</span></a> (<a id="P:505" class="idref" href="#P:505"><span class="id" title="binder">P</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ex_middle"><span class="id" title="definition">ex_middle</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:506" class="idref" href="#x:506"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:506"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:505"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:506"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:507" class="idref" href="#x:507"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:507"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#P:505"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:507"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.min_exists" class="idref" href="#CantorOrdinal.min_exists"><span class="id" title="lemma">min_exists</span></a> (<a id="P:508" class="idref" href="#P:508"><span class="id" title="binder">P</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="x:509" class="idref" href="#x:509"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ex_middle"><span class="id" title="definition">ex_middle</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:509"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#P:508"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:509"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:510" class="idref" href="#y:510"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:510"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:508"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:510"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="z:511" class="idref" href="#z:511"><span class="id" title="binder">z</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:511"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#P:508"><span class="id" title="variable">P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:511"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:510"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:511"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab441"></a><h3 class="section">Definition</h3>
+
+<div class="paragraph"> </div>
+
+ We say that <span class="inlinecode"><span class="id" title="var">x</span></span> is the limit of <span class="inlinecode"><span class="id" title="var">f</span></span>, or the supremum of the <span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">i</span>)</span> if 
+<span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">i</span>)&lt;<span class="id" title="var">x</span></span> (we could use less-or-equal here as <span class="inlinecode"><span class="id" title="var">f</span></span> will be strictly increasing)
+and if moreover, every ordinal less than <span class="inlinecode"><span class="id" title="var">x</span></span> is bounded by some <span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">i</span>)</span>.
+Note that <span class="inlinecode"><span class="id" title="var">x</span></span> is then the least upper bound. The trouble is that each <span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">i</span>)</span>  
+may be NF and <span class="inlinecode"><span class="id" title="var">x</span></span> is not. Thus , we give an  alternate definition. 
+Trouble is: a function may have more then one limit (at most one of them 
+being NF).
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="CantorOrdinal.Tf" class="idref" href="#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.limit_v1" class="idref" href="#CantorOrdinal.limit_v1"><span class="id" title="definition">limit_v1</span></a> (<a id="f:514" class="idref" href="#f:514"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:515" class="idref" href="#x:515"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:516" class="idref" href="#n:516"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:514"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:516"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:515"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:517" class="idref" href="#y:517"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#y:517"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:515"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:518" class="idref" href="#n:518"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:517"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:514"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:518"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.limit_v2" class="idref" href="#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> (<a id="f:519" class="idref" href="#f:519"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:520" class="idref" href="#x:520"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:521" class="idref" href="#n:521"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:519"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:521"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:520"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:522" class="idref" href="#y:522"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:522"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:522"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:520"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:523" class="idref" href="#n:523"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:522"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:519"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:523"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_unique1" class="idref" href="#CantorOrdinal.limit_unique1"><span class="id" title="lemma">limit_unique1</span></a> (<a id="f:524" class="idref" href="#f:524"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:525" class="idref" href="#x:525"><span class="id" title="binder">x</span></a> <a id="x':526" class="idref" href="#x':526"><span class="id" title="binder">x'</span></a> :<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v1"><span class="id" title="definition">limit_v1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:524"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:525"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v1"><span class="id" title="definition">limit_v1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:524"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':526"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:525"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':526"><span class="id" title="variable">x'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_unique2" class="idref" href="#CantorOrdinal.limit_unique2"><span class="id" title="lemma">limit_unique2</span></a> (<a id="f:527" class="idref" href="#f:527"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:528" class="idref" href="#x:528"><span class="id" title="binder">x</span></a> <a id="x':529" class="idref" href="#x':529"><span class="id" title="binder">x'</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:527"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:528"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:527"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':529"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:528"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':529"><span class="id" title="variable">x'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:528"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':529"><span class="id" title="variable">x'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.omega_plus_n" class="idref" href="#CantorOrdinal.omega_plus_n"><span class="id" title="definition">omega_plus_n</span></a> <a id="n:530" class="idref" href="#n:530"><span class="id" title="binder">n</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 0 (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:530"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_omega_plus_n" class="idref" href="#CantorOrdinal.nf_omega_plus_n"><span class="id" title="lemma">nf_omega_plus_n</span></a> <a id="n:531" class="idref" href="#n:531"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> ( <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_plus_n"><span class="id" title="definition">omega_plus_n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:531"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_CE1" class="idref" href="#CantorOrdinal.limit_CE1"><span class="id" title="lemma">limit_CE1</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v1"><span class="id" title="definition">limit_v1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_plus_n"><span class="id" title="definition">omega_plus_n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_CE2" class="idref" href="#CantorOrdinal.limit_CE2"><span class="id" title="lemma">limit_CE2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_plus_n"><span class="id" title="definition">omega_plus_n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 1 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_CE3" class="idref" href="#CantorOrdinal.limit_CE3"><span class="id" title="lemma">limit_CE3</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_plus_n"><span class="id" title="definition">omega_plus_n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab442"></a><h3 class="section">The normal form</h3>
+ To each ordinal, one can associate another ordinal that is NF.
+However,  this is in general incompatible with other operations  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.toNF" class="idref" href="#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a id="x:532" class="idref" href="#x:532"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:532"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#toNF:533"><span class="id" title="definition">toNF</span></a> <span class="id" title="var">a</span>) <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#toNF:533"><span class="id" title="definition">toNF</span></a> <span class="id" title="var">b</span> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_toNF" class="idref" href="#CantorOrdinal.nf_toNF"><span class="id" title="lemma">nf_toNF</span></a> <a id="x:535" class="idref" href="#x:535"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:535"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_nz" class="idref" href="#CantorOrdinal.toNF_nz"><span class="id" title="lemma">toNF_nz</span></a> <a id="x:536" class="idref" href="#x:536"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:536"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:536"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_nf" class="idref" href="#CantorOrdinal.toNF_nf"><span class="id" title="lemma">toNF_nf</span></a> <a id="x:537" class="idref" href="#x:537"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:537"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:537"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:537"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_mon" class="idref" href="#CantorOrdinal.toNF_mon"><span class="id" title="lemma">toNF_mon</span></a> <a id="x:538" class="idref" href="#x:538"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#x:538"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:538"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_ex1" class="idref" href="#CantorOrdinal.toNF_ex1"><span class="id" title="lemma">toNF_ex1</span></a> <a id="x:539" class="idref" href="#x:539"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#x:539"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:539"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_ex2" class="idref" href="#CantorOrdinal.toNF_ex2"><span class="id" title="lemma">toNF_ex2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 1 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_succ" class="idref" href="#CantorOrdinal.toNF_succ"><span class="id" title="lemma">toNF_succ</span></a> (<a id="x:540" class="idref" href="#x:540"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:540"><span class="id" title="variable">x</span></a>) <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:540"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.toNF_pred" class="idref" href="#CantorOrdinal.toNF_pred"><span class="id" title="lemma">toNF_pred</span></a> (<a id="x:541" class="idref" href="#x:541"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:541"><span class="id" title="variable">x</span></a>) <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF"><span class="id" title="definition">toNF</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:541"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab443"></a><h3 class="section">Realizing the limit</h3>
+ This is a simplification of the code given for the type T3 below.
+We define a function <span class="inlinecode"><span class="id" title="var">F</span>(<span class="id" title="var">x</span>)</span>; so that for any limit ordinal <span class="inlinecode"><span class="id" title="var">x</span></span>, if <span class="inlinecode"><span class="id" title="var">f</span>=</span> <span class="inlinecode"><span class="id" title="var">F</span>(<span class="id" title="var">x</span>)</span>,
+then <span class="inlinecode"><span class="id" title="var">f</span></span> is stictly increasing (of type <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">T1</span></span>), and its limit is <span class="inlinecode"><span class="id" title="var">x</span></span>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.fincP" class="idref" href="#CantorOrdinal.fincP"><span class="id" title="lemma">fincP</span></a> (<a id="f:542" class="idref" href="#f:542"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:543" class="idref" href="#n:543"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:542"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:543"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:542"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:543"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:544" class="idref" href="#n:544"><span class="id" title="binder">n</span></a> <a id="m:545" class="idref" href="#m:545"><span class="id" title="binder">m</span></a>, (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:544"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:545"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:542"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:544"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:542"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:545"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.limit_of" class="idref" href="#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> (<a id="f:546" class="idref" href="#f:546"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:547" class="idref" href="#x:547"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:548" class="idref" href="#n:548"><span class="id" title="binder">n</span></a> <a id="m:549" class="idref" href="#m:549"><span class="id" title="binder">m</span></a>, (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:548"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:549"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:546"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:548"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:546"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:549"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">),</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:546"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:547"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:547"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_unique" class="idref" href="#CantorOrdinal.limit_unique"><span class="id" title="lemma">limit_unique</span></a> <a id="f:550" class="idref" href="#f:550"><span class="id" title="binder">f</span></a> <a id="x:551" class="idref" href="#x:551"><span class="id" title="binder">x</span></a> <a id="y:552" class="idref" href="#y:552"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:550"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:551"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:550"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:552"><span class="id" title="variable">y</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:551"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:552"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_lub" class="idref" href="#CantorOrdinal.limit_lub"><span class="id" title="lemma">limit_lub</span></a> <a id="f:553" class="idref" href="#f:553"><span class="id" title="binder">f</span></a> <a id="x:554" class="idref" href="#x:554"><span class="id" title="binder">x</span></a> <a id="y:555" class="idref" href="#y:555"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:553"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:554"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:556" class="idref" href="#n:556"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:553"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:556"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:555"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:555"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:554"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:555"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.phi1" class="idref" href="#CantorOrdinal.phi1"><span class="id" title="definition">phi1</span></a> <a id="a:557" class="idref" href="#a:557"><span class="id" title="binder">a</span></a> (<a id="f:558" class="idref" href="#f:558"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) := <span class="id" title="keyword">fun</span> <a id="n:559" class="idref" href="#n:559"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#a:557"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:558"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:559"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.phi2" class="idref" href="#CantorOrdinal.phi2"><span class="id" title="definition">phi2</span></a> (<a id="f:560" class="idref" href="#f:560"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf"><span class="id" title="abbreviation">Tf</span></a>) := <span class="id" title="keyword">fun</span> <a id="n:561" class="idref" href="#n:561"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:560"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:561"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.phi3" class="idref" href="#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> <a id="a:562" class="idref" href="#a:562"><span class="id" title="binder">a</span></a>:= <span class="id" title="keyword">fun</span> <a id="n:563" class="idref" href="#n:563"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:562"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:563"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit1" class="idref" href="#CantorOrdinal.limit1"><span class="id" title="lemma">limit1</span></a> <a id="a:564" class="idref" href="#a:564"><span class="id" title="binder">a</span></a> <a id="b:565" class="idref" href="#b:565"><span class="id" title="binder">b</span></a> <a id="f:566" class="idref" href="#f:566"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:564"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:566"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:565"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi1"><span class="id" title="definition">phi1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:564"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:566"><span class="id" title="variable">f</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:564"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:565"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit2" class="idref" href="#CantorOrdinal.limit2"><span class="id" title="lemma">limit2</span></a> <a id="b:567" class="idref" href="#b:567"><span class="id" title="binder">b</span></a> <a id="f:568" class="idref" href="#f:568"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:568"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:567"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi2"><span class="id" title="definition">phi2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:568"><span class="id" title="variable">f</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:567"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit3" class="idref" href="#CantorOrdinal.limit3"><span class="id" title="lemma">limit3</span></a> <a id="a:569" class="idref" href="#a:569"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:569"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:569"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:569"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab444"></a><h3 class="section">Normal functions</h3>
+ We say that <span class="inlinecode"><span class="id" title="var">f</span>:<span class="id" title="var">T2</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">T2</span></span> is a normal function  if it is striclly increasing
+and the suppremum of all <span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">y</span>)</span>, for <span class="inlinecode"><span class="id" title="var">y</span>&lt;<span class="id" title="var">x</span></span> is <span class="inlinecode"><span class="id" title="var">f</span>(<span class="id" title="var">x</span>)</span> whenever <span class="inlinecode"><span class="id" title="var">x</span></span> is limit.
+Everything is assumed NF. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.limit_fct" class="idref" href="#CantorOrdinal.limit_fct"><span class="id" title="definition">limit_fct</span></a> <a id="x:570" class="idref" href="#x:570"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:570"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> <span class="id" title="var">a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <span class="id" title="var">a</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span><span class="id" title="notation">.-1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred"><span class="id" title="definition">T1pred</span></a> <span class="id" title="var">a</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi2"><span class="id" title="definition">phi2</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#limit_fct:571"><span class="id" title="definition">limit_fct</span></a> <span class="id" title="var">a</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span><span class="id" title="notation">.-1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>)  (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi2"><span class="id" title="definition">phi2</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#limit_fct:571"><span class="id" title="definition">limit_fct</span></a> <span class="id" title="var">a</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#limit_fct:571"><span class="id" title="definition">limit_fct</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.limit_prop" class="idref" href="#CantorOrdinal.limit_prop"><span class="id" title="lemma">limit_prop</span></a> <a id="x:573" class="idref" href="#x:573"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:573"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:573"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_fct"><span class="id" title="definition">limit_fct</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:573"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#x:573"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.sup" class="idref" href="#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> (<a id="f:576" class="idref" href="#f:576"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) <a id="x:577" class="idref" href="#x:577"><span class="id" title="binder">x</span></a> <a id="z:578" class="idref" href="#z:578"><span class="id" title="binder">z</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:578"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:579" class="idref" href="#y:579"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:579"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:579"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:577"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:576"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:579"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:578"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="z':580" class="idref" href="#z':580"><span class="id" title="binder">z'</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':580"><span class="id" title="variable">z'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':580"><span class="id" title="variable">z'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:578"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:581" class="idref" href="#y:581"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:581"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:581"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:577"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':580"><span class="id" title="variable">z'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:576"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:581"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.normal" class="idref" href="#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> <a id="f:582" class="idref" href="#f:582"><span class="id" title="binder">f</span></a>:=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <span class="id" title="keyword">∀</span> <a id="x:583" class="idref" href="#x:583"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:583"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:582"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:583"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:584" class="idref" href="#x:584"><span class="id" title="binder">x</span></a> <a id="y:585" class="idref" href="#y:585"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:584"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:585"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:584"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:585"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:582"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:584"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:582"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:585"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:586" class="idref" href="#x:586"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:586"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:586"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:582"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:586"><span class="id" title="variable">x</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:582"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:586"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sup_unique" class="idref" href="#CantorOrdinal.sup_unique"><span class="id" title="lemma">sup_unique</span></a> <a id="f:587" class="idref" href="#f:587"><span class="id" title="binder">f</span></a> <a id="x:588" class="idref" href="#x:588"><span class="id" title="binder">x</span></a> <a id="z:589" class="idref" href="#z:589"><span class="id" title="binder">z</span></a> <a id="z':590" class="idref" href="#z':590"><span class="id" title="binder">z'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:587"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:588"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:589"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:587"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:588"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':590"><span class="id" title="variable">z'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:589"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':590"><span class="id" title="variable">z'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sup_Oalpha_zero" class="idref" href="#CantorOrdinal.sup_Oalpha_zero"><span class="id" title="lemma">sup_Oalpha_zero</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sup_Oalpha_succ" class="idref" href="#CantorOrdinal.sup_Oalpha_succ"><span class="id" title="lemma">sup_Oalpha_succ</span></a> <a id="x:591" class="idref" href="#x:591"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:591"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:591"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#x:591"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.sup_Oalpha_limit" class="idref" href="#CantorOrdinal.sup_Oalpha_limit"><span class="id" title="lemma">sup_Oalpha_limit</span></a> <a id="x:592" class="idref" href="#x:592"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:592"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:592"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:592"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:592"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Identity is normal, composition of normal functions is normal, 
+addition is normal when the firtst argument is fixed. A normal function maps limit ordinals to limit ordinls 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.normal_id" class="idref" href="#CantorOrdinal.normal_id"><span class="id" title="lemma">normal_id</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.normal_limit" class="idref" href="#CantorOrdinal.normal_limit"><span class="id" title="lemma">normal_limit</span></a> <a id="f:593" class="idref" href="#f:593"><span class="id" title="binder">f</span></a> <a id="x:594" class="idref" href="#x:594"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:593"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:594"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:594"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:593"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:594"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_normal" class="idref" href="#CantorOrdinal.add_normal"><span class="id" title="lemma">add_normal</span></a> <a id="a:595" class="idref" href="#a:595"><span class="id" title="binder">a</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:595"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add"><span class="id" title="definition">T1add</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:595"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.normal_compose" class="idref" href="#CantorOrdinal.normal_compose"><span class="id" title="lemma">normal_compose</span></a> <a id="f:596" class="idref" href="#f:596"><span class="id" title="binder">f</span></a> <a id="g:597" class="idref" href="#g:597"><span class="id" title="binder">g</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:596"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:597"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:596"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:597"><span class="id" title="variable">g</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab445"></a><h2 class="section">multiplication</h2>
+
+<div class="paragraph"> </div>
+
+ There is a unique way to define multiplication (for NF arguments)
+compatible with our interpretation of <span class="inlinecode"><span class="id" title="var">cons</span></span>. In the case where <span class="inlinecode"><span class="id" title="var">a</span></span> and <span class="inlinecode"><span class="id" title="var">a'</span></span> 
+are zero, we could use zero or <span class="inlinecode"><span class="id" title="var">b</span></span> instead of <span class="inlinecode"><span class="id" title="var">b'</span></span>. With the current 
+implementation, multiplication is associative, and there is a distributivity 
+law 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1mul" class="idref" href="#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> (<a id="c1:598" class="idref" href="#c1:598"><span class="id" title="binder">c1</span></a> <a id="c2:599" class="idref" href="#c2:599"><span class="id" title="binder">c2</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">c2</span>}:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c2:599"><span class="id" title="variable">c2</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c1:598"><span class="id" title="variable">c1</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a'</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> (<span class="id" title="var">n</span><span class="id" title="notation">×</span><span class="id" title="var">n'</span> <span class="id" title="notation">+</span> <span class="id" title="var">n</span> <span class="id" title="notation">+</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a'</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span>  (<span class="id" title="var">n</span><span class="id" title="notation">×</span><span class="id" title="var">n'</span> <span class="id" title="notation">+</span> <span class="id" title="var">n</span> <span class="id" title="notation">+</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> (<span class="id" title="var">a</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <span class="id" title="var">a'</span>) <span class="id" title="var">n'</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>   <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><br/>
+<span class="id" title="keyword">where</span>  <a id="48d91ed92bfa929de85077cab5690e4f" class="idref" href="#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">&quot;</span></a>c1 * c2" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1mul:600"><span class="id" title="definition">T1mul</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_na" class="idref" href="#CantorOrdinal.mul_na"><span class="id" title="lemma">mul_na</span></a> <a id="n:603" class="idref" href="#n:603"><span class="id" title="binder">n</span></a> <a id="b:604" class="idref" href="#b:604"><span class="id" title="binder">b</span></a> <a id="x:605" class="idref" href="#x:605"><span class="id" title="binder">x</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:603"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:604"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:605"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:603"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:605"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1muln0" class="idref" href="#CantorOrdinal.T1muln0"><span class="id" title="lemma">T1muln0</span></a> <a id="x:606" class="idref" href="#x:606"><span class="id" title="binder">x</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:606"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul0n" class="idref" href="#CantorOrdinal.T1mul0n"><span class="id" title="lemma">T1mul0n</span></a> <a id="x:607" class="idref" href="#x:607"><span class="id" title="binder">x</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:607"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_int" class="idref" href="#CantorOrdinal.mul_int"><span class="id" title="lemma">mul_int</span></a> <a id="n:608" class="idref" href="#n:608"><span class="id" title="binder">n</span></a> <a id="m:609" class="idref" href="#m:609"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:608"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:609"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:608"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:609"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_phi0" class="idref" href="#CantorOrdinal.mul_phi0"><span class="id" title="lemma">mul_phi0</span></a> <a id="a:610" class="idref" href="#a:610"><span class="id" title="binder">a</span></a> <a id="b:611" class="idref" href="#b:611"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:610"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:611"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:610"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:611"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul_eq0" class="idref" href="#CantorOrdinal.T1mul_eq0"><span class="id" title="lemma">T1mul_eq0</span></a> <a id="x:612" class="idref" href="#x:612"><span class="id" title="binder">x</span></a> <a id="y:613" class="idref" href="#y:613"><span class="id" title="binder">y</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:612"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:613"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:612"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:613"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul_eq1" class="idref" href="#CantorOrdinal.T1mul_eq1"><span class="id" title="lemma">T1mul_eq1</span></a> <a id="a:614" class="idref" href="#a:614"><span class="id" title="binder">a</span></a> <a id="b:615" class="idref" href="#b:615"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:614"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:614"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:615"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:614"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:615"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_distr" class="idref" href="#CantorOrdinal.mul_distr"><span class="id" title="lemma">mul_distr</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add"><span class="id" title="definition">T1add</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mulA" class="idref" href="#CantorOrdinal.mulA"><span class="id" title="lemma">mulA</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Note that in some case the product of <span class="inlinecode"><span class="id" title="var">x</span></span> and one is not <span class="inlinecode"><span class="id" title="var">x</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1muln1" class="idref" href="#CantorOrdinal.T1muln1"><span class="id" title="lemma">T1muln1</span></a> <a id="x:616" class="idref" href="#x:616"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:616"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:616"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:616"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul1n" class="idref" href="#CantorOrdinal.T1mul1n"><span class="id" title="lemma">T1mul1n</span></a> <a id="x:617" class="idref" href="#x:617"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:617"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:617"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul1nCE" class="idref" href="#CantorOrdinal.T1mul1nCE"><span class="id" title="lemma">T1mul1nCE</span></a> (<a id="x:618" class="idref" href="#x:618"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#x:618"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:618"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1muln1_CE" class="idref" href="#CantorOrdinal.T1muln1_CE"><span class="id" title="lemma">T1muln1_CE</span></a> <a id="x:619" class="idref" href="#x:619"><span class="id" title="binder">x</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:619"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:619"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:619"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_succ" class="idref" href="#CantorOrdinal.mul_succ"><span class="id" title="lemma">mul_succ</span></a> <a id="x:621" class="idref" href="#x:621"><span class="id" title="binder">x</span></a> <a id="y:622" class="idref" href="#y:622"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:621"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:621"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ"><span class="id" title="definition">T1succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:622"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:621"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:622"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:621"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_mul2l" class="idref" href="#CantorOrdinal.T1lt_mul2l"><span class="id" title="lemma">T1lt_mul2l</span></a> <a id="x:623" class="idref" href="#x:623"><span class="id" title="binder">x</span></a> <a id="y:624" class="idref" href="#y:624"><span class="id" title="binder">y</span></a> <a id="z:625" class="idref" href="#z:625"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:623"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:625"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:623"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:624"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:623"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:625"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:624"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:625"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mulnf0" class="idref" href="#CantorOrdinal.mulnf0"><span class="id" title="lemma">mulnf0</span></a> <a id="a:626" class="idref" href="#a:626"><span class="id" title="binder">a</span></a> <a id="n:627" class="idref" href="#n:627"><span class="id" title="binder">n</span></a> <a id="b:628" class="idref" href="#b:628"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:626"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:628"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:626"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:627"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:628"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:626"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_mul" class="idref" href="#CantorOrdinal.nf_mul"><span class="id" title="lemma">nf_mul</span></a> <a id="a:629" class="idref" href="#a:629"><span class="id" title="binder">a</span></a> <a id="b:630" class="idref" href="#b:630"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:629"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:630"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:629"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:630"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1lt_mul2r" class="idref" href="#CantorOrdinal.T1lt_mul2r"><span class="id" title="lemma">T1lt_mul2r</span></a> <a id="x:636" class="idref" href="#x:636"><span class="id" title="binder">x</span></a> <a id="y:637" class="idref" href="#y:637"><span class="id" title="binder">y</span></a> <a id="z:638" class="idref" href="#z:638"><span class="id" title="binder">z</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:637"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:636"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:638"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:636"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:637"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:638"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_mul2l" class="idref" href="#CantorOrdinal.T1le_mul2l"><span class="id" title="lemma">T1le_mul2l</span></a> <a id="x:644" class="idref" href="#x:644"><span class="id" title="binder">x</span></a> <a id="y:645" class="idref" href="#y:645"><span class="id" title="binder">y</span></a> <a id="z:646" class="idref" href="#z:646"><span class="id" title="binder">z</span></a> :  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:644"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:645"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:644"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:645"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:644"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:646"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:645"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:646"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_mul2r" class="idref" href="#CantorOrdinal.T1le_mul2r"><span class="id" title="lemma">T1le_mul2r</span></a> <a id="x:647" class="idref" href="#x:647"><span class="id" title="binder">x</span></a> <a id="y:648" class="idref" href="#y:648"><span class="id" title="binder">y</span></a> <a id="z:649" class="idref" href="#z:649"><span class="id" title="binder">z</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:648"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:649"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:648"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:647"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:649"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:647"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1eq_mul2l" class="idref" href="#CantorOrdinal.T1eq_mul2l"><span class="id" title="lemma">T1eq_mul2l</span></a> <a id="p:650" class="idref" href="#p:650"><span class="id" title="binder">p</span></a> <a id="m:651" class="idref" href="#m:651"><span class="id" title="binder">m</span></a> <a id="n:652" class="idref" href="#n:652"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#p:650"><span class="id" title="variable">p</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:651"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:652"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:650"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:651"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:650"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:652"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:651"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:652"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_pmulr" class="idref" href="#CantorOrdinal.T1le_pmulr"><span class="id" title="lemma">T1le_pmulr</span></a> <a id="x:653" class="idref" href="#x:653"><span class="id" title="binder">x</span></a> <a id="a:654" class="idref" href="#a:654"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:654"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:653"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:654"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:654"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:653"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="CantorOrdinal.T1le_pmulrCE" class="idref" href="#CantorOrdinal.T1le_pmulrCE"><span class="id" title="lemma">T1le_pmulrCE</span></a> (<a id="x:655" class="idref" href="#x:655"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F1</span></a> ) (<a id="a:656" class="idref" href="#a:656"><span class="id" title="binder">a</span></a>:=<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:656"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:656"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:655"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_pmulrl" class="idref" href="#CantorOrdinal.T1le_pmulrl"><span class="id" title="lemma">T1le_pmulrl</span></a> <a id="x:657" class="idref" href="#x:657"><span class="id" title="binder">x</span></a> <a id="a:658" class="idref" href="#a:658"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:657"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:658"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:657"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:658"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_mulCE" class="idref" href="#CantorOrdinal.T1le_mulCE"><span class="id" title="lemma">T1le_mulCE</span></a> (<a id="m1:659" class="idref" href="#m1:659"><span class="id" title="binder">m1</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) (<a id="m2:660" class="idref" href="#m2:660"><span class="id" title="binder">m2</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>) (<a id="n1:661" class="idref" href="#n1:661"><span class="id" title="binder">n1</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F1</span></a>) (<a id="n2:662" class="idref" href="#n2:662"><span class="id" title="binder">n2</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m1:659"><span class="id" title="variable">m1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:661"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m2:660"><span class="id" title="variable">m2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:662"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:659"><span class="id" title="variable">m1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:660"><span class="id" title="variable">m2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:661"><span class="id" title="variable">n1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:662"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <span class="id" title="notation">==</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_mul" class="idref" href="#CantorOrdinal.T1le_mul"><span class="id" title="lemma">T1le_mul</span></a> <a id="m1:663" class="idref" href="#m1:663"><span class="id" title="binder">m1</span></a> <a id="m2:664" class="idref" href="#m2:664"><span class="id" title="binder">m2</span></a> <a id="n1:665" class="idref" href="#n1:665"><span class="id" title="binder">n1</span></a> <a id="n2:666" class="idref" href="#n2:666"><span class="id" title="binder">n2</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:664"><span class="id" title="variable">m2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m1:663"><span class="id" title="variable">m1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:665"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:664"><span class="id" title="variable">m2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:666"><span class="id" title="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#m1:663"><span class="id" title="variable">m1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m2:664"><span class="id" title="variable">m2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n1:665"><span class="id" title="variable">n1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n2:666"><span class="id" title="variable">n2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab446"></a><h3 class="section">Preparation of the exponention</h3>
+
+The prouct of an integer and omega is omega. This holds in fact for any limit 
+ordinals. We give here a formula for the product of omega and <span class="inlinecode"><span class="id" title="var">x</span></span>, and show 
+that this is a limit ordinal. The converse holds. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_fin_omega" class="idref" href="#CantorOrdinal.mul_fin_omega"><span class="id" title="lemma">mul_fin_omega</span></a> <a id="n:667" class="idref" href="#n:667"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:667"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_int_limit" class="idref" href="#CantorOrdinal.mul_int_limit"><span class="id" title="lemma">mul_int_limit</span></a> <a id="n:668" class="idref" href="#n:668"><span class="id" title="binder">n</span></a> <a id="y:669" class="idref" href="#y:669"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:669"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:668"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:669"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:669"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1mul_omega" class="idref" href="#CantorOrdinal.T1mul_omega"><span class="id" title="lemma">T1mul_omega</span></a> <a id="a:670" class="idref" href="#a:670"><span class="id" title="binder">a</span></a> <a id="n:671" class="idref" href="#n:671"><span class="id" title="binder">n</span></a> <a id="b:672" class="idref" href="#b:672"><span class="id" title="binder">b</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:670"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:671"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:672"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:670"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:671"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:670"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n:671"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:672"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_omega_limit" class="idref" href="#CantorOrdinal.mul_omega_limit"><span class="id" title="lemma">mul_omega_limit</span></a> <a id="x:673" class="idref" href="#x:673"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:673"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:673"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.T1div_by_omega" class="idref" href="#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a id="x:674" class="idref" href="#x:674"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:674"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> (<span class="id" title="var">a</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1div_by_omega:675"><span class="id" title="definition">T1div_by_omega</span></a> <span class="id" title="var">b</span>) <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.div_by_omega_pr" class="idref" href="#CantorOrdinal.div_by_omega_pr"><span class="id" title="lemma">div_by_omega_pr</span></a> <a id="x:677" class="idref" href="#x:677"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:677"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:677"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:677"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:677"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:677"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We show here every ordinal <span class="inlinecode"><span class="id" title="var">x</span></span> is the product of omega and <span class="inlinecode"><span class="id" title="var">y</span></span>, 
+to which an integer is added. We study the behaviour 
+of this decomposition and multiplication 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_div_by_omega" class="idref" href="#CantorOrdinal.nf_div_by_omega"><span class="id" title="lemma">nf_div_by_omega</span></a> <a id="x:678" class="idref" href="#x:678"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:678"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:678"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:678"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_revCE" class="idref" href="#CantorOrdinal.nf_revCE"><span class="id" title="lemma">nf_revCE</span></a> <a id="u:679" class="idref" href="#u:679"><span class="id" title="binder">u</span></a> <a id="v:680" class="idref" href="#v:680"><span class="id" title="binder">v</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:679"><span class="id" title="variable">u</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:680"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.add_simpl3" class="idref" href="#CantorOrdinal.add_simpl3"><span class="id" title="lemma">add_simpl3</span></a> <a id="x:681" class="idref" href="#x:681"><span class="id" title="binder">x</span></a> <a id="y:682" class="idref" href="#y:682"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#y:682"><span class="id" title="variable">y</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:681"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:681"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:682"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:681"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:682"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.plus_int_Ox" class="idref" href="#CantorOrdinal.plus_int_Ox"><span class="id" title="lemma">plus_int_Ox</span></a> <a id="n:683" class="idref" href="#n:683"><span class="id" title="binder">n</span></a> <a id="x:684" class="idref" href="#x:684"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:684"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:683"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:684"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:684"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_rev" class="idref" href="#CantorOrdinal.nf_rev"><span class="id" title="lemma">nf_rev</span></a> <a id="x:685" class="idref" href="#x:685"><span class="id" title="binder">x</span></a> (<a id="u:686" class="idref" href="#u:686"><span class="id" title="binder">u</span></a> := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:685"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>)) (<a id="v:687" class="idref" href="#v:687"><span class="id" title="binder">v</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:685"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>):<br/>
+&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:685"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:686"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:685"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:686"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:687"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_rev_unique" class="idref" href="#CantorOrdinal.nf_rev_unique"><span class="id" title="lemma">nf_rev_unique</span></a> <a id="u:688" class="idref" href="#u:688"><span class="id" title="binder">u</span></a> <a id="v:689" class="idref" href="#v:689"><span class="id" title="binder">v</span></a> (<a id="x:690" class="idref" href="#x:690"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:688"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:689"><span class="id" title="variable">v</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:688"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#u:688"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:690"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:689"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:690"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_rev_sum" class="idref" href="#CantorOrdinal.nf_rev_sum"><span class="id" title="lemma">nf_rev_sum</span></a> <a id="x:699" class="idref" href="#x:699"><span class="id" title="binder">x</span></a> <a id="y:700" class="idref" href="#y:700"><span class="id" title="binder">y</span></a><br/>
+&nbsp;&nbsp;(<a id="u:701" class="idref" href="#u:701"><span class="id" title="binder">u</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:699"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="n:702" class="idref" href="#n:702"><span class="id" title="binder">n</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:699"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>)<br/>
+&nbsp;&nbsp;(<a id="v:703" class="idref" href="#v:703"><span class="id" title="binder">v</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:700"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="m:704" class="idref" href="#m:704"><span class="id" title="binder">m</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:700"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>)<br/>
+&nbsp;&nbsp;(<a id="w:705" class="idref" href="#w:705"><span class="id" title="binder">w</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:699"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:700"><span class="id" title="variable">y</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="p:706" class="idref" href="#p:706"><span class="id" title="binder">p</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:699"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:700"><span class="id" title="variable">y</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:699"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:700"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:703"><span class="id" title="variable">v</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:705"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:701"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:706"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:702"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:704"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:705"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:701"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:703"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:706"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:704"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_sum_omega" class="idref" href="#CantorOrdinal.mul_sum_omega"><span class="id" title="lemma">mul_sum_omega</span></a> <a id="a:707" class="idref" href="#a:707"><span class="id" title="binder">a</span></a> <a id="n:708" class="idref" href="#n:708"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:707"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:707"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:708"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:707"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_rev_prod" class="idref" href="#CantorOrdinal.nf_rev_prod"><span class="id" title="lemma">nf_rev_prod</span></a> <a id="x:709" class="idref" href="#x:709"><span class="id" title="binder">x</span></a> <a id="y:710" class="idref" href="#y:710"><span class="id" title="binder">y</span></a><br/>
+&nbsp;&nbsp;(<a id="u:711" class="idref" href="#u:711"><span class="id" title="binder">u</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:709"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="n:712" class="idref" href="#n:712"><span class="id" title="binder">n</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:709"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>)<br/>
+&nbsp;&nbsp;(<a id="v:713" class="idref" href="#v:713"><span class="id" title="binder">v</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:710"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="m:714" class="idref" href="#m:714"><span class="id" title="binder">m</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:710"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>)<br/>
+&nbsp;&nbsp;(<a id="w:715" class="idref" href="#w:715"><span class="id" title="binder">w</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:709"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:710"><span class="id" title="variable">y</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>) (<a id="p:716" class="idref" href="#p:716"><span class="id" title="binder">p</span></a>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:709"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:710"><span class="id" title="variable">y</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:709"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:710"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:711"><span class="id" title="variable">u</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:712"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:715"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:716"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:715"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:713"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:716"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:712"><span class="id" title="variable">n</span></a><span class="id" title="notation">×</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:714"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:714"><span class="id" title="variable">m</span></a><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:715"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:711"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:713"><span class="id" title="variable">v</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:716"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:715"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:711"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:713"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:711"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:714"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:716"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:712"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab447"></a><h3 class="section">Normality of multiplication</h3>
+
+If <span class="inlinecode"><span class="id" title="var">a</span></span> is a non-zero ordinal, the multiplication by <span class="inlinecode"><span class="id" title="var">a</span></span> is normal.
+This means, if <span class="inlinecode"><span class="id" title="var">b</span></span> is limit, the supremum of all <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">×<span class="id" title="var">c</span></span> <span class="inlinecode"></span> for <span class="inlinecode"><span class="id" title="var">c</span>&lt;<span class="id" title="var">b</span></span> <span class="inlinecode"></span> is <span class="inlinecode"><span class="id" title="var">a</span>×<span class="id" title="var">b</span></span>.
+We show this for omega, and for some special cases. 
+This is equivalent to existence of ordinal division.  
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_omega_pr1" class="idref" href="#CantorOrdinal.mul_omega_pr1"><span class="id" title="lemma">mul_omega_pr1</span></a> <a id="a:717" class="idref" href="#a:717"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:717"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:717"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:717"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:717"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_omega2_pr1" class="idref" href="#CantorOrdinal.mul_omega2_pr1"><span class="id" title="lemma">mul_omega2_pr1</span></a> <a id="a:718" class="idref" href="#a:718"><span class="id" title="binder">a</span></a> (<a id="u:719" class="idref" href="#u:719"><span class="id" title="binder">u</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> 1 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#a:718"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:718"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:718"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#u:719"><span class="id" title="variable">u</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:718"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:719"><span class="id" title="variable">u</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.mul_omega_pr3" class="idref" href="#CantorOrdinal.mul_omega_pr3"><span class="id" title="lemma">mul_omega_pr3</span></a> <a id="a:720" class="idref" href="#a:720"><span class="id" title="binder">a</span></a> <a id="b:721" class="idref" href="#b:721"><span class="id" title="binder">b</span></a> <a id="c:722" class="idref" href="#c:722"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:721"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul"><span class="id" title="definition">T1mul</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#b:721"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:720"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:721"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:722"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab448"></a><h2 class="section">Exponentiation</h2>
+
+<div class="paragraph"> </div>
+
+ In order to compute <span class="inlinecode"></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">^<span class="id" title="var">b</span></span> <span class="inlinecode"></span>, we first write <span class="inlinecode"><span class="id" title="var">b</span></span> as the sum of a limit 
+ordinal and an integer <span class="inlinecode"><span class="id" title="var">n</span></span>. Computing  <span class="inlinecode"></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">^<span class="id" title="var">n</span></span> <span class="inlinecode"></span> is trivial. The limit ordinal
+is omega times <span class="inlinecode"><span class="id" title="var">c</span></span>; if <span class="inlinecode"><span class="id" title="var">a</span></span> is at least one, then <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">^<span class="id" title="var">omega</span></span>  <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">^<span class="id" title="var">d</span></span> <span class="inlinecode"></span>
+for some <span class="inlinecode"><span class="id" title="var">d</span></span>, and the result is <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">^</span> <span class="inlinecode">(<span class="id" title="var">omega</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">c</span>)</span>  <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">^(<span class="id" title="var">d</span>×<span class="id" title="var">c</span>)=<span class="id" title="var">phi0</span>(<span class="id" title="var">d</span>×<span class="id" title="var">c</span>)</span> <span class="inlinecode"></span>.
+This leads to the following definitions.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="CantorOrdinal.exp_F" class="idref" href="#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a id="a:723" class="idref" href="#a:723"><span class="id" title="binder">a</span></a> <a id="n:724" class="idref" href="#n:724"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:724"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">p</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:723"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#exp_F:725"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:723"><span class="id" title="variable">a</span></a> <span class="id" title="var">p</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.exp_O" class="idref" href="#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a id="a:727" class="idref" href="#a:727"><span class="id" title="binder">a</span></a> <a id="b:728" class="idref" href="#b:728"><span class="id" title="binder">b</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:727"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:728"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:727"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:727"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:727"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:728"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:727"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:728"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="CantorOrdinal.T1exp" class="idref" href="#CantorOrdinal.T1exp"><span class="id" title="definition">T1exp</span></a> <a id="a:729" class="idref" href="#a:729"><span class="id" title="binder">a</span></a> <a id="b:730" class="idref" href="#b:730"><span class="id" title="binder">b</span></a>:=<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:729"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:730"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:729"><span class="id" title="variable">a</span></a> ( <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:730"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="663739e56a48dc47ea1955ef538eefc5" class="idref" href="#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">&quot;</span></a>a ^ b" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1exp"><span class="id" title="definition">T1exp</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties of <span class="inlinecode"><span class="id" title="var">exp_O</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_mul1" class="idref" href="#CantorOrdinal.expO_mul1"><span class="id" title="lemma">expO_mul1</span></a> <a id="a:731" class="idref" href="#a:731"><span class="id" title="binder">a</span></a> <a id="b:732" class="idref" href="#b:732"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:731"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:732"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:731"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:732"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_expO" class="idref" href="#CantorOrdinal.nf_expO"><span class="id" title="lemma">nf_expO</span></a> <a id="a:733" class="idref" href="#a:733"><span class="id" title="binder">a</span></a> <a id="b:734" class="idref" href="#b:734"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:733"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:734"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:733"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:734"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_n0" class="idref" href="#CantorOrdinal.expO_n0"><span class="id" title="lemma">expO_n0</span></a> <a id="x:735" class="idref" href="#x:735"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:735"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_1n" class="idref" href="#CantorOrdinal.expO_1n"><span class="id" title="lemma">expO_1n</span></a> <a id="n:736" class="idref" href="#n:736"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n:736"><span class="id" title="variable">n</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_eq0" class="idref" href="#CantorOrdinal.expO_eq0"><span class="id" title="lemma">expO_eq0</span></a> <a id="a:737" class="idref" href="#a:737"><span class="id" title="binder">a</span></a> <a id="b:738" class="idref" href="#b:738"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:737"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:738"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:737"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:738"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_eq1" class="idref" href="#CantorOrdinal.expO_eq1"><span class="id" title="lemma">expO_eq1</span></a> <a id="a:739" class="idref" href="#a:739"><span class="id" title="binder">a</span></a> <a id="b:740" class="idref" href="#b:740"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:739"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:740"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:739"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:740"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expO_add" class="idref" href="#CantorOrdinal.expO_add"><span class="id" title="lemma">expO_add</span></a> <a id="z:744" class="idref" href="#z:744"><span class="id" title="binder">z</span></a> <a id="u:745" class="idref" href="#u:745"><span class="id" title="binder">u</span></a> <a id="v:746" class="idref" href="#v:746"><span class="id" title="binder">v</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:744"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:745"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:744"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:746"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:744"><span class="id" title="variable">z</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#u:745"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:746"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties of <span class="inlinecode"><span class="id" title="var">exp_F</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_expF" class="idref" href="#CantorOrdinal.nf_expF"><span class="id" title="lemma">nf_expF</span></a> <a id="a:747" class="idref" href="#a:747"><span class="id" title="binder">a</span></a> <a id="n:748" class="idref" href="#n:748"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:747"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:747"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:748"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expF_add" class="idref" href="#CantorOrdinal.expF_add"><span class="id" title="lemma">expF_add</span></a> <a id="a:749" class="idref" href="#a:749"><span class="id" title="binder">a</span></a> <a id="n:750" class="idref" href="#n:750"><span class="id" title="binder">n</span></a> <a id="m:751" class="idref" href="#m:751"><span class="id" title="binder">m</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:749"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:750"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:749"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:751"><span class="id" title="variable">m</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:749"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:750"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:751"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expF_mul" class="idref" href="#CantorOrdinal.expF_mul"><span class="id" title="lemma">expF_mul</span></a> <a id="a:752" class="idref" href="#a:752"><span class="id" title="binder">a</span></a> <a id="n:753" class="idref" href="#n:753"><span class="id" title="binder">n</span></a> <a id="m:754" class="idref" href="#m:754"><span class="id" title="binder">m</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:752"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:753"><span class="id" title="variable">n</span></a> <span class="id" title="notation">×</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:754"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:752"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:753"><span class="id" title="variable">n</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#m:754"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expF_1n" class="idref" href="#CantorOrdinal.expF_1n"><span class="id" title="lemma">expF_1n</span></a> <a id="n:755" class="idref" href="#n:755"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n:755"><span class="id" title="variable">n</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expF_eq0" class="idref" href="#CantorOrdinal.expF_eq0"><span class="id" title="lemma">expF_eq0</span></a> <a id="a:756" class="idref" href="#a:756"><span class="id" title="binder">a</span></a> <a id="n:757" class="idref" href="#n:757"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:756"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:757"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:756"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:757"><span class="id" title="variable">n</span></a> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expF_eq1" class="idref" href="#CantorOrdinal.expF_eq1"><span class="id" title="lemma">expF_eq1</span></a> <a id="a:758" class="idref" href="#a:758"><span class="id" title="binder">a</span></a> <a id="n:759" class="idref" href="#n:759"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:758"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:758"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:759"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:758"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:759"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties of <span class="inlinecode"><span class="id" title="var">exp</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.nf_exp" class="idref" href="#CantorOrdinal.nf_exp"><span class="id" title="lemma">nf_exp</span></a> <a id="a:760" class="idref" href="#a:760"><span class="id" title="binder">a</span></a> <a id="b:761" class="idref" href="#b:761"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:760"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:761"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:760"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:761"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp00" class="idref" href="#CantorOrdinal.exp00"><span class="id" title="lemma">exp00</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expx0" class="idref" href="#CantorOrdinal.expx0"><span class="id" title="lemma">expx0</span></a> <a id="x:762" class="idref" href="#x:762"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:762"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expx_pnat" class="idref" href="#CantorOrdinal.expx_pnat"><span class="id" title="lemma">expx_pnat</span></a> <a id="x:763" class="idref" href="#x:763"><span class="id" title="binder">x</span></a> <a id="n:764" class="idref" href="#n:764"><span class="id" title="binder">n</span></a> <a id="b:765" class="idref" href="#b:765"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:763"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:764"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:765"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:763"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:764"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expx_nat" class="idref" href="#CantorOrdinal.expx_nat"><span class="id" title="lemma">expx_nat</span></a> <a id="x:766" class="idref" href="#x:766"><span class="id" title="binder">x</span></a> <a id="n:767" class="idref" href="#n:767"><span class="id" title="binder">n</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:766"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:767"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:766"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:767"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expx1" class="idref" href="#CantorOrdinal.expx1"><span class="id" title="lemma">expx1</span></a> <a id="x:768" class="idref" href="#x:768"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:768"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:768"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:768"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.expx1CE" class="idref" href="#CantorOrdinal.expx1CE"><span class="id" title="lemma">expx1CE</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp2omega" class="idref" href="#CantorOrdinal.exp2omega"><span class="id" title="lemma">exp2omega</span></a> <a id="n:769" class="idref" href="#n:769"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:769"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+2</span><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp1x" class="idref" href="#CantorOrdinal.exp1x"><span class="id" title="lemma">exp1x</span></a> <a id="x:770" class="idref" href="#x:770"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:770"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_eq0" class="idref" href="#CantorOrdinal.exp_eq0"><span class="id" title="lemma">exp_eq0</span></a> <a id="x:771" class="idref" href="#x:771"><span class="id" title="binder">x</span></a> <a id="y:772" class="idref" href="#y:772"><span class="id" title="binder">y</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:771"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:772"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:771"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:772"><span class="id" title="variable">y</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp0nz" class="idref" href="#CantorOrdinal.exp0nz"><span class="id" title="lemma">exp0nz</span></a> <a id="x:773" class="idref" href="#x:773"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:773"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:773"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_eq1" class="idref" href="#CantorOrdinal.exp_eq1"><span class="id" title="lemma">exp_eq1</span></a> <a id="x:774" class="idref" href="#x:774"><span class="id" title="binder">x</span></a> <a id="y:775" class="idref" href="#y:775"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:774"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:775"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:774"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:775"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:774"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:775"><span class="id" title="variable">y</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_int" class="idref" href="#CantorOrdinal.exp_int"><span class="id" title="lemma">exp_int</span></a> <a id="a:776" class="idref" href="#a:776"><span class="id" title="binder">a</span></a> <a id="b:777" class="idref" href="#b:777"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:776"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:777"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:776"><span class="id" title="variable">a</span></a> <span class="id" title="notation">^</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:777"><span class="id" title="variable">b</span></a>%<span class="id" title="var">N</span><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_consCE1" class="idref" href="#CantorOrdinal.exp_consCE1"><span class="id" title="lemma">exp_consCE1</span></a> (<a id="x:778" class="idref" href="#x:778"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> 2) (<a id="a:779" class="idref" href="#a:779"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>) (<a id="n:780" class="idref" href="#n:780"><span class="id" title="binder">n</span></a> := 0)(<a id="b:781" class="idref" href="#b:781"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>): <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:778"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:779"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:780"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:781"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:778"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:779"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:780"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:778"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:781"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pow_omega" class="idref" href="#CantorOrdinal.pow_omega"><span class="id" title="lemma">pow_omega</span></a> <a id="x:782" class="idref" href="#x:782"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:782"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:782"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:782"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab449"></a><h3 class="section">Existence and uniqueness of the Cantor Normal Form</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.cantor_exists" class="idref" href="#CantorOrdinal.cantor_exists"><span class="id" title="lemma">cantor_exists</span></a> <a id="a:783" class="idref" href="#a:783"><span class="id" title="binder">a</span></a> <a id="n:784" class="idref" href="#n:784"><span class="id" title="binder">n</span></a> <a id="b:785" class="idref" href="#b:785"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:783"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:784"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:785"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:783"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:784"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:785"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:783"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:784"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:785"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.cantor_unique" class="idref" href="#CantorOrdinal.cantor_unique"><span class="id" title="lemma">cantor_unique</span></a> <a id="a:786" class="idref" href="#a:786"><span class="id" title="binder">a</span></a> <a id="n:787" class="idref" href="#n:787"><span class="id" title="binder">n</span></a> <a id="b:788" class="idref" href="#b:788"><span class="id" title="binder">b</span></a> <a id="a':789" class="idref" href="#a':789"><span class="id" title="binder">a'</span></a> <a id="n':790" class="idref" href="#n':790"><span class="id" title="binder">n'</span></a> <a id="b':791" class="idref" href="#b':791"><span class="id" title="binder">b'</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:786"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:787"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:788"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':789"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':790"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':791"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:786"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:787"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:788"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':789"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':790"><span class="id" title="variable">n'</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':791"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:786"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':789"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:787"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':790"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:788"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':791"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.cantor_CE1" class="idref" href="#CantorOrdinal.cantor_CE1"><span class="id" title="lemma">cantor_CE1</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.cantorCE2" class="idref" href="#CantorOrdinal.cantorCE2"><span class="id" title="lemma">cantorCE2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.cantorCE3" class="idref" href="#CantorOrdinal.cantorCE3"><span class="id" title="lemma">cantorCE3</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>  <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad"><span class="id" title="definition">T1bad</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1log_prod" class="idref" href="#CantorOrdinal.T1log_prod"><span class="id" title="lemma">T1log_prod</span></a> <a id="a:792" class="idref" href="#a:792"><span class="id" title="binder">a</span></a> <a id="b:793" class="idref" href="#b:793"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:792"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:793"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a>(<a class="idref" href="gaia_hydras.nfwfgaia.html#a:792"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:793"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:792"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:793"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1log_exp0" class="idref" href="#CantorOrdinal.T1log_exp0"><span class="id" title="lemma">T1log_exp0</span></a> <a id="x:794" class="idref" href="#x:794"><span class="id" title="binder">x</span></a> <a id="n:795" class="idref" href="#n:795"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:794"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:794"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:795"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:794"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:795"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1log_exp1" class="idref" href="#CantorOrdinal.T1log_exp1"><span class="id" title="lemma">T1log_exp1</span></a> <a id="z:796" class="idref" href="#z:796"><span class="id" title="binder">z</span></a> <a id="x:797" class="idref" href="#x:797"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:796"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:797"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:796"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#z:796"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:797"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:796"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:797"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1log_exp2" class="idref" href="#CantorOrdinal.T1log_exp2"><span class="id" title="lemma">T1log_exp2</span></a> <a id="z:798" class="idref" href="#z:798"><span class="id" title="binder">z</span></a> <a id="u:799" class="idref" href="#u:799"><span class="id" title="binder">u</span></a> <a id="v:800" class="idref" href="#v:800"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:798"><span class="id" title="variable">z</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:798"><span class="id" title="variable">z</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite"><span class="id" title="definition">T1finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:798"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:799"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#z:798"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:799"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:800"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:799"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_FO" class="idref" href="#CantorOrdinal.exp_FO"><span class="id" title="lemma">exp_FO</span></a> <a id="z:801" class="idref" href="#z:801"><span class="id" title="binder">z</span></a> <a id="n:802" class="idref" href="#n:802"><span class="id" title="binder">n</span></a> <a id="v:803" class="idref" href="#v:803"><span class="id" title="binder">v</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#v:803"><span class="id" title="variable">v</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:801"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:802"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:801"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:803"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:801"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:803"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_FO1" class="idref" href="#CantorOrdinal.exp_FO1"><span class="id" title="lemma">exp_FO1</span></a> <a id="z:804" class="idref" href="#z:804"><span class="id" title="binder">z</span></a> <a id="v:805" class="idref" href="#v:805"><span class="id" title="binder">v</span></a> <a id="n:806" class="idref" href="#n:806"><span class="id" title="binder">n</span></a> <a id="m:807" class="idref" href="#m:807"><span class="id" title="binder">m</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:804"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:805"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:805"><span class="id" title="variable">v</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:806"><span class="id" title="variable">n</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:804"><span class="id" title="variable">z</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#v:805"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:806"><span class="id" title="variable">n</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:804"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:807"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:804"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:805"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:804"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:807"><span class="id" title="variable">m</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#n:806"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_FO2" class="idref" href="#CantorOrdinal.exp_FO2"><span class="id" title="lemma">exp_FO2</span></a> <a id="z:808" class="idref" href="#z:808"><span class="id" title="binder">z</span></a> <a id="m:809" class="idref" href="#m:809"><span class="id" title="binder">m</span></a> <a id="u:810" class="idref" href="#u:810"><span class="id" title="binder">u</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:808"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:809"><span class="id" title="variable">m</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:808"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:809"><span class="id" title="variable">m</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#u:810"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:808"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:810"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_FO3" class="idref" href="#CantorOrdinal.exp_FO3"><span class="id" title="lemma">exp_FO3</span></a> <a id="z:812" class="idref" href="#z:812"><span class="id" title="binder">z</span></a> <a id="x:813" class="idref" href="#x:813"><span class="id" title="binder">x</span></a> <a id="u:814" class="idref" href="#u:814"><span class="id" title="binder">u</span></a> (<a id="w:815" class="idref" href="#w:815"><span class="id" title="binder">w</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega"><span class="id" title="definition">T1div_by_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split"><span class="id" title="definition">T1split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:813"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:812"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#w:815"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#w:815"><span class="id" title="variable">w</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:812"><span class="id" title="variable">z</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O"><span class="id" title="definition">exp_O</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#z:812"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:813"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#u:814"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0"><span class="id" title="definition">phi0</span></a>( <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log"><span class="id" title="definition">T1log</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#z:812"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:813"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega"><span class="id" title="definition">T1omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:814"><span class="id" title="variable">u</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab450"></a><h3 class="section">Basic Properties</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_sum" class="idref" href="#CantorOrdinal.exp_sum"><span class="id" title="lemma">exp_sum</span></a> <a id="x:816" class="idref" href="#x:816"><span class="id" title="binder">x</span></a> <a id="y:817" class="idref" href="#y:817"><span class="id" title="binder">y</span></a> <a id="z:818" class="idref" href="#z:818"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:816"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:817"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:818"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:816"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d"><span class="id" title="notation">+</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:817"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:818"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:816"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:818"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:817"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.exp_prod" class="idref" href="#CantorOrdinal.exp_prod"><span class="id" title="lemma">exp_prod</span></a> <a id="x:819" class="idref" href="#x:819"><span class="id" title="binder">x</span></a> <a id="y:820" class="idref" href="#y:820"><span class="id" title="binder">y</span></a> <a id="z:821" class="idref" href="#z:821"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:821"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:819"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:820"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:821"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:819"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:820"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:821"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:819"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:820"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pow_mon1" class="idref" href="#CantorOrdinal.pow_mon1"><span class="id" title="lemma">pow_mon1</span></a> <a id="x:823" class="idref" href="#x:823"><span class="id" title="binder">x</span></a> <a id="y:824" class="idref" href="#y:824"><span class="id" title="binder">y</span></a> <a id="z:825" class="idref" href="#z:825"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:823"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:824"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:825"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:823"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#y:824"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:825"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:823"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:824"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:823"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:825"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pow_mon2" class="idref" href="#CantorOrdinal.pow_mon2"><span class="id" title="lemma">pow_mon2</span></a> <a id="x:826" class="idref" href="#x:826"><span class="id" title="binder">x</span></a> <a id="y:827" class="idref" href="#y:827"><span class="id" title="binder">y</span></a> <a id="z:828" class="idref" href="#z:828"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:826"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:827"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:828"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:826"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:826"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#y:827"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:828"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:826"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:827"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:826"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:828"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.T1le_pmull" class="idref" href="#CantorOrdinal.T1le_pmull"><span class="id" title="lemma">T1le_pmull</span></a> <a id="x:829" class="idref" href="#x:829"><span class="id" title="binder">x</span></a> <a id="a:830" class="idref" href="#a:830"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:829"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:830"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:829"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f"><span class="id" title="notation">×</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:830"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CantorOrdinal.pow_mon3" class="idref" href="#CantorOrdinal.pow_mon3"><span class="id" title="lemma">pow_mon3</span></a> <a id="x:831" class="idref" href="#x:831"><span class="id" title="binder">x</span></a> <a id="y:832" class="idref" href="#y:832"><span class="id" title="binder">y</span></a> <a id="z:833" class="idref" href="#z:833"><span class="id" title="binder">z</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf"><span class="id" title="definition">T1nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:831"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:831"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:832"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:831"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:833"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:832"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5"><span class="id" title="notation">^</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:833"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal"><span class="id" title="module">CantorOrdinal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Export</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal"><span class="id" title="module">CantorOrdinal</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab451"></a><h1 class="section">The type T2</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Gamma0" class="idref" href="#Gamma0"><span class="id" title="module">Gamma0</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab452"></a><h2 class="section">Definition and Equality</h2>
+
+<div class="paragraph"> </div>
+
+ This is like T1 with one more argument 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Gamma0.T2" class="idref" href="#Gamma0.T2"><span class="id" title="inductive">T2</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="Gamma0.zero" class="idref" href="#Gamma0.zero"><span class="id" title="constructor">zero</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2:835"><span class="id" title="inductive">T2</span></a><br/>
+| <a id="Gamma0.cons" class="idref" href="#Gamma0.cons"><span class="id" title="constructor">cons</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2:835"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2:835"><span class="id" title="inductive">T2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2:835"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2:835"><span class="id" title="inductive">T2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">g0_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">g0_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">g0</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2eq" class="idref" href="#Gamma0.T2eq"><span class="id" title="definition">T2eq</span></a> <a id="x:837" class="idref" href="#x:837"><span class="id" title="binder">x</span></a> <a id="y:838" class="idref" href="#y:838"><span class="id" title="binder">y</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">x</span>} :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:837"><span class="id" title="variable">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#y:838"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2eq:839"><span class="id" title="definition">T2eq</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2eq:839"><span class="id" title="definition">T2eq</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a>  <span class="id" title="var">n</span><span class="id" title="notation">==</span> <span class="id" title="var">n'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2eq:839"><span class="id" title="definition">T2eq</span></a> <span class="id" title="var">c</span> <span class="id" title="var">c'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2eqP" class="idref" href="#Gamma0.T2eqP"><span class="id" title="lemma">T2eqP</span></a> : <span class="id" title="definition">Equality.axiom</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2eq"><span class="id" title="definition">T2eq</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="Gamma0.T2_eqMixin" class="idref" href="#Gamma0.T2_eqMixin"><span class="id" title="definition">T2_eqMixin</span></a> := <span class="id" title="abbreviation">EqMixin</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2eqP"><span class="id" title="lemma">T2eqP</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <a id="Gamma0.T2_eqType" class="idref" href="#Gamma0.T2_eqType"><span class="id" title="definition">T2_eqType</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <span class="id" title="abbreviation">EqType</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_eqMixin"><span class="id" title="definition">T2_eqMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2eqP"><span class="id" title="lemma">T2eqP</span></a> {<span class="id" title="var">x</span> <span class="id" title="var">y</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2eqE" class="idref" href="#Gamma0.T2eqE"><span class="id" title="lemma">T2eqE</span></a> <a id="a:842" class="idref" href="#a:842"><span class="id" title="binder">a</span></a> <a id="b:843" class="idref" href="#b:843"><span class="id" title="binder">b</span></a> <a id="n:844" class="idref" href="#n:844"><span class="id" title="binder">n</span></a> <a id="d:845" class="idref" href="#d:845"><span class="id" title="binder">d</span></a> <a id="a':846" class="idref" href="#a':846"><span class="id" title="binder">a'</span></a> <a id="b':847" class="idref" href="#b':847"><span class="id" title="binder">b'</span></a> <a id="n':848" class="idref" href="#n':848"><span class="id" title="binder">n'</span></a> <a id="d':849" class="idref" href="#d':849"><span class="id" title="binder">d'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:842"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:843"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:844"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:845"><span class="id" title="variable">d</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':846"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':847"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':848"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':849"><span class="id" title="variable">d'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:842"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':846"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:843"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':847"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:844"><span class="id" title="variable">n</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':848"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:845"><span class="id" title="variable">d</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':849"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ We write <span class="inlinecode"><span class="id" title="var">psi</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> instead of <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode">0</span> <span class="inlinecode">0</span>; we introduce omega and 
+epsilon0. We consider also the size, this is really the depth of the object 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="d8473b352e1ea62d853482a112bf8ee4" class="idref" href="#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">&quot;</span></a>[ x , y ]" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) :<span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2nat" class="idref" href="#Gamma0.T2nat"><span class="id" title="definition">T2nat</span></a> <a id="p:850" class="idref" href="#p:850"><span class="id" title="binder">p</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:850"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="d9e84494be2f615021318f367a1326d0" class="idref" href="#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">&quot;</span></a>\F n" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nat"><span class="id" title="definition">T2nat</span></a> <span class="id" title="var">n</span>)(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 29) : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.psi" class="idref" href="#Gamma0.psi"><span class="id" title="definition">psi</span></a> <a id="a:852" class="idref" href="#a:852"><span class="id" title="binder">a</span></a> <a id="b:853" class="idref" href="#b:853"><span class="id" title="binder">b</span></a>  := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:852"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:853"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.one" class="idref" href="#Gamma0.one"><span class="id" title="definition">one</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.omega" class="idref" href="#Gamma0.omega"><span class="id" title="definition">omega</span></a>  := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.epsilon0" class="idref" href="#Gamma0.epsilon0"><span class="id" title="definition">epsilon0</span></a>  := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T1T2" class="idref" href="#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> (<a id="c:854" class="idref" href="#c:854"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:854"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">CantorOrdinal.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1T2:855"><span class="id" title="definition">T1T2</span></a> <span class="id" title="var">a</span>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1T2:855"><span class="id" title="definition">T1T2</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.size" class="idref" href="#Gamma0.size"><span class="id" title="definition">size</span></a> <a id="x:857" class="idref" href="#x:857"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:857"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><span class="id" title="definition">maxn</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:858"><span class="id" title="definition">size</span></a> <span class="id" title="var">a</span>) (<span class="id" title="definition">maxn</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:858"><span class="id" title="definition">size</span></a> <span class="id" title="var">b</span>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:858"><span class="id" title="definition">size</span></a> <span class="id" title="var">c</span>))<span class="id" title="notation">).+1</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.size_prop1" class="idref" href="#Gamma0.size_prop1"><span class="id" title="lemma">size_prop1</span></a> <a id="a:860" class="idref" href="#a:860"><span class="id" title="binder">a</span></a> <a id="b:861" class="idref" href="#b:861"><span class="id" title="binder">b</span></a> <a id="n:862" class="idref" href="#n:862"><span class="id" title="binder">n</span></a> <a id="c:863" class="idref" href="#c:863"><span class="id" title="binder">c</span></a> (<a id="l:864" class="idref" href="#l:864"><span class="id" title="binder">l</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:860"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:861"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:862"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:863"><span class="id" title="variable">c</span></a>)):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:860"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:864"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:861"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:864"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:863"><span class="id" title="variable">c</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:864"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:860"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:861"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:864"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.size_prop" class="idref" href="#Gamma0.size_prop"><span class="id" title="lemma">size_prop</span></a> <a id="a:865" class="idref" href="#a:865"><span class="id" title="binder">a</span></a> <a id="b:866" class="idref" href="#b:866"><span class="id" title="binder">b</span></a> <a id="n:867" class="idref" href="#n:867"><span class="id" title="binder">n</span></a> <a id="c:868" class="idref" href="#c:868"><span class="id" title="binder">c</span></a> <a id="a':869" class="idref" href="#a':869"><span class="id" title="binder">a'</span></a> <a id="b':870" class="idref" href="#b':870"><span class="id" title="binder">b'</span></a> <a id="n':871" class="idref" href="#n':871"><span class="id" title="binder">n'</span></a> <a id="c':872" class="idref" href="#c':872"><span class="id" title="binder">c'</span></a> <br/>
+&nbsp;&nbsp;&nbsp;(<a id="l:873" class="idref" href="#l:873"><span class="id" title="binder">l</span></a> := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:865"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:866"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:867"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:868"><span class="id" title="variable">c</span></a>) <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':869"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':870"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':871"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':872"><span class="id" title="variable">c'</span></a>))%<span class="id" title="var">N</span>) :<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:868"><span class="id" title="variable">c</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':872"><span class="id" title="variable">c'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:865"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:866"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':870"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':869"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:865"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:866"><span class="id" title="variable">b</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':870"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:866"><span class="id" title="variable">b</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':869"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':870"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:865"><span class="id" title="variable">a</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':869"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:873"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab453"></a><h2 class="section">Order</h2>
+ Comparing ordinals is complicated. We are looking for the fixpoint of 
+some complicated expression <span class="inlinecode"><span class="id" title="var">F</span>(<span class="id" title="var">a</span>,<span class="id" title="var">b</span>)</span>, in which the psi-parts of <span class="inlinecode"><span class="id" title="var">a</span></span> and <span class="inlinecode"><span class="id" title="var">b</span></span> 
+appear as arguments of <span class="inlinecode"><span class="id" title="var">F</span></span>. Thus, a definition by induction is impossible. 
+However if <span class="inlinecode"><span class="id" title="var">l</span></span> is the some of the size if the arguments of <span class="inlinecode"><span class="id" title="var">F</span></span>, then all calls 
+of <span class="inlinecode"><span class="id" title="var">F</span></span> have a smaller value. Thus, we use a definition by induction on <span class="inlinecode"><span class="id" title="var">l</span></span>.
+All proofs will be by induction on <span class="inlinecode"><span class="id" title="var">l</span></span> as well.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.lt_rec" class="idref" href="#Gamma0.lt_rec"><span class="id" title="definition">lt_rec</span></a> <a id="f:874" class="idref" href="#f:874"><span class="id" title="binder">f</span></a> <a id="x:875" class="idref" href="#x:875"><span class="id" title="binder">x</span></a> <a id="y:876" class="idref" href="#y:876"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:875"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:876"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">b</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <span class="id" title="notation">==</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>) <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">((</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span><span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span> <span class="id" title="notation">&lt;</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <span class="id" title="notation">==</span> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:874"><span class="id" title="variable">f</span></a> <span class="id" title="var">c</span> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:876"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2lta" class="idref" href="#Gamma0.T2lta"><span class="id" title="definition">T2lta</span></a> <a id="k:880" class="idref" href="#k:880"><span class="id" title="binder">k</span></a>:=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:880"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">k</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.lt_rec"><span class="id" title="definition">lt_rec</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T2lta:881"><span class="id" title="definition">T2lta</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:880"><span class="id" title="variable">k</span></a>) <span class="id" title="keyword">else</span> <span class="id" title="keyword">fun</span> <a id="x:883" class="idref" href="#x:883"><span class="id" title="binder">x</span></a> <a id="y:884" class="idref" href="#y:884"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2lt" class="idref" href="#Gamma0.T2lt"><span class="id" title="definition">T2lt</span></a> <a id="a:885" class="idref" href="#a:885"><span class="id" title="binder">a</span></a> <a id="b:886" class="idref" href="#b:886"><span class="id" title="binder">b</span></a> :=  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2lta"><span class="id" title="definition">T2lta</span></a> <span class="id" title="notation">(</span><span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:885"><span class="id" title="variable">a</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:886"><span class="id" title="variable">b</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:885"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:886"><span class="id" title="variable">b</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2le" class="idref" href="#Gamma0.T2le"><span class="id" title="definition">T2le</span></a> (<a id="x:887" class="idref" href="#x:887"><span class="id" title="binder">x</span></a> <a id="y:888" class="idref" href="#y:888"><span class="id" title="binder">y</span></a> :<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:887"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:888"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt"><span class="id" title="definition">T2lt</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:887"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:888"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Gamma0.::g0_scope:x_'&lt;'_x" class="idref" href="#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt; y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt"><span class="id" title="definition">T2lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="bed9251d191f4ad1afb65df9e03eb72e" class="idref" href="#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">&quot;</span></a>x &lt;= y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2le"><span class="id" title="definition">T2le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="a8834fd6e3c1640c213de7020e0510e2" class="idref" href="#a8834fd6e3c1640c213de7020e0510e2"><span class="id" title="notation">&quot;</span></a>x &gt;= y" := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Gamma0.::g0_scope:x_'&gt;'_x" class="idref" href="#Gamma0.::g0_scope:x_'&gt;'_x"><span class="id" title="notation">&quot;</span></a>x &gt; y"  := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>)  : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Main result: Our comparison is a fix-point 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltE" class="idref" href="#Gamma0.T2ltE"><span class="id" title="lemma">T2ltE</span></a> <a id="x:889" class="idref" href="#x:889"><span class="id" title="binder">x</span></a> <a id="y:890" class="idref" href="#y:890"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#x:889"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:890"><span class="id" title="variable">y</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.lt_rec"><span class="id" title="definition">lt_rec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt"><span class="id" title="definition">T2lt</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:889"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:890"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+This is how we compare two psi-terms 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.lt_psi" class="idref" href="#Gamma0.lt_psi"><span class="id" title="definition">lt_psi</span></a>  <a id="a:894" class="idref" href="#a:894"><span class="id" title="binder">a</span></a> <a id="b:895" class="idref" href="#b:895"><span class="id" title="binder">b</span></a> <a id="a':896" class="idref" href="#a':896"><span class="id" title="binder">a'</span></a> <a id="b':897" class="idref" href="#b':897"><span class="id" title="binder">b'</span></a>:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':896"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:895"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':896"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':897"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':896"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:895"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':897"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':896"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:895"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':897"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':896"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:894"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:895"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':897"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_psi" class="idref" href="#Gamma0.T2lt_psi"><span class="id" title="lemma">T2lt_psi</span></a> <a id="a:898" class="idref" href="#a:898"><span class="id" title="binder">a</span></a> <a id="b:899" class="idref" href="#b:899"><span class="id" title="binder">b</span></a> <a id="a':900" class="idref" href="#a':900"><span class="id" title="binder">a'</span></a> <a id="b':901" class="idref" href="#b':901"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:898"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:899"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':900"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':901"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.lt_psi"><span class="id" title="definition">lt_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:898"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:899"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':900"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':901"><span class="id" title="variable">b'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_consE" class="idref" href="#Gamma0.T2lt_consE"><span class="id" title="lemma">T2lt_consE</span></a> <a id="a:902" class="idref" href="#a:902"><span class="id" title="binder">a</span></a> <a id="b:903" class="idref" href="#b:903"><span class="id" title="binder">b</span></a> <a id="n:904" class="idref" href="#n:904"><span class="id" title="binder">n</span></a> <a id="c:905" class="idref" href="#c:905"><span class="id" title="binder">c</span></a> <a id="a':906" class="idref" href="#a':906"><span class="id" title="binder">a'</span></a> <a id="b':907" class="idref" href="#b':907"><span class="id" title="binder">b'</span></a> <a id="n':908" class="idref" href="#n':908"><span class="id" title="binder">n'</span></a> <a id="c':909" class="idref" href="#c':909"><span class="id" title="binder">c'</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:902"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:903"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:904"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:905"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':906"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':907"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':908"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':909"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.lt_psi"><span class="id" title="definition">lt_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:902"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:903"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':906"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':907"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:902"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':906"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:903"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b':907"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:904"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':908"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:904"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':908"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:905"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':909"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Less-or-equal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_consE" class="idref" href="#Gamma0.T2le_consE"><span class="id" title="lemma">T2le_consE</span></a> <a id="a:910" class="idref" href="#a:910"><span class="id" title="binder">a</span></a> <a id="b:911" class="idref" href="#b:911"><span class="id" title="binder">b</span></a> <a id="n:912" class="idref" href="#n:912"><span class="id" title="binder">n</span></a> <a id="c:913" class="idref" href="#c:913"><span class="id" title="binder">c</span></a> <a id="a':914" class="idref" href="#a':914"><span class="id" title="binder">a'</span></a> <a id="b':915" class="idref" href="#b':915"><span class="id" title="binder">b'</span></a> <a id="n':916" class="idref" href="#n':916"><span class="id" title="binder">n'</span></a> <a id="c':917" class="idref" href="#c':917"><span class="id" title="binder">c'</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:910"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:911"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:912"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:913"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':914"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':915"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':916"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':917"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.lt_psi"><span class="id" title="definition">lt_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:910"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:911"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':914"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':915"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:910"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':914"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:911"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b':915"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:912"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':916"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:912"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':916"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:913"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':917"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltn0" class="idref" href="#Gamma0.T2ltn0"><span class="id" title="lemma">T2ltn0</span></a> <a id="x:918" class="idref" href="#x:918"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:918"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt0n" class="idref" href="#Gamma0.T2lt0n"><span class="id" title="lemma">T2lt0n</span></a> <a id="x:919" class="idref" href="#x:919"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:919"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:919"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le0n" class="idref" href="#Gamma0.T2le0n"><span class="id" title="lemma">T2le0n</span></a> <a id="x:920" class="idref" href="#x:920"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:920"><span class="id" title="variable">x</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2len0" class="idref" href="#Gamma0.T2len0"><span class="id" title="lemma">T2len0</span></a> <a id="x:921" class="idref" href="#x:921"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:921"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:921"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.omega_lt_epsilon0" class="idref" href="#Gamma0.omega_lt_epsilon0"><span class="id" title="lemma">omega_lt_epsilon0</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.epsilon0"><span class="id" title="definition">epsilon0</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltnn" class="idref" href="#Gamma0.T2ltnn"><span class="id" title="lemma">T2ltnn</span></a> <a id="x:922" class="idref" href="#x:922"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:922"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:922"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_ne" class="idref" href="#Gamma0.T2lt_ne"><span class="id" title="lemma">T2lt_ne</span></a> <a id="a:923" class="idref" href="#a:923"><span class="id" title="binder">a</span></a> <a id="b:924" class="idref" href="#b:924"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:923"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:924"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:923"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:924"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_ne'" class="idref" href="#Gamma0.T2lt_ne'"><span class="id" title="lemma">T2lt_ne'</span></a> <a id="a:925" class="idref" href="#a:925"><span class="id" title="binder">a</span></a> <a id="b:926" class="idref" href="#b:926"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:925"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:926"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:926"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:925"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltW" class="idref" href="#Gamma0.T2ltW"><span class="id" title="lemma">T2ltW</span></a> <a id="a:927" class="idref" href="#a:927"><span class="id" title="binder">a</span></a> <a id="b:928" class="idref" href="#b:928"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:927"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:928"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:927"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:928"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_eqVlt" class="idref" href="#Gamma0.T2le_eqVlt"><span class="id" title="lemma">T2le_eqVlt</span></a> <a id="a:929" class="idref" href="#a:929"><span class="id" title="binder">a</span></a> <a id="b:930" class="idref" href="#b:930"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:929"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:930"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:929"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:930"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:929"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:930"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_neAle" class="idref" href="#Gamma0.T2lt_neAle"><span class="id" title="lemma">T2lt_neAle</span></a> <a id="a:931" class="idref" href="#a:931"><span class="id" title="binder">a</span></a> <a id="b:932" class="idref" href="#b:932"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:931"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:932"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:931"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:932"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:931"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:932"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lenn" class="idref" href="#Gamma0.T2lenn"><span class="id" title="lemma">T2lenn</span></a> <a id="x:933" class="idref" href="#x:933"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:933"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:933"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T2lenn</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ge1" class="idref" href="#Gamma0.T2ge1"><span class="id" title="lemma">T2ge1</span></a> <a id="x:934" class="idref" href="#x:934"><span class="id" title="binder">x</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:934"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:934"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt1" class="idref" href="#Gamma0.T2lt1"><span class="id" title="lemma">T2lt1</span></a> <a id="x:935" class="idref" href="#x:935"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:935"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:935"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2nat_inc" class="idref" href="#Gamma0.T2nat_inc"><span class="id" title="lemma">T2nat_inc</span></a> <a id="n:936" class="idref" href="#n:936"><span class="id" title="binder">n</span></a> <a id="p:937" class="idref" href="#p:937"><span class="id" title="binder">p</span></a> : (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:936"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:937"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:936"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:937"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.psi_lt1" class="idref" href="#Gamma0.psi_lt1"><span class="id" title="lemma">psi_lt1</span></a> <a id="a:938" class="idref" href="#a:938"><span class="id" title="binder">a</span></a> <a id="b:939" class="idref" href="#b:939"><span class="id" title="binder">b</span></a> <a id="c:940" class="idref" href="#c:940"><span class="id" title="binder">c</span></a>  <a id="n:941" class="idref" href="#n:941"><span class="id" title="binder">n</span></a> <a id="a':942" class="idref" href="#a':942"><span class="id" title="binder">a'</span></a> <a id="b':943" class="idref" href="#b':943"><span class="id" title="binder">b'</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:938"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:939"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:941"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:940"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':942"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':943"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:938"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:939"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':942"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':943"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.psi_lt2" class="idref" href="#Gamma0.psi_lt2"><span class="id" title="lemma">psi_lt2</span></a> <a id="a:944" class="idref" href="#a:944"><span class="id" title="binder">a</span></a> <a id="b:945" class="idref" href="#b:945"><span class="id" title="binder">b</span></a> <a id="n:946" class="idref" href="#n:946"><span class="id" title="binder">n</span></a> <a id="c:947" class="idref" href="#c:947"><span class="id" title="binder">c</span></a> <a id="n':948" class="idref" href="#n':948"><span class="id" title="binder">n'</span></a> <a id="c':949" class="idref" href="#c':949"><span class="id" title="binder">c'</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:944"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:945"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':948"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':949"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:944"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:945"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:946"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:947"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n':948"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:946"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':948"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:946"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':949"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:947"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T1T2_inj" class="idref" href="#Gamma0.T1T2_inj"><span class="id" title="lemma">T1T2_inj</span></a> <a id="n:950" class="idref" href="#n:950"><span class="id" title="binder">n</span></a> <a id="p:951" class="idref" href="#p:951"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:950"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:951"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:950"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:951"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T1T2_inc" class="idref" href="#Gamma0.T1T2_inc"><span class="id" title="lemma">T1T2_inc</span></a> <a id="n:952" class="idref" href="#n:952"><span class="id" title="binder">n</span></a> <a id="p:953" class="idref" href="#p:953"><span class="id" title="binder">p</span></a> : (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:952"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:953"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">ca</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:952"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:953"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">g0</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+First two non-trivial results 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_anti" class="idref" href="#Gamma0.T2lt_anti"><span class="id" title="lemma">T2lt_anti</span></a> <a id="b:954" class="idref" href="#b:954"><span class="id" title="binder">b</span></a> <a id="a:955" class="idref" href="#a:955"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:955"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:954"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:954"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:955"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_trichotomy" class="idref" href="#Gamma0.T2lt_trichotomy"><span class="id" title="lemma">T2lt_trichotomy</span></a> <a id="a:958" class="idref" href="#a:958"><span class="id" title="binder">a</span></a> <a id="b:959" class="idref" href="#b:959"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:958"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:959"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:958"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:959"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:959"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:958"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+what follows is the same as for T1 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2leNgt" class="idref" href="#Gamma0.T2leNgt"><span class="id" title="lemma">T2leNgt</span></a> <a id="a:962" class="idref" href="#a:962"><span class="id" title="binder">a</span></a> <a id="b:963" class="idref" href="#b:963"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:962"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:963"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:963"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:962"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltNge" class="idref" href="#Gamma0.T2ltNge"><span class="id" title="lemma">T2ltNge</span></a> <a id="a:964" class="idref" href="#a:964"><span class="id" title="binder">a</span></a> <a id="b:965" class="idref" href="#b:965"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:964"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:965"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:965"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:964"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2eq_le" class="idref" href="#Gamma0.T2eq_le"><span class="id" title="lemma">T2eq_le</span></a> <a id="m:966" class="idref" href="#m:966"><span class="id" title="binder">m</span></a> <a id="n:967" class="idref" href="#n:967"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:966"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:967"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:966"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:967"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:967"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:966"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_total" class="idref" href="#Gamma0.T2le_total"><span class="id" title="lemma">T2le_total</span></a> <a id="m:968" class="idref" href="#m:968"><span class="id" title="binder">m</span></a> <a id="n:969" class="idref" href="#n:969"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:968"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:969"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:969"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:968"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Gamma0.T2ltn_xor_geq" class="idref" href="#Gamma0.T2ltn_xor_geq"><span class="id" title="inductive">T2ltn_xor_geq</span></a> <a id="m:970" class="idref" href="#m:970"><span class="id" title="binder">m</span></a> <a id="n:971" class="idref" href="#n:971"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Gamma0.T2LtnNotGeq" class="idref" href="#Gamma0.T2LtnNotGeq"><span class="id" title="constructor">T2LtnNotGeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:970"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:971"><span class="id" title="variable">n</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2ltn_xor_geq:972"><span class="id" title="inductive">T2ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:970"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:971"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0.T2GeqNotLtn" class="idref" href="#Gamma0.T2GeqNotLtn"><span class="id" title="constructor">T2GeqNotLtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:971"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:970"><span class="id" title="variable">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2ltn_xor_geq:972"><span class="id" title="inductive">T2ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:970"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:971"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Gamma0.T2leq_xor_gtn" class="idref" href="#Gamma0.T2leq_xor_gtn"><span class="id" title="inductive">T2leq_xor_gtn</span></a> <a id="m:974" class="idref" href="#m:974"><span class="id" title="binder">m</span></a> <a id="n:975" class="idref" href="#n:975"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Gamma0.T2GeqNotGtn" class="idref" href="#Gamma0.T2GeqNotGtn"><span class="id" title="constructor">T2GeqNotGtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:974"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:975"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2leq_xor_gtn:976"><span class="id" title="inductive">T2leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:974"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:975"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0.T2GtnNotLeq" class="idref" href="#Gamma0.T2GtnNotLeq"><span class="id" title="constructor">T2GtnNotLeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:975"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:974"><span class="id" title="variable">m</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T2leq_xor_gtn:976"><span class="id" title="inductive">T2leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:974"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:975"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Gamma0.compare_T2" class="idref" href="#Gamma0.compare_T2"><span class="id" title="inductive">compare_T2</span></a> <a id="m:978" class="idref" href="#m:978"><span class="id" title="binder">m</span></a> <a id="n:979" class="idref" href="#n:979"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Gamma0.CompareT2Lt" class="idref" href="#Gamma0.CompareT2Lt"><span class="id" title="constructor">CompareT2Lt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T2:980"><span class="id" title="inductive">compare_T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0.CompareT2Gt" class="idref" href="#Gamma0.CompareT2Gt"><span class="id" title="constructor">CompareT2Gt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T2:980"><span class="id" title="inductive">compare_T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0.CompareT2Eq" class="idref" href="#Gamma0.CompareT2Eq"><span class="id" title="constructor">CompareT2Eq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T2:980"><span class="id" title="inductive">compare_T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:978"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:979"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2leP" class="idref" href="#Gamma0.T2leP"><span class="id" title="lemma">T2leP</span></a> <a id="x:982" class="idref" href="#x:982"><span class="id" title="binder">x</span></a> <a id="y:983" class="idref" href="#y:983"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2leq_xor_gtn"><span class="id" title="inductive">T2leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:982"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:983"><span class="id" title="variable">y</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:982"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:983"><span class="id" title="variable">y</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#y:983"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:982"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltP" class="idref" href="#Gamma0.T2ltP"><span class="id" title="lemma">T2ltP</span></a> <a id="m:984" class="idref" href="#m:984"><span class="id" title="binder">m</span></a> <a id="n:985" class="idref" href="#n:985"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltn_xor_geq"><span class="id" title="inductive">T2ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:984"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:985"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:985"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:984"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:984"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:985"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2ltgtP" class="idref" href="#Gamma0.T2ltgtP"><span class="id" title="lemma">T2ltgtP</span></a> <a id="m:986" class="idref" href="#m:986"><span class="id" title="binder">m</span></a> <a id="n:987" class="idref" href="#n:987"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.compare_T2"><span class="id" title="inductive">compare_T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:986"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:987"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:986"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:987"><span class="id" title="variable">n</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:987"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:986"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:986"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:987"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_psi_aux" class="idref" href="#Gamma0.T2lt_psi_aux"><span class="id" title="lemma">T2lt_psi_aux</span></a> <a id="a:988" class="idref" href="#a:988"><span class="id" title="binder">a</span></a> <a id="b:989" class="idref" href="#b:989"><span class="id" title="binder">b</span></a> <a id="a':990" class="idref" href="#a':990"><span class="id" title="binder">a'</span></a> <a id="b':991" class="idref" href="#b':991"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:988"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':990"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:989"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':990"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':991"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:988"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:989"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':990"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':991"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2gt1" class="idref" href="#Gamma0.T2gt1"><span class="id" title="lemma">T2gt1</span></a> <a id="x:992" class="idref" href="#x:992"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:992"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:992"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:992"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Second non-trivial result 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Gamma0.T2lt_trans" class="idref" href="#Gamma0.T2lt_trans"><span class="id" title="lemma">T2lt_trans</span></a> <a id="b:993" class="idref" href="#b:993"><span class="id" title="binder">b</span></a> <a id="a:994" class="idref" href="#a:994"><span class="id" title="binder">a</span></a> <a id="c:995" class="idref" href="#c:995"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:994"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:993"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:993"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:995"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:994"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:995"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_le_trans" class="idref" href="#Gamma0.T2lt_le_trans"><span class="id" title="lemma">T2lt_le_trans</span></a> <a id="b:999" class="idref" href="#b:999"><span class="id" title="binder">b</span></a> <a id="a:1000" class="idref" href="#a:1000"><span class="id" title="binder">a</span></a> <a id="c:1001" class="idref" href="#c:1001"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1000"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:999"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:999"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1001"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1000"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1001"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_lt_trans" class="idref" href="#Gamma0.T2le_lt_trans"><span class="id" title="lemma">T2le_lt_trans</span></a> <a id="b:1002" class="idref" href="#b:1002"><span class="id" title="binder">b</span></a> <a id="a:1003" class="idref" href="#a:1003"><span class="id" title="binder">a</span></a> <a id="c:1004" class="idref" href="#c:1004"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1003"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1002"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1002"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1004"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1003"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1004"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_trans" class="idref" href="#Gamma0.T2le_trans"><span class="id" title="lemma">T2le_trans</span></a> <a id="b:1005" class="idref" href="#b:1005"><span class="id" title="binder">b</span></a> <a id="a:1006" class="idref" href="#a:1006"><span class="id" title="binder">a</span></a> <a id="c:1007" class="idref" href="#c:1007"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1005"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1005"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1007"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1007"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_psi1" class="idref" href="#Gamma0.T2le_psi1"><span class="id" title="lemma">T2le_psi1</span></a> <a id="a:1008" class="idref" href="#a:1008"><span class="id" title="binder">a</span></a> <a id="b:1009" class="idref" href="#b:1009"><span class="id" title="binder">b</span></a> <a id="n:1010" class="idref" href="#n:1010"><span class="id" title="binder">n</span></a> <a id="c:1011" class="idref" href="#c:1011"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1008"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1009"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1008"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1009"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1010"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1011"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_psi_b" class="idref" href="#Gamma0.T2lt_psi_b"><span class="id" title="lemma">T2lt_psi_b</span></a> <a id="a:1012" class="idref" href="#a:1012"><span class="id" title="binder">a</span></a> <a id="b:1013" class="idref" href="#b:1013"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1013"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1012"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1013"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_psi_a" class="idref" href="#Gamma0.T2lt_psi_a"><span class="id" title="lemma">T2lt_psi_a</span></a> <a id="a:1014" class="idref" href="#a:1014"><span class="id" title="binder">a</span></a> <a id="b:1015" class="idref" href="#b:1015"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1014"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1014"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1015"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab454"></a><h2 class="section">Normal form</h2>
+ Same as in T1.
+TODO:: show that compraison is well-founded for NF
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2nf" class="idref" href="#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a id="x:1016" class="idref" href="#x:1016"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1016"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2nf:1017"><span class="id" title="definition">T2nf</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2nf:1017"><span class="id" title="definition">T2nf</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2nf:1017"><span class="id" title="definition">T2nf</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <span class="id" title="var">c</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2nf_cons_cons" class="idref" href="#Gamma0.T2nf_cons_cons"><span class="id" title="lemma">T2nf_cons_cons</span></a> <a id="a:1019" class="idref" href="#a:1019"><span class="id" title="binder">a</span></a> <a id="b:1020" class="idref" href="#b:1020"><span class="id" title="binder">b</span></a> <a id="n:1021" class="idref" href="#n:1021"><span class="id" title="binder">n</span></a> <a id="a':1022" class="idref" href="#a':1022"><span class="id" title="binder">a'</span></a> <a id="b':1023" class="idref" href="#b':1023"><span class="id" title="binder">b'</span></a> <a id="n':1024" class="idref" href="#n':1024"><span class="id" title="binder">n'</span></a> <a id="c':1025" class="idref" href="#c':1025"><span class="id" title="binder">c'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a>(<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1019"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1020"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1021"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1022"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1023"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1024"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1025"><span class="id" title="variable">c'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1022"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1023"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1019"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1020"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1019"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1020"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a>(<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1022"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1023"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1024"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1025"><span class="id" title="variable">c'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_psi" class="idref" href="#Gamma0.nf_psi"><span class="id" title="lemma">nf_psi</span></a> <a id="a:1026" class="idref" href="#a:1026"><span class="id" title="binder">a</span></a> <a id="b:1027" class="idref" href="#b:1027"><span class="id" title="binder">b</span></a> <a id="n:1028" class="idref" href="#n:1028"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1026"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1027"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1028"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1026"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1027"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2nf_consE" class="idref" href="#Gamma0.T2nf_consE"><span class="id" title="lemma">T2nf_consE</span></a> <a id="a:1029" class="idref" href="#a:1029"><span class="id" title="binder">a</span></a> <a id="b:1030" class="idref" href="#b:1030"><span class="id" title="binder">b</span></a> <a id="n:1031" class="idref" href="#n:1031"><span class="id" title="binder">n</span></a> <a id="c:1032" class="idref" href="#c:1032"><span class="id" title="binder">c</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1029"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1030"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1031"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1032"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1029"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1030"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1032"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1032"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1029"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1030"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_omega" class="idref" href="#Gamma0.nf_omega"><span class="id" title="lemma">nf_omega</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_one" class="idref" href="#Gamma0.nf_one"><span class="id" title="lemma">nf_one</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_finite" class="idref" href="#Gamma0.nf_finite"><span class="id" title="lemma">nf_finite</span></a> <a id="n:1033" class="idref" href="#n:1033"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1033"><span class="id" title="variable">n</span></a>).  <br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab455"></a><h2 class="section">Successor and predecessor</h2>
+ Same as for T1 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_tail" class="idref" href="#Gamma0.lt_tail"><span class="id" title="lemma">lt_tail</span></a> <a id="a:1034" class="idref" href="#a:1034"><span class="id" title="binder">a</span></a> <a id="b:1035" class="idref" href="#b:1035"><span class="id" title="binder">b</span></a> <a id="n:1036" class="idref" href="#n:1036"><span class="id" title="binder">n</span></a> <a id="c:1037" class="idref" href="#c:1037"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1034"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1035"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1036"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1037"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1037"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1034"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1035"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1036"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1037"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T1T2range1" class="idref" href="#Gamma0.T1T2range1"><span class="id" title="lemma">T1T2range1</span></a> <a id="x:1038" class="idref" href="#x:1038"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1038"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T1T2range2" class="idref" href="#Gamma0.T1T2range2"><span class="id" title="lemma">T1T2range2</span></a> <a id="x:1039" class="idref" href="#x:1039"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1039"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1039"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:1040" class="idref" href="#y:1040"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1039"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2"><span class="id" title="definition">T1T2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1040"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2finite" class="idref" href="#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a id="x:1041" class="idref" href="#x:1041"><span class="id" title="binder">x</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1041"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2limit" class="idref" href="#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <a id="x:1043" class="idref" href="#x:1043"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1043"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">c</span><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2limit:1044"><span class="id" title="definition">T2limit</span></a> <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2split" class="idref" href="#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a id="x:1046" class="idref" href="#x:1046"><span class="id" title="binder">x</span></a>:=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1046"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#T2split:1047"><span class="id" title="definition">T2split</span></a> <span class="id" title="var">c</span> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1046"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2nf_finite" class="idref" href="#Gamma0.T2nf_finite"><span class="id" title="lemma">T2nf_finite</span></a> <a id="a:1049" class="idref" href="#a:1049"><span class="id" title="binder">a</span></a> <a id="b:1050" class="idref" href="#b:1050"><span class="id" title="binder">b</span></a> <a id="n:1051" class="idref" href="#n:1051"><span class="id" title="binder">n</span></a> <a id="c:1052" class="idref" href="#c:1052"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1049"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1050"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1049"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1050"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1051"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1052"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1052"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_finite" class="idref" href="#Gamma0.split_finite"><span class="id" title="lemma">split_finite</span></a> <a id="x:1053" class="idref" href="#x:1053"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1053"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1053"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2finite2" class="idref" href="#Gamma0.T2finite2"><span class="id" title="lemma">T2finite2</span></a> <a id="x:1054" class="idref" href="#x:1054"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1054"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1054"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1054"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1054"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.omega_least_inf1" class="idref" href="#Gamma0.omega_least_inf1"><span class="id" title="lemma">omega_least_inf1</span></a> <a id="x:1055" class="idref" href="#x:1055"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1055"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1055"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.omega_least_inf2" class="idref" href="#Gamma0.omega_least_inf2"><span class="id" title="lemma">omega_least_inf2</span></a> <a id="x:1056" class="idref" href="#x:1056"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1056"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1056"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_limit" class="idref" href="#Gamma0.split_limit"><span class="id" title="lemma">split_limit</span></a> <a id="x:1057" class="idref" href="#x:1057"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1057"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1057"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1057"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2is_succ" class="idref" href="#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> <a id="x:1058" class="idref" href="#x:1058"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1058"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T2is_succ:1059"><span class="id" title="definition">T2is_succ</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2succ" class="idref" href="#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a id="x:1061" class="idref" href="#x:1061"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1061"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+2</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T2succ:1062"><span class="id" title="definition">T2succ</span></a> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2pred" class="idref" href="#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a id="x:1064" class="idref" href="#x:1064"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1064"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T2pred:1065"><span class="id" title="definition">T2pred</span></a> <span class="id" title="var">c</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_is_succ" class="idref" href="#Gamma0.split_is_succ"><span class="id" title="lemma">split_is_succ</span></a> <a id="x:1067" class="idref" href="#x:1067"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1067"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1067"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_succ" class="idref" href="#Gamma0.split_succ"><span class="id" title="lemma">split_succ</span></a> <a id="x:1068" class="idref" href="#x:1068"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1068"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1068"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_pred" class="idref" href="#Gamma0.split_pred"><span class="id" title="lemma">split_pred</span></a> <a id="x:1069" class="idref" href="#x:1069"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1069"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1069"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_le" class="idref" href="#Gamma0.split_le"><span class="id" title="lemma">split_le</span></a> <a id="x:1070" class="idref" href="#x:1070"><span class="id" title="binder">x</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1070"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1070"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_split" class="idref" href="#Gamma0.nf_split"><span class="id" title="lemma">nf_split</span></a> <a id="x:1071" class="idref" href="#x:1071"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1071"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1071"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2finite_succ" class="idref" href="#Gamma0.T2finite_succ"><span class="id" title="lemma">T2finite_succ</span></a> <a id="x:1072" class="idref" href="#x:1072"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1072"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1072"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T1succ_nat" class="idref" href="#Gamma0.T1succ_nat"><span class="id" title="lemma">T1succ_nat</span></a> <a id="n:1073" class="idref" href="#n:1073"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1073"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1073"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.limit_pr1" class="idref" href="#Gamma0.limit_pr1"><span class="id" title="lemma">limit_pr1</span></a> <a id="x:1074" class="idref" href="#x:1074"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1074"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1074"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1074"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.limit_pr" class="idref" href="#Gamma0.limit_pr"><span class="id" title="lemma">limit_pr</span></a> <a id="x:1075" class="idref" href="#x:1075"><span class="id" title="binder">x</span></a> <a id="y:1076" class="idref" href="#y:1076"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1075"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1076"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1075"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1076"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1075"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_psi_b" class="idref" href="#Gamma0.T2le_psi_b"><span class="id" title="lemma">T2le_psi_b</span></a> <a id="a:1077" class="idref" href="#a:1077"><span class="id" title="binder">a</span></a> <a id="b:1078" class="idref" href="#b:1078"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1078"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1077"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1078"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.pred_le" class="idref" href="#Gamma0.pred_le"><span class="id" title="lemma">pred_le</span></a> <a id="a:1079" class="idref" href="#a:1079"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1079"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1079"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.pred_lt" class="idref" href="#Gamma0.pred_lt"><span class="id" title="lemma">pred_lt</span></a> <a id="a:1080" class="idref" href="#a:1080"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1080"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1080"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1080"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_lt" class="idref" href="#Gamma0.succ_lt"><span class="id" title="lemma">succ_lt</span></a> <a id="a:1081" class="idref" href="#a:1081"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1081"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1081"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_succ" class="idref" href="#Gamma0.nf_succ"><span class="id" title="lemma">nf_succ</span></a> <a id="a:1082" class="idref" href="#a:1082"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1082"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1082"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_pred" class="idref" href="#Gamma0.nf_pred"><span class="id" title="lemma">nf_pred</span></a> <a id="a:1083" class="idref" href="#a:1083"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1083"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1083"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_pred" class="idref" href="#Gamma0.succ_pred"><span class="id" title="lemma">succ_pred</span></a> <a id="x:1084" class="idref" href="#x:1084"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1084"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1084"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1084"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1084"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_p1" class="idref" href="#Gamma0.succ_p1"><span class="id" title="lemma">succ_p1</span></a> <a id="x:1085" class="idref" href="#x:1085"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ"><span class="id" title="definition">T2is_succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1085"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.pred_succ" class="idref" href="#Gamma0.pred_succ"><span class="id" title="lemma">pred_succ</span></a> <a id="x:1086" class="idref" href="#x:1086"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1086"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1086"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1086"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_inj" class="idref" href="#Gamma0.succ_inj"><span class="id" title="lemma">succ_inj</span></a> <a id="x:1087" class="idref" href="#x:1087"><span class="id" title="binder">x</span></a> <a id="y:1088" class="idref" href="#y:1088"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1087"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1088"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1087"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1088"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1087"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1088"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_succ" class="idref" href="#Gamma0.lt_succ_succ"><span class="id" title="lemma">lt_succ_succ</span></a> <a id="x:1089" class="idref" href="#x:1089"><span class="id" title="binder">x</span></a> <a id="y:1090" class="idref" href="#y:1090"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1089"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1090"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1089"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1090"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.le_succ_succ" class="idref" href="#Gamma0.le_succ_succ"><span class="id" title="lemma">le_succ_succ</span></a> <a id="x:1091" class="idref" href="#x:1091"><span class="id" title="binder">x</span></a> <a id="y:1092" class="idref" href="#y:1092"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1091"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1092"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1091"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1092"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_succE" class="idref" href="#Gamma0.lt_succ_succE"><span class="id" title="lemma">lt_succ_succE</span></a> <a id="x:1093" class="idref" href="#x:1093"><span class="id" title="binder">x</span></a> <a id="y:1094" class="idref" href="#y:1094"><span class="id" title="binder">y</span></a>:  <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1093"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1094"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1093"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1094"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1093"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1094"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.le_succ_succE" class="idref" href="#Gamma0.le_succ_succE"><span class="id" title="lemma">le_succ_succE</span></a> <a id="x:1095" class="idref" href="#x:1095"><span class="id" title="binder">x</span></a> <a id="y:1096" class="idref" href="#y:1096"><span class="id" title="binder">y</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1095"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1096"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1095"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1096"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1095"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1096"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_le_1" class="idref" href="#Gamma0.lt_succ_le_1"><span class="id" title="lemma">lt_succ_le_1</span></a> <a id="a:1097" class="idref" href="#a:1097"><span class="id" title="binder">a</span></a> <a id="b:1098" class="idref" href="#b:1098"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1097"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1098"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1097"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1098"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_le_2" class="idref" href="#Gamma0.lt_succ_le_2"><span class="id" title="lemma">lt_succ_le_2</span></a> <a id="a:1099" class="idref" href="#a:1099"><span class="id" title="binder">a</span></a> <a id="b:1100" class="idref" href="#b:1100"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1099"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1099"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1100"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1099"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1100"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_le_3" class="idref" href="#Gamma0.lt_succ_le_3"><span class="id" title="lemma">lt_succ_le_3</span></a> <a id="a:1101" class="idref" href="#a:1101"><span class="id" title="binder">a</span></a> <a id="b:1102" class="idref" href="#b:1102"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1101"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1101"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1102"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1101"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1102"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.lt_succ_le_4" class="idref" href="#Gamma0.lt_succ_le_4"><span class="id" title="lemma">lt_succ_le_4</span></a> <a id="a:1103" class="idref" href="#a:1103"><span class="id" title="binder">a</span></a> <a id="b:1104" class="idref" href="#b:1104"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1104"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1103"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1104"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1103"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1104"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_nz" class="idref" href="#Gamma0.succ_nz"><span class="id" title="lemma">succ_nz</span></a> <a id="x:1105" class="idref" href="#x:1105"><span class="id" title="binder">x</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1105"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_psi" class="idref" href="#Gamma0.succ_psi"><span class="id" title="lemma">succ_psi</span></a> <a id="a:1106" class="idref" href="#a:1106"><span class="id" title="binder">a</span></a> <a id="b:1107" class="idref" href="#b:1107"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1106"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1107"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1106"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1107"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1106"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1107"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_psi_lt" class="idref" href="#Gamma0.succ_psi_lt"><span class="id" title="lemma">succ_psi_lt</span></a> <a id="x:1108" class="idref" href="#x:1108"><span class="id" title="binder">x</span></a> <a id="a:1109" class="idref" href="#a:1109"><span class="id" title="binder">a</span></a> <a id="b:1110" class="idref" href="#b:1110"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1109"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1110"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1108"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1109"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1110"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1108"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1109"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1110"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_psi_lt2" class="idref" href="#Gamma0.succ_psi_lt2"><span class="id" title="lemma">succ_psi_lt2</span></a> <a id="a:1111" class="idref" href="#a:1111"><span class="id" title="binder">a</span></a> <a id="b:1112" class="idref" href="#b:1112"><span class="id" title="binder">b</span></a> <a id="x:1113" class="idref" href="#x:1113"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1111"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1112"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1111"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1112"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1113"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1111"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1112"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1113"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab456"></a><h2 class="section">Addition</h2>
+ same as for T1 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2add" class="idref" href="#Gamma0.T2add"><span class="id" title="definition">T2add</span></a> <a id="x:1114" class="idref" href="#x:1114"><span class="id" title="binder">x</span></a> <a id="y:1115" class="idref" href="#y:1115"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1114"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1115"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1115"><span class="id" title="variable">y</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<span class="id" title="var">c</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1115"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">+</span><span class="id" title="var">n'</span><span class="id" title="notation">).+1</span> <span class="id" title="var">c'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1114"><span class="id" title="variable">x</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1115"><span class="id" title="variable">y</span></a><br/>
+&nbsp;<span class="id" title="keyword">where</span> <a id="513cd892ae2a37a8e1be319c411cb220" class="idref" href="#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">&quot;</span></a>x + y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T2add:1116"><span class="id" title="definition">T2add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Gamma0.T2sub" class="idref" href="#Gamma0.T2sub"><span class="id" title="definition">T2sub</span></a> <a id="x:1119" class="idref" href="#x:1119"><span class="id" title="binder">x</span></a> <a id="y:1120" class="idref" href="#y:1120"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1119"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1120"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1119"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1120"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1119"><span class="id" title="variable">x</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span><span class="id" title="notation">&lt;</span><span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span><span class="id" title="notation">==</span><span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> (<span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">-</span><span class="id" title="var">n'</span><span class="id" title="notation">).-1</span>) <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">c</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">c'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span> <span class="id" title="notation">-</span> <span class="id" title="var">n'</span><span class="id" title="notation">).-1</span> <span class="id" title="var">c</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1119"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><br/>
+<span class="id" title="keyword">where</span> <a id="Gamma0.::g0_scope:x_'-'_x" class="idref" href="#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">&quot;</span></a>a - b" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T2sub:1121"><span class="id" title="definition">T2sub</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2subn0" class="idref" href="#Gamma0.T2subn0"><span class="id" title="lemma">T2subn0</span></a> <a id="x:1124" class="idref" href="#x:1124"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1124"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1124"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2sub0n" class="idref" href="#Gamma0.T2sub0n"><span class="id" title="lemma">T2sub0n</span></a> <a id="x:1125" class="idref" href="#x:1125"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1125"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.minus_lt" class="idref" href="#Gamma0.minus_lt"><span class="id" title="lemma">minus_lt</span></a> <a id="a:1126" class="idref" href="#a:1126"><span class="id" title="binder">a</span></a> <a id="b:1127" class="idref" href="#b:1127"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1126"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1127"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1126"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1127"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2subnn" class="idref" href="#Gamma0.T2subnn"><span class="id" title="lemma">T2subnn</span></a> <a id="x:1128" class="idref" href="#x:1128"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1128"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1128"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.minus_le" class="idref" href="#Gamma0.minus_le"><span class="id" title="lemma">minus_le</span></a> <a id="a:1129" class="idref" href="#a:1129"><span class="id" title="binder">a</span></a> <a id="b:1130" class="idref" href="#b:1130"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1129"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1130"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1129"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1130"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_sub" class="idref" href="#Gamma0.nf_sub"><span class="id" title="lemma">nf_sub</span></a> <a id="a:1131" class="idref" href="#a:1131"><span class="id" title="binder">a</span></a> <a id="b:1132" class="idref" href="#b:1132"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1131"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1132"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1131"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1132"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_int" class="idref" href="#Gamma0.sub_int"><span class="id" title="lemma">sub_int</span></a> <a id="n:1133" class="idref" href="#n:1133"><span class="id" title="binder">n</span></a> <a id="m:1134" class="idref" href="#m:1134"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1133"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1134"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1133"><span class="id" title="variable">n</span></a> <span class="id" title="notation">-</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1134"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.succ_is_add_one" class="idref" href="#Gamma0.succ_is_add_one"><span class="id" title="lemma">succ_is_add_one</span></a> <a id="a:1135" class="idref" href="#a:1135"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1135"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1135"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add1Nfin" class="idref" href="#Gamma0.add1Nfin"><span class="id" title="lemma">add1Nfin</span></a> <a id="a:1136" class="idref" href="#a:1136"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1136"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1136"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1136"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub1Nfin" class="idref" href="#Gamma0.sub1Nfin"><span class="id" title="lemma">sub1Nfin</span></a> <a id="a:1137" class="idref" href="#a:1137"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite"><span class="id" title="definition">T2finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1137"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1137"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1137"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub1a" class="idref" href="#Gamma0.sub1a"><span class="id" title="lemma">sub1a</span></a> <a id="x:1138" class="idref" href="#x:1138"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1138"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1138"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1138"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1138"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub1b" class="idref" href="#Gamma0.sub1b"><span class="id" title="lemma">sub1b</span></a> <a id="x:1139" class="idref" href="#x:1139"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1139"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1139"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1139"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2add0n" class="idref" href="#Gamma0.T2add0n"><span class="id" title="lemma">T2add0n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2add"><span class="id" title="definition">T2add</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2addn0" class="idref" href="#Gamma0.T2addn0"><span class="id" title="lemma">T2addn0</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2add"><span class="id" title="definition">T2add</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_int" class="idref" href="#Gamma0.add_int"><span class="id" title="lemma">add_int</span></a> <a id="n:1140" class="idref" href="#n:1140"><span class="id" title="binder">n</span></a> <a id="m:1141" class="idref" href="#m:1141"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1140"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1141"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1140"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1141"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_fin_omega" class="idref" href="#Gamma0.add_fin_omega"><span class="id" title="lemma">add_fin_omega</span></a> <a id="n:1142" class="idref" href="#n:1142"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1142"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.split_add" class="idref" href="#Gamma0.split_add"><span class="id" title="lemma">split_add</span></a> <a id="x:1143" class="idref" href="#x:1143"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> :=<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1143"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1143"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1143"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_to_cons" class="idref" href="#Gamma0.add_to_cons"><span class="id" title="lemma">add_to_cons</span></a> <a id="a:1144" class="idref" href="#a:1144"><span class="id" title="binder">a</span></a> <a id="b:1145" class="idref" href="#b:1145"><span class="id" title="binder">b</span></a> <a id="n:1146" class="idref" href="#n:1146"><span class="id" title="binder">n</span></a> <a id="c:1147" class="idref" href="#c:1147"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#c:1147"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1144"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1145"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1144"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1145"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1146"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1147"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1144"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1145"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1146"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1147"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_add" class="idref" href="#Gamma0.nf_add"><span class="id" title="lemma">nf_add</span></a> <a id="a:1148" class="idref" href="#a:1148"><span class="id" title="binder">a</span></a> <a id="b:1149" class="idref" href="#b:1149"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1148"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1149"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1148"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1149"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2add_eq0" class="idref" href="#Gamma0.T2add_eq0"><span class="id" title="lemma">T2add_eq0</span></a> <a id="m:1156" class="idref" href="#m:1156"><span class="id" title="binder">m</span></a> <a id="n:1157" class="idref" href="#n:1157"><span class="id" title="binder">n</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1156"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1157"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1156"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1157"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_le1" class="idref" href="#Gamma0.add_le1"><span class="id" title="lemma">add_le1</span></a> <a id="a:1158" class="idref" href="#a:1158"><span class="id" title="binder">a</span></a> <a id="b:1159" class="idref" href="#b:1159"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1158"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1158"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1159"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_le2" class="idref" href="#Gamma0.add_le2"><span class="id" title="lemma">add_le2</span></a> <a id="a:1160" class="idref" href="#a:1160"><span class="id" title="binder">a</span></a> <a id="b:1161" class="idref" href="#b:1161"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1161"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1160"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1161"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_le1" class="idref" href="#Gamma0.sub_le1"><span class="id" title="lemma">sub_le1</span></a> <a id="a:1162" class="idref" href="#a:1162"><span class="id" title="binder">a</span></a> <a id="b:1163" class="idref" href="#b:1163"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1162"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1162"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1163"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1162"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_pr" class="idref" href="#Gamma0.sub_pr"><span class="id" title="lemma">sub_pr</span></a> <a id="a:1164" class="idref" href="#a:1164"><span class="id" title="binder">a</span></a> <a id="b:1165" class="idref" href="#b:1165"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1165"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1164"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1165"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1164"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1165"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_inj" class="idref" href="#Gamma0.add_inj"><span class="id" title="lemma">add_inj</span></a> <a id="a:1166" class="idref" href="#a:1166"><span class="id" title="binder">a</span></a> <a id="b:1167" class="idref" href="#b:1167"><span class="id" title="binder">b</span></a> <a id="c:1168" class="idref" href="#c:1168"><span class="id" title="binder">c</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1167"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1168"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1166"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1167"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1166"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1168"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1167"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1168"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_pr1" class="idref" href="#Gamma0.sub_pr1"><span class="id" title="lemma">sub_pr1</span></a> <a id="a:1169" class="idref" href="#a:1169"><span class="id" title="binder">a</span></a> <a id="b:1170" class="idref" href="#b:1170"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1170"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1169"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1170"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1170"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1169"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1170"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1169"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.omega_minus_one" class="idref" href="#Gamma0.omega_minus_one"><span class="id" title="lemma">omega_minus_one</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.omega"><span class="id" title="definition">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_nz" class="idref" href="#Gamma0.sub_nz"><span class="id" title="lemma">sub_nz</span></a> <a id="a:1171" class="idref" href="#a:1171"><span class="id" title="binder">a</span></a> <a id="b:1172" class="idref" href="#b:1172"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1172"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1171"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1172"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1172"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1171"><span class="id" title="variable">a</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2addA" class="idref" href="#Gamma0.T2addA"><span class="id" title="lemma">T2addA</span></a> <a id="c1:1173" class="idref" href="#c1:1173"><span class="id" title="binder">c1</span></a> <a id="c2:1174" class="idref" href="#c2:1174"><span class="id" title="binder">c2</span></a> <a id="c3:1175" class="idref" href="#c3:1175"><span class="id" title="binder">c3</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c1:1173"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c2:1174"><span class="id" title="variable">c2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:1175"><span class="id" title="variable">c3</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c1:1173"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c2:1174"><span class="id" title="variable">c2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:1175"><span class="id" title="variable">c3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_add2l" class="idref" href="#Gamma0.T2le_add2l"><span class="id" title="lemma">T2le_add2l</span></a>  <a id="p:1176" class="idref" href="#p:1176"><span class="id" title="binder">p</span></a> <a id="m:1177" class="idref" href="#m:1177"><span class="id" title="binder">m</span></a> <a id="n:1178" class="idref" href="#n:1178"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1176"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1177"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1176"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1178"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1177"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1178"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_add2l" class="idref" href="#Gamma0.T2lt_add2l"><span class="id" title="lemma">T2lt_add2l</span></a>  <a id="p:1179" class="idref" href="#p:1179"><span class="id" title="binder">p</span></a> <a id="m:1180" class="idref" href="#m:1180"><span class="id" title="binder">m</span></a> <a id="n:1181" class="idref" href="#n:1181"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1179"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1180"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1179"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1181"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1180"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1181"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2lt_add2r" class="idref" href="#Gamma0.T2lt_add2r"><span class="id" title="lemma">T2lt_add2r</span></a>  <a id="p:1182" class="idref" href="#p:1182"><span class="id" title="binder">p</span></a> <a id="m:1183" class="idref" href="#m:1183"><span class="id" title="binder">m</span></a> <a id="n:1184" class="idref" href="#n:1184"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1183"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1182"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1184"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1182"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1183"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1184"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2le_add2r" class="idref" href="#Gamma0.T2le_add2r"><span class="id" title="lemma">T2le_add2r</span></a>  <a id="p:1185" class="idref" href="#p:1185"><span class="id" title="binder">p</span></a> <a id="m:1186" class="idref" href="#m:1186"><span class="id" title="binder">m</span></a> <a id="n:1187" class="idref" href="#n:1187"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1186"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1187"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1186"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1185"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1187"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1185"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.T2eq_add2l" class="idref" href="#Gamma0.T2eq_add2l"><span class="id" title="lemma">T2eq_add2l</span></a>  <a id="p:1188" class="idref" href="#p:1188"><span class="id" title="binder">p</span></a> <a id="m:1189" class="idref" href="#m:1189"><span class="id" title="binder">m</span></a> <a id="n:1190" class="idref" href="#n:1190"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1188"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1189"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1188"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1190"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1189"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1190"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_le3" class="idref" href="#Gamma0.add_le3"><span class="id" title="lemma">add_le3</span></a> <a id="a:1191" class="idref" href="#a:1191"><span class="id" title="binder">a</span></a> <a id="b:1192" class="idref" href="#b:1192"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1191"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1191"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1192"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1192"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.add_le4" class="idref" href="#Gamma0.add_le4"><span class="id" title="lemma">add_le4</span></a> <a id="a:1193" class="idref" href="#a:1193"><span class="id" title="binder">a</span></a> <a id="b:1194" class="idref" href="#b:1194"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1194"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1193"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1193"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1194"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.sub_pr1r" class="idref" href="#Gamma0.sub_pr1r"><span class="id" title="lemma">sub_pr1r</span></a> <a id="a:1195" class="idref" href="#a:1195"><span class="id" title="binder">a</span></a> <a id="b:1196" class="idref" href="#b:1196"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1195"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1195"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1196"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1195"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1196"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2ap" class="idref" href="#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a id="x:1197" class="idref" href="#x:1197"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1197"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">c</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_pr0" class="idref" href="#Gamma0.ap_pr0"><span class="id" title="lemma">ap_pr0</span></a> <a id="a:1199" class="idref" href="#a:1199"><span class="id" title="binder">a</span></a> <a id="b:1200" class="idref" href="#b:1200"><span class="id" title="binder">b</span></a> (<a id="x:1201" class="idref" href="#x:1201"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1199"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1200"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>) <a id="u:1202" class="idref" href="#u:1202"><span class="id" title="binder">u</span></a> <a id="v:1203" class="idref" href="#v:1203"><span class="id" title="binder">v</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#u:1202"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1201"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1203"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1201"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1202"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1203"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1201"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_limit" class="idref" href="#Gamma0.ap_limit"><span class="id" title="lemma">ap_limit</span></a> <a id="x:1206" class="idref" href="#x:1206"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1206"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1206"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit"><span class="id" title="definition">T2limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1206"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_pr1" class="idref" href="#Gamma0.ap_pr1"><span class="id" title="lemma">ap_pr1</span></a> <a id="c:1207" class="idref" href="#c:1207"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:1208" class="idref" href="#a:1208"><span class="id" title="binder">a</span></a> <a id="b:1209" class="idref" href="#b:1209"><span class="id" title="binder">b</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1208"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1207"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1209"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1207"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1208"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1209"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1207"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1207"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1207"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_pr2" class="idref" href="#Gamma0.ap_pr2"><span class="id" title="lemma">ap_pr2</span></a> <a id="c:1210" class="idref" href="#c:1210"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:1211" class="idref" href="#a:1211"><span class="id" title="binder">a</span></a> <a id="b:1212" class="idref" href="#b:1212"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1211"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1212"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1211"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1212"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1211"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1212"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1210"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_pr3" class="idref" href="#Gamma0.ap_pr3"><span class="id" title="lemma">ap_pr3</span></a> <a id="a:1214" class="idref" href="#a:1214"><span class="id" title="binder">a</span></a> <a id="b:1215" class="idref" href="#b:1215"><span class="id" title="binder">b</span></a> <a id="y:1216" class="idref" href="#y:1216"><span class="id" title="binder">y</span></a> (<a id="x:1217" class="idref" href="#x:1217"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1214"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1215"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1216"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1217"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1216"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1217"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1217"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.ap_pr4" class="idref" href="#Gamma0.ap_pr4"><span class="id" title="lemma">ap_pr4</span></a> <a id="x:1218" class="idref" href="#x:1218"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:1219" class="idref" href="#b:1219"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1219"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1218"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1219"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1218"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1218"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1218"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1218"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab457"></a><h2 class="section">The function phi</h2>
+ We consider he some funciton phi 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2_pr1" class="idref" href="#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a id="x:1220" class="idref" href="#x:1220"><span class="id" title="binder">x</span></a>:= <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1220"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <span class="id" title="var">a</span> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2_pr2" class="idref" href="#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> <a id="x:1222" class="idref" href="#x:1222"><span class="id" title="binder">x</span></a>:= <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1222"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <span class="id" title="var">b</span> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.T2finite1" class="idref" href="#Gamma0.T2finite1"><span class="id" title="definition">T2finite1</span></a> <a id="x:1224" class="idref" href="#x:1224"><span class="id" title="binder">x</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1224"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <span class="id" title="var">a</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <span class="id" title="var">c</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.phi" class="idref" href="#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a id="a:1226" class="idref" href="#a:1226"><span class="id" title="binder">a</span></a> <a id="b:1227" class="idref" href="#b:1227"><span class="id" title="binder">b</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1227"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">n</span> <span class="id" title="var">k</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">k</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1227"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1227"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite1"><span class="id" title="definition">T2finite1</span></a> <span class="id" title="var">k</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span> 0 (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <span class="id" title="var">k</span>) <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1227"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1226"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1227"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_ap" class="idref" href="#Gamma0.phi_ap"><span class="id" title="lemma">phi_ap</span></a> <a id="x:1229" class="idref" href="#x:1229"><span class="id" title="binder">x</span></a> <a id="y:1230" class="idref" href="#y:1230"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1229"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1230"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1229"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1230"><span class="id" title="variable">y</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1229"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1230"><span class="id" title="variable">y</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_le1" class="idref" href="#Gamma0.phi_le1"><span class="id" title="lemma">phi_le1</span></a> <a id="a:1231" class="idref" href="#a:1231"><span class="id" title="binder">a</span></a> <a id="b:1232" class="idref" href="#b:1232"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1231"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1231"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1232"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_le2" class="idref" href="#Gamma0.phi_le2"><span class="id" title="lemma">phi_le2</span></a> <a id="a:1233" class="idref" href="#a:1233"><span class="id" title="binder">a</span></a> <a id="b:1234" class="idref" href="#b:1234"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1233"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1234"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1234"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_le3" class="idref" href="#Gamma0.phi_le3"><span class="id" title="lemma">phi_le3</span></a> <a id="a:1235" class="idref" href="#a:1235"><span class="id" title="binder">a</span></a> <a id="b:1236" class="idref" href="#b:1236"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1235"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1235"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1236"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1235"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1236"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1236"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_fix1" class="idref" href="#Gamma0.phi_fix1"><span class="id" title="lemma">phi_fix1</span></a> <a id="a:1237" class="idref" href="#a:1237"><span class="id" title="binder">a</span></a> <a id="u:1238" class="idref" href="#u:1238"><span class="id" title="binder">u</span></a> <a id="v:1239" class="idref" href="#v:1239"><span class="id" title="binder">v</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1237"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1238"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1237"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:1238"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:1239"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:1238"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1239"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_fix2" class="idref" href="#Gamma0.phi_fix2"><span class="id" title="lemma">phi_fix2</span></a> <a id="a:1240" class="idref" href="#a:1240"><span class="id" title="binder">a</span></a> <a id="b:1241" class="idref" href="#b:1241"><span class="id" title="binder">b</span></a> (<a id="u:1242" class="idref" href="#u:1242"><span class="id" title="binder">u</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1241"><span class="id" title="variable">b</span></a>) (<a id="v:1243" class="idref" href="#v:1243"><span class="id" title="binder">v</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1241"><span class="id" title="variable">b</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1240"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1241"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1241"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1241"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:1242"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:1243"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1240"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1242"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_succ" class="idref" href="#Gamma0.phi_succ"><span class="id" title="lemma">phi_succ</span></a> <a id="a:1244" class="idref" href="#a:1244"><span class="id" title="binder">a</span></a> <a id="u:1245" class="idref" href="#u:1245"><span class="id" title="binder">u</span></a> <a id="v:1246" class="idref" href="#v:1246"><span class="id" title="binder">v</span></a> <a id="n:1247" class="idref" href="#n:1247"><span class="id" title="binder">n</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1244"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1245"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1244"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1245"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1246"><span class="id" title="variable">v</span></a> 0 (<a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1247"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1244"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1245"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1246"><span class="id" title="variable">v</span></a> 0 (<a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1247"><span class="id" title="variable">n</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">phi</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> is either <span class="inlinecode"><span class="id" title="var">b</span></span>, <span class="inlinecode"><span class="id" title="var">psi</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> or <span class="inlinecode"><span class="id" title="var">psi</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">(<span class="id" title="var">b</span>-1)</span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_cases" class="idref" href="#Gamma0.phi_cases"><span class="id" title="lemma">phi_cases</span></a> <a id="a:1248" class="idref" href="#a:1248"><span class="id" title="binder">a</span></a> <a id="b:1249" class="idref" href="#b:1249"><span class="id" title="binder">b</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1248"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1248"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1248"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1248"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1248"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred"><span class="id" title="definition">T2pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1249"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.nf_phi" class="idref" href="#Gamma0.nf_phi"><span class="id" title="lemma">nf_phi</span></a> <a id="x:1250" class="idref" href="#x:1250"><span class="id" title="binder">x</span></a> <a id="y:1251" class="idref" href="#y:1251"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1250"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1251"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1250"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1251"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_principalR" class="idref" href="#Gamma0.phi_principalR"><span class="id" title="lemma">phi_principalR</span></a> <a id="a:1252" class="idref" href="#a:1252"><span class="id" title="binder">a</span></a> <a id="b:1253" class="idref" href="#b:1253"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="c:1254" class="idref" href="#c:1254"><span class="id" title="binder">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1252"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1253"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1254"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Gamma0.phi_spec1" class="idref" href="#Gamma0.phi_spec1"><span class="id" title="lemma">phi_spec1</span></a> <a id="a:1255" class="idref" href="#a:1255"><span class="id" title="binder">a</span></a> <a id="b:1256" class="idref" href="#b:1256"><span class="id" title="binder">b</span></a> <a id="c:1257" class="idref" href="#c:1257"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1257"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1255"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1257"><span class="id" title="variable">c</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1255"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1256"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1255"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1256"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_spec2" class="idref" href="#Gamma0.phi_spec2"><span class="id" title="lemma">phi_spec2</span></a> <a id="a:1258" class="idref" href="#a:1258"><span class="id" title="binder">a</span></a> <a id="x:1259" class="idref" href="#x:1259"><span class="id" title="binder">x</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1258"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1259"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="c:1260" class="idref" href="#c:1260"><span class="id" title="binder">c</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1260"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1260"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1258"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1260"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1259"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1259"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1258"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1259"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_spec3" class="idref" href="#Gamma0.phi_spec3"><span class="id" title="lemma">phi_spec3</span></a> <a id="a:1261" class="idref" href="#a:1261"><span class="id" title="binder">a</span></a> <a id="x:1262" class="idref" href="#x:1262"><span class="id" title="binder">x</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1261"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1262"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="c:1263" class="idref" href="#c:1263"><span class="id" title="binder">c</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1263"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1263"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1261"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1263"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1262"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1262"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1261"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b:1264" class="idref" href="#b:1264"><span class="id" title="binder">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1262"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1261"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1264"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_spec4a" class="idref" href="#Gamma0.phi_spec4a"><span class="id" title="lemma">phi_spec4a</span></a> <a id="u:1265" class="idref" href="#u:1265"><span class="id" title="binder">u</span></a> <a id="v:1266" class="idref" href="#v:1266"><span class="id" title="binder">v</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1265"><span class="id" title="variable">u</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:1265"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:1266"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:1265"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1266"><span class="id" title="variable">v</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_spec4b" class="idref" href="#Gamma0.phi_spec4b"><span class="id" title="lemma">phi_spec4b</span></a> <a id="x:1267" class="idref" href="#x:1267"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1267"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_spec4c" class="idref" href="#Gamma0.phi_spec4c"><span class="id" title="lemma">phi_spec4c</span></a> <a id="x:1268" class="idref" href="#x:1268"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1268"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1268"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1268"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="b:1269" class="idref" href="#b:1269"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1268"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1269"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.no_critical" class="idref" href="#Gamma0.no_critical"><span class="id" title="lemma">no_critical</span></a> <a id="a:1270" class="idref" href="#a:1270"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1270"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1270"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_ab_le1" class="idref" href="#Gamma0.phi_ab_le1"><span class="id" title="lemma">phi_ab_le1</span></a> <a id="a:1271" class="idref" href="#a:1271"><span class="id" title="binder">a</span></a> <a id="b:1272" class="idref" href="#b:1272"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1272"><span class="id" title="variable">b</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1271"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1272"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_ab_le2" class="idref" href="#Gamma0.phi_ab_le2"><span class="id" title="lemma">phi_ab_le2</span></a>  <a id="a:1273" class="idref" href="#a:1273"><span class="id" title="binder">a</span></a> <a id="b:1274" class="idref" href="#b:1274"><span class="id" title="binder">b</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1273"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1273"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1274"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inv1" class="idref" href="#Gamma0.phi_inv1"><span class="id" title="lemma">phi_inv1</span></a> <a id="a:1275" class="idref" href="#a:1275"><span class="id" title="binder">a</span></a> <a id="b:1276" class="idref" href="#b:1276"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1275"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1275"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="n:1277" class="idref" href="#n:1277"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2"><span class="id" title="definition">T2_pr2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a>) 0 (<a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1277"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1275"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1"><span class="id" title="definition">T2_pr1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1276"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Monotonicity is non-trivial 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_mono_a" class="idref" href="#Gamma0.phi_mono_a"><span class="id" title="lemma">phi_mono_a</span></a> <a id="a:1278" class="idref" href="#a:1278"><span class="id" title="binder">a</span></a> <a id="b:1279" class="idref" href="#b:1279"><span class="id" title="binder">b</span></a> <a id="b':1280" class="idref" href="#b':1280"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1279"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1279"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1280"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1278"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1279"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1278"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1280"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_mono_b" class="idref" href="#Gamma0.phi_mono_b"><span class="id" title="lemma">phi_mono_b</span></a> <a id="a:1283" class="idref" href="#a:1283"><span class="id" title="binder">a</span></a> <a id="b:1284" class="idref" href="#b:1284"><span class="id" title="binder">b</span></a> <a id="b':1285" class="idref" href="#b':1285"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1284"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1284"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1285"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1283"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1284"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1283"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1285"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_mono_c" class="idref" href="#Gamma0.phi_mono_c"><span class="id" title="lemma">phi_mono_c</span></a> <a id="a:1286" class="idref" href="#a:1286"><span class="id" title="binder">a</span></a> <a id="b:1287" class="idref" href="#b:1287"><span class="id" title="binder">b</span></a> <a id="b':1288" class="idref" href="#b':1288"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1287"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1288"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1286"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1287"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1286"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1288"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1287"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1288"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inj" class="idref" href="#Gamma0.phi_inj"><span class="id" title="lemma">phi_inj</span></a> <a id="a:1289" class="idref" href="#a:1289"><span class="id" title="binder">a</span></a> <a id="b:1290" class="idref" href="#b:1290"><span class="id" title="binder">b</span></a> <a id="b':1291" class="idref" href="#b':1291"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1290"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1291"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1289"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1290"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1289"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1291"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1290"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1291"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inj1" class="idref" href="#Gamma0.phi_inj1"><span class="id" title="lemma">phi_inj1</span></a> <a id="a:1292" class="idref" href="#a:1292"><span class="id" title="binder">a</span></a> <a id="b:1293" class="idref" href="#b:1293"><span class="id" title="binder">b</span></a> <a id="b':1294" class="idref" href="#b':1294"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1293"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1294"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1292"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1293"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1292"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1294"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1293"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1294"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Two lemmas for equal or less-than 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_eqE" class="idref" href="#Gamma0.phi_eqE"><span class="id" title="lemma">phi_eqE</span></a> <a id="a:1295" class="idref" href="#a:1295"><span class="id" title="binder">a</span></a> <a id="b:1296" class="idref" href="#b:1296"><span class="id" title="binder">b</span></a> <a id="a':1297" class="idref" href="#a':1297"><span class="id" title="binder">a'</span></a> <a id="b':1298" class="idref" href="#b':1298"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1295"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1297"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1296"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1298"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1295"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1296"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1297"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1298"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1295"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1297"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1296"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1297"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1298"><span class="id" title="variable">b'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1297"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1295"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1295"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1296"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1298"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1296"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1298"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_ltE" class="idref" href="#Gamma0.phi_ltE"><span class="id" title="lemma">phi_ltE</span></a> <a id="a:1299" class="idref" href="#a:1299"><span class="id" title="binder">a</span></a> <a id="b:1300" class="idref" href="#b:1300"><span class="id" title="binder">b</span></a> <a id="a':1301" class="idref" href="#a':1301"><span class="id" title="binder">a'</span></a> <a id="b':1302" class="idref" href="#b':1302"><span class="id" title="binder">b'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1299"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1301"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1300"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf"><span class="id" title="definition">T2nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1302"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1299"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1300"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1301"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1302"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1299"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1301"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1300"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1301"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1302"><span class="id" title="variable">b'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1301"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1299"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1299"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1300"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1302"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1300"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1302"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Every <span class="inlinecode"><span class="id" title="var">x</span></span> is uniquely a phi with some conditions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inv0" class="idref" href="#Gamma0.phi_inv0"><span class="id" title="lemma">phi_inv0</span></a>  <a id="a:1303" class="idref" href="#a:1303"><span class="id" title="binder">a</span></a> <a id="b:1304" class="idref" href="#b:1304"><span class="id" title="binder">b</span></a> <a id="a':1305" class="idref" href="#a':1305"><span class="id" title="binder">a'</span></a> <a id="b':1306" class="idref" href="#b':1306"><span class="id" title="binder">b'</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1303"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1304"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1305"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1306"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1304"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1303"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1304"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1306"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1305"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1306"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1303"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1305"><span class="id" title="variable">a'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inv2" class="idref" href="#Gamma0.phi_inv2"><span class="id" title="lemma">phi_inv2</span></a> <a id="a:1307" class="idref" href="#a:1307"><span class="id" title="binder">a</span></a> <a id="b:1308" class="idref" href="#b:1308"><span class="id" title="binder">b</span></a> <a id="a':1309" class="idref" href="#a':1309"><span class="id" title="binder">a'</span></a> <a id="b':1310" class="idref" href="#b':1310"><span class="id" title="binder">b'</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1307"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1308"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1309"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1310"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1308"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1307"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1308"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1310"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1309"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1310"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1308"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1310"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.phi_inv3" class="idref" href="#Gamma0.phi_inv3"><span class="id" title="lemma">phi_inv3</span></a> <a id="x:1311" class="idref" href="#x:1311"><span class="id" title="binder">x</span></a>: <br/>
+&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap"><span class="id" title="definition">T2ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1311"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a> <a id="a:1312" class="idref" href="#a:1312"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="b:1313" class="idref" href="#b:1313"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1311"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1312"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1313"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1313"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1311"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a>  (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1312"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1311"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&amp;</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1313"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1311"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Expression psi in terms of phi 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.psi_phi_aux" class="idref" href="#Gamma0.psi_phi_aux"><span class="id" title="definition">psi_phi_aux</span></a> <a id="a:1314" class="idref" href="#a:1314"><span class="id" title="binder">a</span></a> <a id="b:1315" class="idref" href="#b:1315"><span class="id" title="binder">b</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="b':1316" class="idref" href="#b':1316"><span class="id" title="binder">b'</span></a>, <a id="n:1317" class="idref" href="#n:1317"><span class="id" title="binder">n</span></a>) := <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2split"><span class="id" title="definition">T2split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1315"><span class="id" title="variable">b</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1314"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1316"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1316"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ"><span class="id" title="definition">T2succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1315"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1315"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0.psi_phi" class="idref" href="#Gamma0.psi_phi"><span class="id" title="definition">psi_phi</span></a> <a id="a:1318" class="idref" href="#a:1318"><span class="id" title="binder">a</span></a> <a id="b:1319" class="idref" href="#b:1319"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1318"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.psi_phi_aux"><span class="id" title="definition">psi_phi_aux</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1318"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1319"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gamma0.psi_phi1" class="idref" href="#Gamma0.psi_phi1"><span class="id" title="lemma">psi_phi1</span></a> <a id="a:1320" class="idref" href="#a:1320"><span class="id" title="binder">a</span></a> <a id="b:1321" class="idref" href="#b:1321"><span class="id" title="binder">b</span></a> (<a id="c:1322" class="idref" href="#c:1322"><span class="id" title="binder">c</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.psi_phi_aux"><span class="id" title="definition">psi_phi_aux</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1320"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1321"><span class="id" title="variable">b</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1322"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1320"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1322"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Gamma0"><span class="id" title="module">Gamma0</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab458"></a><h1 class="section">Ackermann</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Ackermann" class="idref" href="#Ackermann"><span class="id" title="module">Ackermann</span></a>.<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">ak_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ak_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ak</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ak_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Ackermann.T3" class="idref" href="#Ackermann.T3"><span class="id" title="inductive">T3</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="Ackermann.zero" class="idref" href="#Ackermann.zero"><span class="id" title="constructor">zero</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a><br/>
+| <a id="Ackermann.cons" class="idref" href="#Ackermann.cons"><span class="id" title="constructor">cons</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3:1323"><span class="id" title="inductive">T3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3eq" class="idref" href="#Ackermann.T3eq"><span class="id" title="definition">T3eq</span></a> <a id="x:1325" class="idref" href="#x:1325"><span class="id" title="binder">x</span></a> <a id="y:1326" class="idref" href="#y:1326"><span class="id" title="binder">y</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">x</span>} :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1325"><span class="id" title="variable">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1326"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span> <span class="id" title="var">n'</span> <span class="id" title="var">d'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3eq:1327"><span class="id" title="definition">T3eq</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3eq:1327"><span class="id" title="definition">T3eq</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3eq:1327"><span class="id" title="definition">T3eq</span></a> <span class="id" title="var">c</span> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><span class="id" title="notation">==</span> <span class="id" title="var">n'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3eq:1327"><span class="id" title="definition">T3eq</span></a> <span class="id" title="var">d</span> <span class="id" title="var">d'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3eqP" class="idref" href="#Ackermann.T3eqP"><span class="id" title="lemma">T3eqP</span></a> : <span class="id" title="definition">Equality.axiom</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3eq"><span class="id" title="definition">T3eq</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <a id="Ackermann.T3_eqMixin" class="idref" href="#Ackermann.T3_eqMixin"><span class="id" title="definition">T3_eqMixin</span></a> := <span class="id" title="abbreviation">EqMixin</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3eqP"><span class="id" title="lemma">T3eqP</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <a id="Ackermann.T3_eqType" class="idref" href="#Ackermann.T3_eqType"><span class="id" title="definition">T3_eqType</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <span class="id" title="abbreviation">EqType</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3_eqMixin"><span class="id" title="definition">T3_eqMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3eqP"><span class="id" title="lemma">T3eqP</span></a> {<span class="id" title="var">x</span> <span class="id" title="var">y</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3eqE" class="idref" href="#Ackermann.T3eqE"><span class="id" title="lemma">T3eqE</span></a> <a id="a:1330" class="idref" href="#a:1330"><span class="id" title="binder">a</span></a> <a id="b:1331" class="idref" href="#b:1331"><span class="id" title="binder">b</span></a> <a id="c:1332" class="idref" href="#c:1332"><span class="id" title="binder">c</span></a> <a id="n:1333" class="idref" href="#n:1333"><span class="id" title="binder">n</span></a> <a id="d:1334" class="idref" href="#d:1334"><span class="id" title="binder">d</span></a> <a id="a':1335" class="idref" href="#a':1335"><span class="id" title="binder">a'</span></a> <a id="b':1336" class="idref" href="#b':1336"><span class="id" title="binder">b'</span></a> <a id="c':1337" class="idref" href="#c':1337"><span class="id" title="binder">c'</span></a> <a id="n':1338" class="idref" href="#n':1338"><span class="id" title="binder">n'</span></a> <a id="d':1339" class="idref" href="#d':1339"><span class="id" title="binder">d'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1330"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1331"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1332"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1333"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1334"><span class="id" title="variable">d</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1335"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1336"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1337"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1338"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1339"><span class="id" title="variable">d'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1330"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1335"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1331"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1336"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1332"><span class="id" title="variable">c</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1337"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1333"><span class="id" title="variable">n</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1338"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1334"><span class="id" title="variable">d</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1339"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="c590f386bdc613e117a7531b9fbcc613" class="idref" href="#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">&quot;</span></a>[ x , y , z ]" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) :<span class="id" title="var">ak_scope</span>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3nat" class="idref" href="#Ackermann.T3nat"><span class="id" title="definition">T3nat</span></a> <a id="p:1340" class="idref" href="#p:1340"><span class="id" title="binder">p</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1340"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="7fdc35ce1b6f711389e0dbafa7795fb3" class="idref" href="#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">&quot;</span></a>\F n" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nat"><span class="id" title="definition">T3nat</span></a> <span class="id" title="var">n</span>)(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 29) : <span class="id" title="var">ak_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.size" class="idref" href="#Ackermann.size"><span class="id" title="definition">size</span></a> <a id="x:1342" class="idref" href="#x:1342"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1342"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><span class="id" title="definition">maxn</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:1343"><span class="id" title="definition">size</span></a> <span class="id" title="var">a</span>) (<span class="id" title="definition">maxn</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:1343"><span class="id" title="definition">size</span></a> <span class="id" title="var">b</span>) (<span class="id" title="definition">maxn</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:1343"><span class="id" title="definition">size</span></a> <span class="id" title="var">c</span>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#size:1343"><span class="id" title="definition">size</span></a> <span class="id" title="var">d</span>)))<span class="id" title="notation">).+1</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_a" class="idref" href="#Ackermann.size_a"><span class="id" title="lemma">size_a</span></a> <a id="a:1345" class="idref" href="#a:1345"><span class="id" title="binder">a</span></a> <a id="b:1346" class="idref" href="#b:1346"><span class="id" title="binder">b</span></a> <a id="c:1347" class="idref" href="#c:1347"><span class="id" title="binder">c</span></a> <a id="n:1348" class="idref" href="#n:1348"><span class="id" title="binder">n</span></a> <a id="d:1349" class="idref" href="#d:1349"><span class="id" title="binder">d</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1345"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1345"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1346"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1347"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1348"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1349"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_b" class="idref" href="#Ackermann.size_b"><span class="id" title="lemma">size_b</span></a> <a id="a:1350" class="idref" href="#a:1350"><span class="id" title="binder">a</span></a> <a id="b:1351" class="idref" href="#b:1351"><span class="id" title="binder">b</span></a> <a id="c:1352" class="idref" href="#c:1352"><span class="id" title="binder">c</span></a> <a id="n:1353" class="idref" href="#n:1353"><span class="id" title="binder">n</span></a> <a id="d:1354" class="idref" href="#d:1354"><span class="id" title="binder">d</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1351"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1350"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1351"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1352"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1353"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1354"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_c" class="idref" href="#Ackermann.size_c"><span class="id" title="lemma">size_c</span></a> <a id="a:1355" class="idref" href="#a:1355"><span class="id" title="binder">a</span></a> <a id="b:1356" class="idref" href="#b:1356"><span class="id" title="binder">b</span></a> <a id="c:1357" class="idref" href="#c:1357"><span class="id" title="binder">c</span></a> <a id="n:1358" class="idref" href="#n:1358"><span class="id" title="binder">n</span></a> <a id="d:1359" class="idref" href="#d:1359"><span class="id" title="binder">d</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1357"><span class="id" title="variable">c</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1355"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1356"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1357"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1358"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1359"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_d" class="idref" href="#Ackermann.size_d"><span class="id" title="lemma">size_d</span></a> <a id="a:1360" class="idref" href="#a:1360"><span class="id" title="binder">a</span></a> <a id="b:1361" class="idref" href="#b:1361"><span class="id" title="binder">b</span></a> <a id="c:1362" class="idref" href="#c:1362"><span class="id" title="binder">c</span></a> <a id="n:1363" class="idref" href="#n:1363"><span class="id" title="binder">n</span></a> <a id="d:1364" class="idref" href="#d:1364"><span class="id" title="binder">d</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1364"><span class="id" title="variable">d</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1360"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1361"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1362"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1363"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1364"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_psi" class="idref" href="#Ackermann.size_psi"><span class="id" title="lemma">size_psi</span></a> <a id="a:1365" class="idref" href="#a:1365"><span class="id" title="binder">a</span></a> <a id="b:1366" class="idref" href="#b:1366"><span class="id" title="binder">b</span></a> <a id="c:1367" class="idref" href="#c:1367"><span class="id" title="binder">c</span></a> <a id="n:1368" class="idref" href="#n:1368"><span class="id" title="binder">n</span></a> <a id="d:1369" class="idref" href="#d:1369"><span class="id" title="binder">d</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1365"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1366"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1367"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1365"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1366"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1367"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1368"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1369"><span class="id" title="variable">d</span></a>))%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_prop1" class="idref" href="#Ackermann.size_prop1"><span class="id" title="lemma">size_prop1</span></a> <a id="a:1370" class="idref" href="#a:1370"><span class="id" title="binder">a</span></a> <a id="b:1371" class="idref" href="#b:1371"><span class="id" title="binder">b</span></a> <a id="c:1372" class="idref" href="#c:1372"><span class="id" title="binder">c</span></a> <a id="n:1373" class="idref" href="#n:1373"><span class="id" title="binder">n</span></a> <a id="d:1374" class="idref" href="#d:1374"><span class="id" title="binder">d</span></a> (<a id="l:1375" class="idref" href="#l:1375"><span class="id" title="binder">l</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1370"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1371"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1372"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1373"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1374"><span class="id" title="variable">d</span></a>)):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1370"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1375"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1371"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1375"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1372"><span class="id" title="variable">c</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1375"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1374"><span class="id" title="variable">d</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1375"><span class="id" title="variable">l</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1370"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1371"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1372"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1375"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.size_prop" class="idref" href="#Ackermann.size_prop"><span class="id" title="lemma">size_prop</span></a> <a id="a:1376" class="idref" href="#a:1376"><span class="id" title="binder">a</span></a> <a id="b:1377" class="idref" href="#b:1377"><span class="id" title="binder">b</span></a> <a id="c:1378" class="idref" href="#c:1378"><span class="id" title="binder">c</span></a> <a id="n:1379" class="idref" href="#n:1379"><span class="id" title="binder">n</span></a> <a id="d:1380" class="idref" href="#d:1380"><span class="id" title="binder">d</span></a> <a id="a':1381" class="idref" href="#a':1381"><span class="id" title="binder">a'</span></a> <a id="b':1382" class="idref" href="#b':1382"><span class="id" title="binder">b'</span></a> <a id="c':1383" class="idref" href="#c':1383"><span class="id" title="binder">c'</span></a> <a id="n':1384" class="idref" href="#n':1384"><span class="id" title="binder">n'</span></a> <a id="d':1385" class="idref" href="#d':1385"><span class="id" title="binder">d'</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a id="l:1386" class="idref" href="#l:1386"><span class="id" title="binder">l</span></a> := ((<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1376"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1378"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1379"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1380"><span class="id" title="variable">d</span></a>) <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1381"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1383"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1384"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1385"><span class="id" title="variable">d'</span></a>)))%<span class="id" title="var">N</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1381"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1376"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1378"><span class="id" title="variable">c</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1383"><span class="id" title="variable">c'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1380"><span class="id" title="variable">d</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1385"><span class="id" title="variable">d'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1376"><span class="id" title="variable">a</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1381"><span class="id" title="variable">a'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1376"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1378"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1381"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1383"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1376"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1377"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1378"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1383"><span class="id" title="variable">c'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1378"><span class="id" title="variable">c</span></a> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1381"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1382"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1383"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#l:1386"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">)]</span></a>%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab459"></a><h2 class="section">Comparison</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.lt_psi_rec" class="idref" href="#Ackermann.lt_psi_rec"><span class="id" title="definition">lt_psi_rec</span></a> <a id="f:1387" class="idref" href="#f:1387"><span class="id" title="binder">f</span></a> <a id="a:1388" class="idref" href="#a:1388"><span class="id" title="binder">a</span></a> <a id="b:1389" class="idref" href="#b:1389"><span class="id" title="binder">b</span></a> <a id="c:1390" class="idref" href="#c:1390"><span class="id" title="binder">c</span></a> <a id="a':1391" class="idref" href="#a':1391"><span class="id" title="binder">a'</span></a> <a id="b':1392" class="idref" href="#b':1392"><span class="id" title="binder">b'</span></a> <a id="c':1393" class="idref" href="#c':1393"><span class="id" title="binder">c'</span></a> (<a id="x:1394" class="idref" href="#x:1394"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1390"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>)(<a id="x':1395" class="idref" href="#x':1395"><span class="id" title="binder">x'</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>):=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1390"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1390"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':1395"><span class="id" title="variable">x'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1389"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':1395"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1390"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x':1395"><span class="id" title="variable">x'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1392"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1387"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1391"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1388"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1394"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1393"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.lt_rec" class="idref" href="#Ackermann.lt_rec"><span class="id" title="definition">lt_rec</span></a> <a id="f:1396" class="idref" href="#f:1396"><span class="id" title="binder">f</span></a> <a id="x:1397" class="idref" href="#x:1397"><span class="id" title="binder">x</span></a> <a id="y:1398" class="idref" href="#y:1398"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1397"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1398"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span> <span class="id" title="var">n'</span> <span class="id" title="var">d'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi_rec"><span class="id" title="definition">lt_psi_rec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1396"><span class="id" title="variable">f</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span> <span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span><span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">c</span><span class="id" title="notation">==</span><span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span> <span class="id" title="notation">&lt;</span> <span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span> <span class="id" title="notation">==</span> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1396"><span class="id" title="variable">f</span></a> <span class="id" title="var">d</span> <span class="id" title="var">d'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <br/>
+&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1398"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span> <span class="id" title="var">n'</span> <span class="id" title="var">d'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3lta" class="idref" href="#Ackermann.T3lta"><span class="id" title="definition">T3lta</span></a> <a id="k:1402" class="idref" href="#k:1402"><span class="id" title="binder">k</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">k</span>}:=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:1402"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">k</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.lt_rec"><span class="id" title="definition">lt_rec</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T3lta:1403"><span class="id" title="definition">T3lta</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:1402"><span class="id" title="variable">k</span></a>) <span class="id" title="keyword">else</span> <span class="id" title="keyword">fun</span> <a id="x:1405" class="idref" href="#x:1405"><span class="id" title="binder">x</span></a> <a id="y:1406" class="idref" href="#y:1406"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3lt" class="idref" href="#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a> <a id="a:1407" class="idref" href="#a:1407"><span class="id" title="binder">a</span></a> <a id="b:1408" class="idref" href="#b:1408"><span class="id" title="binder">b</span></a> :=  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lta"><span class="id" title="definition">T3lta</span></a> <span class="id" title="notation">(</span><span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1407"><span class="id" title="variable">a</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">+</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1408"><span class="id" title="variable">b</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1407"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1408"><span class="id" title="variable">b</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3le" class="idref" href="#Ackermann.T3le"><span class="id" title="definition">T3le</span></a> (<a id="x:1409" class="idref" href="#x:1409"><span class="id" title="binder">x</span></a> <a id="y:1410" class="idref" href="#y:1410"><span class="id" title="binder">y</span></a> :<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1409"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1410"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1409"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1410"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Ackermann.::ak_scope:x_'&lt;'_x" class="idref" href="#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt; y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">ak_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="aac3646df53c7919a108e9925f5268d7" class="idref" href="#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">&quot;</span></a>x &lt;= y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3le"><span class="id" title="definition">T3le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">ak_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="752c7e761a34d95cac9b695b31aec25e" class="idref" href="#752c7e761a34d95cac9b695b31aec25e"><span class="id" title="notation">&quot;</span></a>x &gt;= y" := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ak_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Ackermann.::ak_scope:x_'&gt;'_x" class="idref" href="#Ackermann.::ak_scope:x_'&gt;'_x"><span class="id" title="notation">&quot;</span></a>x &gt; y"  := (<span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>)  : <span class="id" title="var">ak_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltE" class="idref" href="#Ackermann.T3ltE"><span class="id" title="lemma">T3ltE</span></a> <a id="x:1411" class="idref" href="#x:1411"><span class="id" title="binder">x</span></a> <a id="y:1412" class="idref" href="#y:1412"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1411"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1412"><span class="id" title="variable">y</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.lt_rec"><span class="id" title="definition">lt_rec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1411"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1412"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.lt_psi" class="idref" href="#Ackermann.lt_psi"><span class="id" title="definition">lt_psi</span></a> (<a id="a:1416" class="idref" href="#a:1416"><span class="id" title="binder">a</span></a> <a id="b:1417" class="idref" href="#b:1417"><span class="id" title="binder">b</span></a> <a id="c:1418" class="idref" href="#c:1418"><span class="id" title="binder">c</span></a> <a id="a':1419" class="idref" href="#a':1419"><span class="id" title="binder">a'</span></a> <a id="b':1420" class="idref" href="#b':1420"><span class="id" title="binder">b'</span></a> <a id="c':1421" class="idref" href="#c':1421"><span class="id" title="binder">c'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a>):=<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1420"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1419"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1416"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1417"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1418"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1421"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi" class="idref" href="#Ackermann.T3lt_psi"><span class="id" title="lemma">T3lt_psi</span></a> <a id="a:1422" class="idref" href="#a:1422"><span class="id" title="binder">a</span></a> <a id="b:1423" class="idref" href="#b:1423"><span class="id" title="binder">b</span></a> <a id="c:1424" class="idref" href="#c:1424"><span class="id" title="binder">c</span></a>  <a id="a':1425" class="idref" href="#a':1425"><span class="id" title="binder">a'</span></a> <a id="b':1426" class="idref" href="#b':1426"><span class="id" title="binder">b'</span></a> <a id="c':1427" class="idref" href="#c':1427"><span class="id" title="binder">c'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1422"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1423"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1424"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1425"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1426"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1427"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi"><span class="id" title="definition">lt_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1422"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1423"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1424"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1425"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1426"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1427"><span class="id" title="variable">c'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_consE" class="idref" href="#Ackermann.T3lt_consE"><span class="id" title="lemma">T3lt_consE</span></a> <a id="a:1428" class="idref" href="#a:1428"><span class="id" title="binder">a</span></a> <a id="b:1429" class="idref" href="#b:1429"><span class="id" title="binder">b</span></a> <a id="c:1430" class="idref" href="#c:1430"><span class="id" title="binder">c</span></a> <a id="n:1431" class="idref" href="#n:1431"><span class="id" title="binder">n</span></a> <a id="d:1432" class="idref" href="#d:1432"><span class="id" title="binder">d</span></a>  <a id="a':1433" class="idref" href="#a':1433"><span class="id" title="binder">a'</span></a> <a id="b':1434" class="idref" href="#b':1434"><span class="id" title="binder">b'</span></a> <a id="c':1435" class="idref" href="#c':1435"><span class="id" title="binder">c'</span></a> <a id="n':1436" class="idref" href="#n':1436"><span class="id" title="binder">n'</span></a> <a id="d':1437" class="idref" href="#d':1437"><span class="id" title="binder">d'</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1428"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1429"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1430"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1431"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1432"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1433"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1434"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1435"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1436"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1437"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1428"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1429"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1430"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1433"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1434"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1435"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1428"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1429"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1430"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1433"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1434"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1435"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1431"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1436"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1431"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1436"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d:1432"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1437"><span class="id" title="variable">d'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltn0" class="idref" href="#Ackermann.T3ltn0"><span class="id" title="lemma">T3ltn0</span></a> <a id="x:1438" class="idref" href="#x:1438"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1438"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt0n" class="idref" href="#Ackermann.T3lt0n"><span class="id" title="lemma">T3lt0n</span></a> <a id="x:1439" class="idref" href="#x:1439"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1439"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1439"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltnn" class="idref" href="#Ackermann.T3ltnn"><span class="id" title="lemma">T3ltnn</span></a> <a id="x:1440" class="idref" href="#x:1440"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1440"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1440"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_ne" class="idref" href="#Ackermann.T3lt_ne"><span class="id" title="lemma">T3lt_ne</span></a> <a id="a:1441" class="idref" href="#a:1441"><span class="id" title="binder">a</span></a> <a id="b:1442" class="idref" href="#b:1442"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1441"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1442"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1441"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1442"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_ne'" class="idref" href="#Ackermann.T3lt_ne'"><span class="id" title="lemma">T3lt_ne'</span></a> <a id="a:1443" class="idref" href="#a:1443"><span class="id" title="binder">a</span></a> <a id="b:1444" class="idref" href="#b:1444"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1443"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1444"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1444"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1443"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltW" class="idref" href="#Ackermann.T3ltW"><span class="id" title="lemma">T3ltW</span></a> <a id="a:1445" class="idref" href="#a:1445"><span class="id" title="binder">a</span></a> <a id="b:1446" class="idref" href="#b:1446"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1445"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1446"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1445"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1446"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_eqVlt" class="idref" href="#Ackermann.T3le_eqVlt"><span class="id" title="lemma">T3le_eqVlt</span></a> <a id="a:1447" class="idref" href="#a:1447"><span class="id" title="binder">a</span></a> <a id="b:1448" class="idref" href="#b:1448"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1447"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1448"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1447"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1448"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1447"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1448"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_neAle" class="idref" href="#Ackermann.T3lt_neAle"><span class="id" title="lemma">T3lt_neAle</span></a> <a id="a:1449" class="idref" href="#a:1449"><span class="id" title="binder">a</span></a> <a id="b:1450" class="idref" href="#b:1450"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1449"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1450"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1449"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1450"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1449"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1450"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.one" class="idref" href="#Ackermann.one"><span class="id" title="definition">one</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.omega" class="idref" href="#Ackermann.omega"><span class="id" title="definition">omega</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.epsilon0" class="idref" href="#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a> :=  <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3bad" class="idref" href="#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le0n" class="idref" href="#Ackermann.T3le0n"><span class="id" title="lemma">T3le0n</span></a> <a id="x:1451" class="idref" href="#x:1451"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1451"><span class="id" title="variable">x</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3len0" class="idref" href="#Ackermann.T3len0"><span class="id" title="lemma">T3len0</span></a> <a id="x:1452" class="idref" href="#x:1452"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1452"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1452"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ge1" class="idref" href="#Ackermann.T3ge1"><span class="id" title="lemma">T3ge1</span></a> <a id="x:1453" class="idref" href="#x:1453"><span class="id" title="binder">x</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1453"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1453"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt1" class="idref" href="#Ackermann.T3lt1"><span class="id" title="lemma">T3lt1</span></a> <a id="x:1454" class="idref" href="#x:1454"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1454"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1454"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lcp0_pr" class="idref" href="#Ackermann.T3lcp0_pr"><span class="id" title="lemma">T3lcp0_pr</span></a> <a id="x:1455" class="idref" href="#x:1455"><span class="id" title="binder">x</span></a> <a id="y:1456" class="idref" href="#y:1456"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1455"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1456"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1456"><span class="id" title="variable">y</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.finite_ltP" class="idref" href="#Ackermann.finite_ltP"><span class="id" title="lemma">finite_ltP</span></a> <a id="n:1457" class="idref" href="#n:1457"><span class="id" title="binder">n</span></a> <a id="p:1458" class="idref" href="#p:1458"><span class="id" title="binder">p</span></a> : (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1457"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1458"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1457"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1458"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_anti" class="idref" href="#Ackermann.T3lt_anti"><span class="id" title="lemma">T3lt_anti</span></a> <a id="b:1459" class="idref" href="#b:1459"><span class="id" title="binder">b</span></a> <a id="a:1460" class="idref" href="#a:1460"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1460"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1459"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1459"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1460"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_trichotomy" class="idref" href="#Ackermann.T3lt_trichotomy"><span class="id" title="lemma">T3lt_trichotomy</span></a> <a id="a:1463" class="idref" href="#a:1463"><span class="id" title="binder">a</span></a> <a id="b:1464" class="idref" href="#b:1464"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1463"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1464"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1463"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1464"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1464"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1463"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lenn" class="idref" href="#Ackermann.T3lenn"><span class="id" title="lemma">T3lenn</span></a> <a id="x:1467" class="idref" href="#x:1467"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1467"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1467"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T3lenn</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3leNgt" class="idref" href="#Ackermann.T3leNgt"><span class="id" title="lemma">T3leNgt</span></a> <a id="a:1468" class="idref" href="#a:1468"><span class="id" title="binder">a</span></a> <a id="b:1469" class="idref" href="#b:1469"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1468"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1469"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1469"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1468"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltNge" class="idref" href="#Ackermann.T3ltNge"><span class="id" title="lemma">T3ltNge</span></a> <a id="a:1470" class="idref" href="#a:1470"><span class="id" title="binder">a</span></a> <a id="b:1471" class="idref" href="#b:1471"><span class="id" title="binder">b</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1470"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1471"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1471"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1470"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3eq_le" class="idref" href="#Ackermann.T3eq_le"><span class="id" title="lemma">T3eq_le</span></a> <a id="m:1472" class="idref" href="#m:1472"><span class="id" title="binder">m</span></a> <a id="n:1473" class="idref" href="#n:1473"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1472"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1473"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1472"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1473"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1473"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1472"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Ackermann.T3ltn_xor_geq" class="idref" href="#Ackermann.T3ltn_xor_geq"><span class="id" title="inductive">T3ltn_xor_geq</span></a> <a id="m:1474" class="idref" href="#m:1474"><span class="id" title="binder">m</span></a> <a id="n:1475" class="idref" href="#n:1475"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Ackermann.T3LtnNotGeq" class="idref" href="#Ackermann.T3LtnNotGeq"><span class="id" title="constructor">T3LtnNotGeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1474"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1475"><span class="id" title="variable">n</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3ltn_xor_geq:1476"><span class="id" title="inductive">T3ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1474"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1475"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a id="Ackermann.T3GeqNotLtn" class="idref" href="#Ackermann.T3GeqNotLtn"><span class="id" title="constructor">T3GeqNotLtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1475"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1474"><span class="id" title="variable">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3ltn_xor_geq:1476"><span class="id" title="inductive">T3ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1474"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1475"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Ackermann.T3leq_xor_gtn" class="idref" href="#Ackermann.T3leq_xor_gtn"><span class="id" title="inductive">T3leq_xor_gtn</span></a> <a id="m:1478" class="idref" href="#m:1478"><span class="id" title="binder">m</span></a> <a id="n:1479" class="idref" href="#n:1479"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Ackermann.T3GeqNotGtn" class="idref" href="#Ackermann.T3GeqNotGtn"><span class="id" title="constructor">T3GeqNotGtn</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1478"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1479"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3leq_xor_gtn:1480"><span class="id" title="inductive">T3leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1478"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1479"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Ackermann.T3GtnNotLeq" class="idref" href="#Ackermann.T3GtnNotLeq"><span class="id" title="constructor">T3GtnNotLeq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1479"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1478"><span class="id" title="variable">m</span></a>  : <a class="idref" href="gaia_hydras.nfwfgaia.html#T3leq_xor_gtn:1480"><span class="id" title="inductive">T3leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1478"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1479"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">CoInductive</span> <a id="Ackermann.compare_T3" class="idref" href="#Ackermann.compare_T3"><span class="id" title="inductive">compare_T3</span></a> <a id="m:1482" class="idref" href="#m:1482"><span class="id" title="binder">m</span></a> <a id="n:1483" class="idref" href="#n:1483"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Ackermann.CompareT3Lt" class="idref" href="#Ackermann.CompareT3Lt"><span class="id" title="constructor">CompareT3Lt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T3:1484"><span class="id" title="inductive">compare_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Ackermann.CompareT3Gt" class="idref" href="#Ackermann.CompareT3Gt"><span class="id" title="constructor">CompareT3Gt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T3:1484"><span class="id" title="inductive">compare_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a id="Ackermann.CompareT3Eq" class="idref" href="#Ackermann.CompareT3Eq"><span class="id" title="constructor">CompareT3Eq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#compare_T3:1484"><span class="id" title="inductive">compare_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1482"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1483"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3leP" class="idref" href="#Ackermann.T3leP"><span class="id" title="lemma">T3leP</span></a> <a id="x:1486" class="idref" href="#x:1486"><span class="id" title="binder">x</span></a> <a id="y:1487" class="idref" href="#y:1487"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3leq_xor_gtn"><span class="id" title="inductive">T3leq_xor_gtn</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1486"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1487"><span class="id" title="variable">y</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:1486"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1487"><span class="id" title="variable">y</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#y:1487"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1486"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltP" class="idref" href="#Ackermann.T3ltP"><span class="id" title="lemma">T3ltP</span></a> <a id="m:1488" class="idref" href="#m:1488"><span class="id" title="binder">m</span></a> <a id="n:1489" class="idref" href="#n:1489"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltn_xor_geq"><span class="id" title="inductive">T3ltn_xor_geq</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1488"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1489"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1489"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1488"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:1488"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1489"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3ltgtP" class="idref" href="#Ackermann.T3ltgtP"><span class="id" title="lemma">T3ltgtP</span></a> <a id="m:1490" class="idref" href="#m:1490"><span class="id" title="binder">m</span></a> <a id="n:1491" class="idref" href="#n:1491"><span class="id" title="binder">n</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.compare_T3"><span class="id" title="inductive">compare_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1490"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1491"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:1490"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1491"><span class="id" title="variable">n</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1491"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1490"><span class="id" title="variable">m</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#m:1490"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1491"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_consE" class="idref" href="#Ackermann.T3le_consE"><span class="id" title="lemma">T3le_consE</span></a> <a id="a:1492" class="idref" href="#a:1492"><span class="id" title="binder">a</span></a> <a id="b:1493" class="idref" href="#b:1493"><span class="id" title="binder">b</span></a> <a id="c:1494" class="idref" href="#c:1494"><span class="id" title="binder">c</span></a> <a id="n:1495" class="idref" href="#n:1495"><span class="id" title="binder">n</span></a> <a id="d:1496" class="idref" href="#d:1496"><span class="id" title="binder">d</span></a>  <a id="a':1497" class="idref" href="#a':1497"><span class="id" title="binder">a'</span></a> <a id="b':1498" class="idref" href="#b':1498"><span class="id" title="binder">b'</span></a> <a id="c':1499" class="idref" href="#c':1499"><span class="id" title="binder">c'</span></a> <a id="n':1500" class="idref" href="#n':1500"><span class="id" title="binder">n'</span></a> <a id="d':1501" class="idref" href="#d':1501"><span class="id" title="binder">d'</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1492"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1493"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1494"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1495"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1496"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1497"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1498"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1499"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1500"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1501"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1492"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1493"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1494"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1497"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1498"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1499"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1492"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1493"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1494"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1497"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1498"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1499"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1495"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1500"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1495"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1500"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d:1496"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1501"><span class="id" title="variable">d'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi'" class="idref" href="#Ackermann.T3lt_psi'"><span class="id" title="lemma">T3lt_psi'</span></a> <a id="a:1502" class="idref" href="#a:1502"><span class="id" title="binder">a</span></a> <a id="b:1503" class="idref" href="#b:1503"><span class="id" title="binder">b</span></a> <a id="c:1504" class="idref" href="#c:1504"><span class="id" title="binder">c</span></a> <a id="a':1505" class="idref" href="#a':1505"><span class="id" title="binder">a'</span></a> <a id="b':1506" class="idref" href="#b':1506"><span class="id" title="binder">b'</span></a> <a id="c':1507" class="idref" href="#c':1507"><span class="id" title="binder">c'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1506"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1505"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1502"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1503"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1504"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1507"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Ackermann.T3lt_trans" class="idref" href="#Ackermann.T3lt_trans"><span class="id" title="lemma">T3lt_trans</span></a> <a id="b:1508" class="idref" href="#b:1508"><span class="id" title="binder">b</span></a> <a id="a:1509" class="idref" href="#a:1509"><span class="id" title="binder">a</span></a> <a id="c:1510" class="idref" href="#c:1510"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1509"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1508"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1508"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1510"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1509"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1510"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_le_trans" class="idref" href="#Ackermann.T3lt_le_trans"><span class="id" title="lemma">T3lt_le_trans</span></a> <a id="b:1520" class="idref" href="#b:1520"><span class="id" title="binder">b</span></a> <a id="a:1521" class="idref" href="#a:1521"><span class="id" title="binder">a</span></a> <a id="c:1522" class="idref" href="#c:1522"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1521"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1520"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1520"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1522"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1521"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1522"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_lt_trans" class="idref" href="#Ackermann.T3le_lt_trans"><span class="id" title="lemma">T3le_lt_trans</span></a> <a id="b:1523" class="idref" href="#b:1523"><span class="id" title="binder">b</span></a> <a id="a:1524" class="idref" href="#a:1524"><span class="id" title="binder">a</span></a> <a id="c:1525" class="idref" href="#c:1525"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1524"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1523"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1523"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1525"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1524"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1525"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_trans" class="idref" href="#Ackermann.T3le_trans"><span class="id" title="lemma">T3le_trans</span></a> <a id="b:1526" class="idref" href="#b:1526"><span class="id" title="binder">b</span></a> <a id="a:1527" class="idref" href="#a:1527"><span class="id" title="binder">a</span></a> <a id="c:1528" class="idref" href="#c:1528"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1527"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1526"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1526"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1528"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1527"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1528"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_anti" class="idref" href="#Ackermann.T3le_anti"><span class="id" title="lemma">T3le_anti</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3le"><span class="id" title="definition">T3le</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_total" class="idref" href="#Ackermann.T3le_total"><span class="id" title="lemma">T3le_total</span></a> <a id="m:1529" class="idref" href="#m:1529"><span class="id" title="binder">m</span></a> <a id="n:1530" class="idref" href="#n:1530"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1529"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1530"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1530"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1529"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_psi" class="idref" href="#Ackermann.T3le_psi"><span class="id" title="lemma">T3le_psi</span></a> <a id="a:1531" class="idref" href="#a:1531"><span class="id" title="binder">a</span></a> <a id="b:1532" class="idref" href="#b:1532"><span class="id" title="binder">b</span></a> <a id="c:1533" class="idref" href="#c:1533"><span class="id" title="binder">c</span></a> <a id="n:1534" class="idref" href="#n:1534"><span class="id" title="binder">n</span></a> <a id="d:1535" class="idref" href="#d:1535"><span class="id" title="binder">d</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1531"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1532"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1533"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1531"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1532"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1533"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1534"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1535"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi_bc" class="idref" href="#Ackermann.T3lt_psi_bc"><span class="id" title="lemma">T3lt_psi_bc</span></a>  <a id="a:1536" class="idref" href="#a:1536"><span class="id" title="binder">a</span></a> <a id="b:1537" class="idref" href="#b:1537"><span class="id" title="binder">b</span></a> <a id="c:1538" class="idref" href="#c:1538"><span class="id" title="binder">c</span></a>: (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1537"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1536"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1537"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1538"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1538"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1536"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1537"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1538"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.psi_lt1" class="idref" href="#Ackermann.psi_lt1"><span class="id" title="lemma">psi_lt1</span></a> <a id="a:1543" class="idref" href="#a:1543"><span class="id" title="binder">a</span></a> <a id="b:1544" class="idref" href="#b:1544"><span class="id" title="binder">b</span></a> <a id="c:1545" class="idref" href="#c:1545"><span class="id" title="binder">c</span></a> <a id="d:1546" class="idref" href="#d:1546"><span class="id" title="binder">d</span></a> <a id="n:1547" class="idref" href="#n:1547"><span class="id" title="binder">n</span></a> <a id="a':1548" class="idref" href="#a':1548"><span class="id" title="binder">a'</span></a> <a id="b':1549" class="idref" href="#b':1549"><span class="id" title="binder">b'</span></a> <a id="c':1550" class="idref" href="#c':1550"><span class="id" title="binder">c'</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1543"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1544"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1545"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1547"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1546"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1548"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1549"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1550"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1543"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1544"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1545"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1548"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1549"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1550"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.psi_lt2" class="idref" href="#Ackermann.psi_lt2"><span class="id" title="lemma">psi_lt2</span></a> <a id="a:1551" class="idref" href="#a:1551"><span class="id" title="binder">a</span></a> <a id="b:1552" class="idref" href="#b:1552"><span class="id" title="binder">b</span></a> <a id="c:1553" class="idref" href="#c:1553"><span class="id" title="binder">c</span></a> <a id="n:1554" class="idref" href="#n:1554"><span class="id" title="binder">n</span></a> <a id="d:1555" class="idref" href="#d:1555"><span class="id" title="binder">d</span></a> <a id="n':1556" class="idref" href="#n':1556"><span class="id" title="binder">n'</span></a> <a id="d':1557" class="idref" href="#d':1557"><span class="id" title="binder">d'</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1551"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1552"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1553"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1556"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1557"><span class="id" title="variable">d'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1551"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1552"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1553"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1554"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1555"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n':1556"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1554"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1556"><span class="id" title="variable">n'</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1554"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1557"><span class="id" title="variable">d'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1555"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi_b" class="idref" href="#Ackermann.T3lt_psi_b"><span class="id" title="lemma">T3lt_psi_b</span></a> <a id="a:1558" class="idref" href="#a:1558"><span class="id" title="binder">a</span></a> <a id="b:1559" class="idref" href="#b:1559"><span class="id" title="binder">b</span></a> <a id="c:1560" class="idref" href="#c:1560"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1559"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1558"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1559"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1560"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi_c" class="idref" href="#Ackermann.T3lt_psi_c"><span class="id" title="lemma">T3lt_psi_c</span></a> <a id="a:1561" class="idref" href="#a:1561"><span class="id" title="binder">a</span></a> <a id="b:1562" class="idref" href="#b:1562"><span class="id" title="binder">b</span></a> <a id="c:1563" class="idref" href="#c:1563"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1563"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1561"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1562"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1563"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_psi_a" class="idref" href="#Ackermann.T3lt_psi_a"><span class="id" title="lemma">T3lt_psi_a</span></a> <a id="a:1564" class="idref" href="#a:1564"><span class="id" title="binder">a</span></a> <a id="b:1565" class="idref" href="#b:1565"><span class="id" title="binder">b</span></a> <a id="c:1566" class="idref" href="#c:1566"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1564"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1564"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1565"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1566"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab460"></a><h2 class="section">Normal form</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3nf" class="idref" href="#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a id="x:1571" class="idref" href="#x:1571"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1571"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="var">d</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3nf:1572"><span class="id" title="definition">T3nf</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3nf:1572"><span class="id" title="definition">T3nf</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3nf:1572"><span class="id" title="definition">T3nf</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3nf:1572"><span class="id" title="definition">T3nf</span></a> <span class="id" title="var">d</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <span class="id" title="var">d</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_0" class="idref" href="#Ackermann.nf_0"><span class="id" title="lemma">nf_0</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_psi" class="idref" href="#Ackermann.nf_psi"><span class="id" title="lemma">nf_psi</span></a> <a id="a:1574" class="idref" href="#a:1574"><span class="id" title="binder">a</span></a> <a id="b:1575" class="idref" href="#b:1575"><span class="id" title="binder">b</span></a> <a id="c:1576" class="idref" href="#c:1576"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1574"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1575"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1576"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1574"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1575"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1576"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_int" class="idref" href="#Ackermann.nf_int"><span class="id" title="lemma">nf_int</span></a> <a id="n:1577" class="idref" href="#n:1577"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1577"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_cons_cons" class="idref" href="#Ackermann.nf_cons_cons"><span class="id" title="lemma">nf_cons_cons</span></a> <a id="a:1578" class="idref" href="#a:1578"><span class="id" title="binder">a</span></a> <a id="b:1579" class="idref" href="#b:1579"><span class="id" title="binder">b</span></a> <a id="c:1580" class="idref" href="#c:1580"><span class="id" title="binder">c</span></a> <a id="n:1581" class="idref" href="#n:1581"><span class="id" title="binder">n</span></a> <a id="a':1582" class="idref" href="#a':1582"><span class="id" title="binder">a'</span></a> <a id="b':1583" class="idref" href="#b':1583"><span class="id" title="binder">b'</span></a> <a id="c':1584" class="idref" href="#c':1584"><span class="id" title="binder">c'</span></a> <a id="n':1585" class="idref" href="#n':1585"><span class="id" title="binder">n'</span></a> <a id="d':1586" class="idref" href="#d':1586"><span class="id" title="binder">d'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1578"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1579"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1580"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1581"><span class="id" title="variable">n</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1582"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1583"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1584"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1585"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1586"><span class="id" title="variable">d'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1582"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1583"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1584"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1578"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1579"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1580"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1578"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1579"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1580"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1582"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1583"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1584"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1585"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1586"><span class="id" title="variable">d'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_consE" class="idref" href="#Ackermann.nf_consE"><span class="id" title="lemma">nf_consE</span></a> <a id="a:1587" class="idref" href="#a:1587"><span class="id" title="binder">a</span></a> <a id="b:1588" class="idref" href="#b:1588"><span class="id" title="binder">b</span></a> <a id="c:1589" class="idref" href="#c:1589"><span class="id" title="binder">c</span></a> <a id="n:1590" class="idref" href="#n:1590"><span class="id" title="binder">n</span></a> <a id="d:1591" class="idref" href="#d:1591"><span class="id" title="binder">d</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1587"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1588"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1589"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1590"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1591"><span class="id" title="variable">d</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1587"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1588"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1589"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1591"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1591"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1587"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1588"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1589"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_Wf" class="idref" href="#Ackermann.nf_Wf"><span class="id" title="lemma">nf_Wf</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#well_founded_P"><span class="id" title="definition">well_founded_P</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Ackermann.lt_not_wf" class="idref" href="#Ackermann.lt_not_wf"><span class="id" title="lemma">lt_not_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt"><span class="id" title="definition">T3lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab461"></a><h2 class="section">Successor Predecessor</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T1_T3" class="idref" href="#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a> (<a id="c:1647" class="idref" href="#c:1647"><span class="id" title="binder">c</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1"><span class="id" title="inductive">CantorOrdinal.T1</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1647"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons"><span class="id" title="constructor">CantorOrdinal.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1_T3:1648"><span class="id" title="definition">T1_T3</span></a> <span class="id" title="var">a</span>) <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T1_T3:1648"><span class="id" title="definition">T1_T3</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_epsilon0" class="idref" href="#Ackermann.lt_epsilon0"><span class="id" title="lemma">lt_epsilon0</span></a> <a id="a:1650" class="idref" href="#a:1650"><span class="id" title="binder">a</span></a> <a id="b:1651" class="idref" href="#b:1651"><span class="id" title="binder">b</span></a> <a id="c:1652" class="idref" href="#c:1652"><span class="id" title="binder">c</span></a> <a id="n:1653" class="idref" href="#n:1653"><span class="id" title="binder">n</span></a> <a id="d:1654" class="idref" href="#d:1654"><span class="id" title="binder">d</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1650"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1651"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1652"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1653"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1654"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1650"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1651"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1652"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T1T3_lt_epsilon0" class="idref" href="#Ackermann.T1T3_lt_epsilon0"><span class="id" title="lemma">T1T3_lt_epsilon0</span></a> <a id="x:1655" class="idref" href="#x:1655"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1655"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">cantor_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">ca</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Ackermann.::cantor_scope:x_'&lt;'_x" class="idref" href="#Ackermann.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt; y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt"><span class="id" title="definition">CantorOrdinal.T1lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">cantor_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T1T3_inc" class="idref" href="#Ackermann.T1T3_inc"><span class="id" title="lemma">T1T3_inc</span></a> <a id="x:1656" class="idref" href="#x:1656"><span class="id" title="binder">x</span></a> <a id="y:1657" class="idref" href="#y:1657"><span class="id" title="binder">y</span></a>: (<a class="idref" href="gaia_hydras.nfwfgaia.html#x:1656"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::cantor_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1657"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">ca</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1656"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1657"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.TT1T3_inj" class="idref" href="#Ackermann.TT1T3_inj"><span class="id" title="lemma">TT1T3_inj</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T1T3_surj" class="idref" href="#Ackermann.T1T3_surj"><span class="id" title="lemma">T1T3_surj</span></a> <a id="x:1658" class="idref" href="#x:1658"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1658"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1658"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:1659" class="idref" href="#y:1659"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1658"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3"><span class="id" title="definition">T1_T3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1659"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.all_zero" class="idref" href="#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <a id="a:1660" class="idref" href="#a:1660"><span class="id" title="binder">a</span></a> <a id="b:1661" class="idref" href="#b:1661"><span class="id" title="binder">b</span></a> <a id="c:1662" class="idref" href="#c:1662"><span class="id" title="binder">c</span></a> :=<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1660"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1661"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1662"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3limit" class="idref" href="#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a id="x:1663" class="idref" href="#x:1663"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1663"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">d</span><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3limit:1664"><span class="id" title="definition">T3limit</span></a> <span class="id" title="var">d</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3finite" class="idref" href="#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a id="x:1666" class="idref" href="#x:1666"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1666"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3split" class="idref" href="#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a id="x:1668" class="idref" href="#x:1668"><span class="id" title="binder">x</span></a>:=<br/>
+&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1668"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#T3split:1669"><span class="id" title="definition">T3split</span></a> <span class="id" title="var">d</span> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1668"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.all_zeroE" class="idref" href="#Ackermann.all_zeroE"><span class="id" title="lemma">all_zeroE</span></a> <a id="a:1671" class="idref" href="#a:1671"><span class="id" title="binder">a</span></a> <a id="b:1672" class="idref" href="#b:1672"><span class="id" title="binder">b</span></a> <a id="c:1673" class="idref" href="#c:1673"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1671"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1672"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1673"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1671"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1672"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1673"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3nf_finite" class="idref" href="#Ackermann.T3nf_finite"><span class="id" title="lemma">T3nf_finite</span></a> <a id="a:1674" class="idref" href="#a:1674"><span class="id" title="binder">a</span></a> <a id="b:1675" class="idref" href="#b:1675"><span class="id" title="binder">b</span></a> <a id="c:1676" class="idref" href="#c:1676"><span class="id" title="binder">c</span></a> <a id="n:1677" class="idref" href="#n:1677"><span class="id" title="binder">n</span></a> <a id="d:1678" class="idref" href="#d:1678"><span class="id" title="binder">d</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1674"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1675"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1676"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1674"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1675"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1676"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1677"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1678"><span class="id" title="variable">d</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#d:1678"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_finite" class="idref" href="#Ackermann.split_finite"><span class="id" title="lemma">split_finite</span></a> <a id="x:1679" class="idref" href="#x:1679"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1679"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1679"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3finite1" class="idref" href="#Ackermann.T3finite1"><span class="id" title="lemma">T3finite1</span></a> <a id="n:1680" class="idref" href="#n:1680"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1680"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3finite2" class="idref" href="#Ackermann.T3finite2"><span class="id" title="lemma">T3finite2</span></a> <a id="x:1681" class="idref" href="#x:1681"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1681"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1681"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1681"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1681"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3gt1" class="idref" href="#Ackermann.T3gt1"><span class="id" title="lemma">T3gt1</span></a> <a id="x:1682" class="idref" href="#x:1682"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1682"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1682"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1682"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.omega_least_inf1" class="idref" href="#Ackermann.omega_least_inf1"><span class="id" title="lemma">omega_least_inf1</span></a> <a id="x:1683" class="idref" href="#x:1683"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1683"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1683"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.omega_least_inf2" class="idref" href="#Ackermann.omega_least_inf2"><span class="id" title="lemma">omega_least_inf2</span></a> <a id="x:1684" class="idref" href="#x:1684"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1684"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1684"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_omega1" class="idref" href="#Ackermann.lt_omega1"><span class="id" title="lemma">lt_omega1</span></a> <a id="c:1685" class="idref" href="#c:1685"><span class="id" title="binder">c</span></a> <a id="n:1686" class="idref" href="#n:1686"><span class="id" title="binder">n</span></a> <a id="d:1687" class="idref" href="#d:1687"><span class="id" title="binder">d</span></a> <a id="a':1688" class="idref" href="#a':1688"><span class="id" title="binder">a'</span></a> <a id="b':1689" class="idref" href="#b':1689"><span class="id" title="binder">b'</span></a> <a id="c':1690" class="idref" href="#c':1690"><span class="id" title="binder">c'</span></a> <a id="n':1691" class="idref" href="#n':1691"><span class="id" title="binder">n'</span></a> <a id="d':1692" class="idref" href="#d':1692"><span class="id" title="binder">d'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1685"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1686"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1687"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a':1688"><span class="id" title="variable">a'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1689"><span class="id" title="variable">b'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1690"><span class="id" title="variable">c'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1691"><span class="id" title="variable">n'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1692"><span class="id" title="variable">d'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1688"><span class="id" title="variable">a'</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1689"><span class="id" title="variable">b'</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1685"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1690"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1685"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c':1690"><span class="id" title="variable">c'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a>(<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1686"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n':1691"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1686"><span class="id" title="variable">n</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#n':1691"><span class="id" title="variable">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#d:1687"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d':1692"><span class="id" title="variable">d'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1685"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1688"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1689"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1690"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_omega2" class="idref" href="#Ackermann.lt_omega2"><span class="id" title="lemma">lt_omega2</span></a> <a id="c:1693" class="idref" href="#c:1693"><span class="id" title="binder">c</span></a> <a id="a':1694" class="idref" href="#a':1694"><span class="id" title="binder">a'</span></a> <a id="b':1695" class="idref" href="#b':1695"><span class="id" title="binder">b'</span></a> <a id="c':1696" class="idref" href="#c':1696"><span class="id" title="binder">c'</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1693"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1694"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1695"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1696"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1694"><span class="id" title="variable">a'</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':1695"><span class="id" title="variable">b'</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1693"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1696"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1693"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':1694"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b':1695"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':1696"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_limit" class="idref" href="#Ackermann.split_limit"><span class="id" title="lemma">split_limit</span></a> <a id="x:1697" class="idref" href="#x:1697"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1697"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1697"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1697"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3is_succ" class="idref" href="#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a id="x:1698" class="idref" href="#x:1698"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1698"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#T3is_succ:1699"><span class="id" title="definition">T3is_succ</span></a> <span class="id" title="var">d</span>  <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3succ" class="idref" href="#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a id="x:1701" class="idref" href="#x:1701"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1701"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span><span class="id" title="notation">.+2</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T3succ:1702"><span class="id" title="definition">T3succ</span></a> <span class="id" title="var">d</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3pred" class="idref" href="#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a id="x:1704" class="idref" href="#x:1704"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1704"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero"><span class="id" title="definition">all_zero</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> (<a class="idref" href="gaia_hydras.nfwfgaia.html#T3pred:1705"><span class="id" title="definition">T3pred</span></a> <span class="id" title="var">d</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_is_succ" class="idref" href="#Ackermann.split_is_succ"><span class="id" title="lemma">split_is_succ</span></a> <a id="x:1707" class="idref" href="#x:1707"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1707"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.2'"><span class="id" title="notation">).2</span></a> <span class="id" title="notation">!=</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1707"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_succ" class="idref" href="#Ackermann.split_succ"><span class="id" title="lemma">split_succ</span></a> <a id="x:1708" class="idref" href="#x:1708"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1708"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1708"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_pred" class="idref" href="#Ackermann.split_pred"><span class="id" title="lemma">split_pred</span></a> <a id="x:1709" class="idref" href="#x:1709"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1709"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1709"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><span class="id" title="notation">.-1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_le" class="idref" href="#Ackermann.split_le"><span class="id" title="lemma">split_le</span></a> <a id="x:1710" class="idref" href="#x:1710"><span class="id" title="binder">x</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1710"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1710"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_split" class="idref" href="#Ackermann.nf_split"><span class="id" title="lemma">nf_split</span></a> <a id="x:1711" class="idref" href="#x:1711"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1711"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1711"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#::pair_scope:x_'.1'"><span class="id" title="notation">).1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3finite_succ" class="idref" href="#Ackermann.T3finite_succ"><span class="id" title="lemma">T3finite_succ</span></a> <a id="x:1712" class="idref" href="#x:1712"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1712"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1712"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T1succ_nat" class="idref" href="#Ackermann.T1succ_nat"><span class="id" title="lemma">T1succ_nat</span></a> <a id="n:1713" class="idref" href="#n:1713"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1713"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1713"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_omega" class="idref" href="#Ackermann.nf_omega"><span class="id" title="lemma">nf_omega</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_finite" class="idref" href="#Ackermann.nf_finite"><span class="id" title="lemma">nf_finite</span></a> <a id="n:1714" class="idref" href="#n:1714"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1714"><span class="id" title="variable">n</span></a>).  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit_pr1" class="idref" href="#Ackermann.limit_pr1"><span class="id" title="lemma">limit_pr1</span></a> <a id="x:1715" class="idref" href="#x:1715"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1715"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1715"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">(+)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1715"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#a60537c464e134477471443dd91ae651"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit_pr" class="idref" href="#Ackermann.limit_pr"><span class="id" title="lemma">limit_pr</span></a> <a id="x:1716" class="idref" href="#x:1716"><span class="id" title="binder">x</span></a> <a id="y:1717" class="idref" href="#y:1717"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1716"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1717"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1716"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1717"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1716"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.pred_le" class="idref" href="#Ackermann.pred_le"><span class="id" title="lemma">pred_le</span></a> <a id="a:1718" class="idref" href="#a:1718"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1718"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1718"><span class="id" title="variable">a</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.pred_lt" class="idref" href="#Ackermann.pred_lt"><span class="id" title="lemma">pred_lt</span></a> <a id="a:1719" class="idref" href="#a:1719"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1719"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1719"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1719"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_lt" class="idref" href="#Ackermann.succ_lt"><span class="id" title="lemma">succ_lt</span></a> <a id="a:1720" class="idref" href="#a:1720"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1720"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1720"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_succ" class="idref" href="#Ackermann.nf_succ"><span class="id" title="lemma">nf_succ</span></a> <a id="a:1721" class="idref" href="#a:1721"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1721"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1721"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_pred" class="idref" href="#Ackermann.nf_pred"><span class="id" title="lemma">nf_pred</span></a> <a id="a:1722" class="idref" href="#a:1722"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1722"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1722"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_pred" class="idref" href="#Ackermann.succ_pred"><span class="id" title="lemma">succ_pred</span></a> <a id="x:1723" class="idref" href="#x:1723"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1723"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1723"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1723"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1723"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_p1" class="idref" href="#Ackermann.succ_p1"><span class="id" title="lemma">succ_p1</span></a> <a id="x:1724" class="idref" href="#x:1724"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1724"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.pred_succ" class="idref" href="#Ackermann.pred_succ"><span class="id" title="lemma">pred_succ</span></a> <a id="x:1725" class="idref" href="#x:1725"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1725"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1725"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1725"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_inj" class="idref" href="#Ackermann.succ_inj"><span class="id" title="lemma">succ_inj</span></a> <a id="x:1726" class="idref" href="#x:1726"><span class="id" title="binder">x</span></a> <a id="y:1727" class="idref" href="#y:1727"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1726"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1727"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1726"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1727"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1726"><span class="id" title="variable">x</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1727"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_succ" class="idref" href="#Ackermann.lt_succ_succ"><span class="id" title="lemma">lt_succ_succ</span></a> <a id="x:1728" class="idref" href="#x:1728"><span class="id" title="binder">x</span></a> <a id="y:1729" class="idref" href="#y:1729"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1728"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1729"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1728"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1729"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.le_succ_succ" class="idref" href="#Ackermann.le_succ_succ"><span class="id" title="lemma">le_succ_succ</span></a> <a id="x:1730" class="idref" href="#x:1730"><span class="id" title="binder">x</span></a> <a id="y:1731" class="idref" href="#y:1731"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1730"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1731"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1730"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1731"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_succE" class="idref" href="#Ackermann.lt_succ_succE"><span class="id" title="lemma">lt_succ_succE</span></a> <a id="x:1732" class="idref" href="#x:1732"><span class="id" title="binder">x</span></a> <a id="y:1733" class="idref" href="#y:1733"><span class="id" title="binder">y</span></a>:  <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1732"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1733"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1732"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1733"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1732"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1733"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.le_succ_succE" class="idref" href="#Ackermann.le_succ_succE"><span class="id" title="lemma">le_succ_succE</span></a> <a id="x:1734" class="idref" href="#x:1734"><span class="id" title="binder">x</span></a> <a id="y:1735" class="idref" href="#y:1735"><span class="id" title="binder">y</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1734"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1735"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1734"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1735"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1734"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1735"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_le_1" class="idref" href="#Ackermann.lt_succ_le_1"><span class="id" title="lemma">lt_succ_le_1</span></a> <a id="a:1736" class="idref" href="#a:1736"><span class="id" title="binder">a</span></a> <a id="b:1737" class="idref" href="#b:1737"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1736"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1737"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1736"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1737"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_le_2" class="idref" href="#Ackermann.lt_succ_le_2"><span class="id" title="lemma">lt_succ_le_2</span></a> <a id="a:1738" class="idref" href="#a:1738"><span class="id" title="binder">a</span></a> <a id="b:1739" class="idref" href="#b:1739"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1738"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1738"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1739"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1738"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1739"><span class="id" title="variable">b</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_le_3" class="idref" href="#Ackermann.lt_succ_le_3"><span class="id" title="lemma">lt_succ_le_3</span></a> <a id="a:1740" class="idref" href="#a:1740"><span class="id" title="binder">a</span></a> <a id="b:1741" class="idref" href="#b:1741"><span class="id" title="binder">b</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1740"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1740"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1741"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1740"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1741"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.lt_succ_le_4" class="idref" href="#Ackermann.lt_succ_le_4"><span class="id" title="lemma">lt_succ_le_4</span></a> <a id="a:1742" class="idref" href="#a:1742"><span class="id" title="binder">a</span></a> <a id="b:1743" class="idref" href="#b:1743"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1743"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1742"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1743"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1742"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1743"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_nz" class="idref" href="#Ackermann.succ_nz"><span class="id" title="lemma">succ_nz</span></a> <a id="x:1744" class="idref" href="#x:1744"><span class="id" title="binder">x</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1744"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psi" class="idref" href="#Ackermann.succ_psi"><span class="id" title="lemma">succ_psi</span></a> <a id="a:1745" class="idref" href="#a:1745"><span class="id" title="binder">a</span></a> <a id="b:1746" class="idref" href="#b:1746"><span class="id" title="binder">b</span></a> <a id="c:1747" class="idref" href="#c:1747"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1745"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1746"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1747"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1745"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1746"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1747"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1745"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1746"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1747"><span class="id" title="variable">c</span></a> 0 <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psi_lt" class="idref" href="#Ackermann.succ_psi_lt"><span class="id" title="lemma">succ_psi_lt</span></a> <a id="x:1748" class="idref" href="#x:1748"><span class="id" title="binder">x</span></a> <a id="a:1749" class="idref" href="#a:1749"><span class="id" title="binder">a</span></a> <a id="b:1750" class="idref" href="#b:1750"><span class="id" title="binder">b</span></a> <a id="c:1751" class="idref" href="#c:1751"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1749"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1750"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1751"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#x:1748"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1749"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1750"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1751"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1748"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1749"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1750"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1751"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psi_lt2" class="idref" href="#Ackermann.succ_psi_lt2"><span class="id" title="lemma">succ_psi_lt2</span></a> <a id="a:1752" class="idref" href="#a:1752"><span class="id" title="binder">a</span></a> <a id="b:1753" class="idref" href="#b:1753"><span class="id" title="binder">b</span></a> <a id="c:1754" class="idref" href="#c:1754"><span class="id" title="binder">c</span></a> <a id="x:1755" class="idref" href="#x:1755"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1752"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1753"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1754"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1752"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1753"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1754"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1755"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1752"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1753"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1754"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1755"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab462"></a><h2 class="section">Addition</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3add" class="idref" href="#Ackermann.T3add"><span class="id" title="definition">T3add</span></a> <a id="x:1756" class="idref" href="#x:1756"><span class="id" title="binder">x</span></a> <a id="y:1757" class="idref" href="#y:1757"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1756"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1757"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span> <span class="id" title="var">n'</span> <span class="id" title="var">d'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1757"><span class="id" title="variable">y</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> (<span class="id" title="var">d</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1757"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">+</span><span class="id" title="var">n'</span><span class="id" title="notation">).+1</span> <span class="id" title="var">d'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1756"><span class="id" title="variable">x</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1757"><span class="id" title="variable">y</span></a><br/>
+&nbsp;<span class="id" title="keyword">where</span> <a id="07c58f65e5105c31dc2f9d874ea182dd" class="idref" href="#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">&quot;</span></a>x + y" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T3add:1758"><span class="id" title="definition">T3add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">ak_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.T3sub" class="idref" href="#Ackermann.T3sub"><span class="id" title="definition">T3sub</span></a> <a id="x:1761" class="idref" href="#x:1761"><span class="id" title="binder">x</span></a> <a id="y:1762" class="idref" href="#y:1762"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1761"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1762"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">c'</span> <span class="id" title="var">n'</span> <span class="id" title="var">d'</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1761"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1762"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c'</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1761"><span class="id" title="variable">x</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span><span class="id" title="notation">&lt;</span><span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<span class="id" title="var">n</span><span class="id" title="notation">==</span><span class="id" title="var">n'</span>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> (<span class="id" title="notation">(</span><span class="id" title="var">n</span><span class="id" title="notation">-</span><span class="id" title="var">n'</span><span class="id" title="notation">).-1</span>) <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">d</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">d'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="notation">(</span><span class="id" title="var">n</span> <span class="id" title="notation">-</span> <span class="id" title="var">n'</span><span class="id" title="notation">).-1</span> <span class="id" title="var">d</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1761"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><br/>
+<span class="id" title="keyword">where</span> <a id="Ackermann.::ak_scope:x_'-'_x" class="idref" href="#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">&quot;</span></a>a - b" := (<a class="idref" href="gaia_hydras.nfwfgaia.html#T3sub:1763"><span class="id" title="definition">T3sub</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">ak_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3subn0" class="idref" href="#Ackermann.T3subn0"><span class="id" title="lemma">T3subn0</span></a> <a id="x:1766" class="idref" href="#x:1766"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1766"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1766"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3sub0n" class="idref" href="#Ackermann.T3sub0n"><span class="id" title="lemma">T3sub0n</span></a> <a id="x:1767" class="idref" href="#x:1767"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1767"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.minus_lt" class="idref" href="#Ackermann.minus_lt"><span class="id" title="lemma">minus_lt</span></a> <a id="a:1768" class="idref" href="#a:1768"><span class="id" title="binder">a</span></a> <a id="b:1769" class="idref" href="#b:1769"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1768"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1769"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1768"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1769"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3subnn" class="idref" href="#Ackermann.T3subnn"><span class="id" title="lemma">T3subnn</span></a> <a id="x:1770" class="idref" href="#x:1770"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1770"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1770"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.minus_le" class="idref" href="#Ackermann.minus_le"><span class="id" title="lemma">minus_le</span></a> <a id="a:1771" class="idref" href="#a:1771"><span class="id" title="binder">a</span></a> <a id="b:1772" class="idref" href="#b:1772"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1771"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1772"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1771"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1772"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_sub" class="idref" href="#Ackermann.nf_sub"><span class="id" title="lemma">nf_sub</span></a> <a id="a:1773" class="idref" href="#a:1773"><span class="id" title="binder">a</span></a> <a id="b:1774" class="idref" href="#b:1774"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1773"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1774"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1773"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1774"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_int" class="idref" href="#Ackermann.sub_int"><span class="id" title="lemma">sub_int</span></a> <a id="n:1775" class="idref" href="#n:1775"><span class="id" title="binder">n</span></a> <a id="m:1776" class="idref" href="#m:1776"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1775"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1776"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1775"><span class="id" title="variable">n</span></a> <span class="id" title="notation">-</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1776"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_is_add_one" class="idref" href="#Ackermann.succ_is_add_one"><span class="id" title="lemma">succ_is_add_one</span></a> <a id="a:1777" class="idref" href="#a:1777"><span class="id" title="binder">a</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1777"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1777"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add1Nfin" class="idref" href="#Ackermann.add1Nfin"><span class="id" title="lemma">add1Nfin</span></a> <a id="a:1778" class="idref" href="#a:1778"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1778"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1778"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1778"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub1Nfin" class="idref" href="#Ackermann.sub1Nfin"><span class="id" title="lemma">sub1Nfin</span></a> <a id="a:1779" class="idref" href="#a:1779"><span class="id" title="binder">a</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite"><span class="id" title="definition">T3finite</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1779"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1779"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1779"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub1a" class="idref" href="#Ackermann.sub1a"><span class="id" title="lemma">sub1a</span></a> <a id="x:1780" class="idref" href="#x:1780"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1780"><span class="id" title="variable">x</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1780"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1780"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1780"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub1b" class="idref" href="#Ackermann.sub1b"><span class="id" title="lemma">sub1b</span></a> <a id="x:1781" class="idref" href="#x:1781"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1781"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1781"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1781"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_1aCE" class="idref" href="#Ackermann.sub_1aCE"><span class="id" title="lemma">sub_1aCE</span></a> (<a id="a:1782" class="idref" href="#a:1782"><span class="id" title="binder">a</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a>) : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1782"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">)</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1782"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_1bCE" class="idref" href="#Ackermann.sub_1bCE"><span class="id" title="lemma">sub_1bCE</span></a> (<a id="a:1783" class="idref" href="#a:1783"><span class="id" title="binder">a</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a>) : <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1783"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1783"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3add0n" class="idref" href="#Ackermann.T3add0n"><span class="id" title="lemma">T3add0n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3add"><span class="id" title="definition">T3add</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3addn0" class="idref" href="#Ackermann.T3addn0"><span class="id" title="lemma">T3addn0</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3add"><span class="id" title="definition">T3add</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_int" class="idref" href="#Ackermann.add_int"><span class="id" title="lemma">add_int</span></a> <a id="n:1784" class="idref" href="#n:1784"><span class="id" title="binder">n</span></a> <a id="m:1785" class="idref" href="#m:1785"><span class="id" title="binder">m</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1784"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1785"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1784"><span class="id" title="variable">n</span></a> <span class="id" title="notation">+</span><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1785"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_fin_omega" class="idref" href="#Ackermann.add_fin_omega"><span class="id" title="lemma">add_fin_omega</span></a> <a id="n:1786" class="idref" href="#n:1786"><span class="id" title="binder">n</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1786"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.fooCE" class="idref" href="#Ackermann.fooCE"><span class="id" title="lemma">fooCE</span></a> (<a id="x:1787" class="idref" href="#x:1787"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1787"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\(</span></a><span class="id" title="keyword">∀</span> <a id="u:1788" class="idref" href="#u:1788"><span class="id" title="binder">u</span></a> <a id="v:1789" class="idref" href="#v:1789"><span class="id" title="binder">v</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1788"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1787"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:1788"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:1789"><span class="id" title="variable">v</span></a><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.split_add" class="idref" href="#Ackermann.split_add"><span class="id" title="lemma">split_add</span></a> <a id="x:1790" class="idref" href="#x:1790"><span class="id" title="binder">x</span></a>: <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> :=<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3split"><span class="id" title="definition">T3split</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1790"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1790"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1790"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <span class="id" title="var">y</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">\</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3"><span class="id" title="notation">F</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_to_cons" class="idref" href="#Ackermann.add_to_cons"><span class="id" title="lemma">add_to_cons</span></a> <a id="a:1791" class="idref" href="#a:1791"><span class="id" title="binder">a</span></a> <a id="b:1792" class="idref" href="#b:1792"><span class="id" title="binder">b</span></a> <a id="c:1793" class="idref" href="#c:1793"><span class="id" title="binder">c</span></a> <a id="n:1794" class="idref" href="#n:1794"><span class="id" title="binder">n</span></a> <a id="d:1795" class="idref" href="#d:1795"><span class="id" title="binder">d</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#d:1795"><span class="id" title="variable">d</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1791"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1792"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1793"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1791"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1792"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1793"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1794"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1795"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1791"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1792"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1793"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1794"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:1795"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.addC_CE" class="idref" href="#Ackermann.addC_CE"><span class="id" title="lemma">addC_CE</span></a> (<a id="a:1796" class="idref" href="#a:1796"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>) (<a id="b:1797" class="idref" href="#b:1797"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1796"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1797"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1796"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1797"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1797"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1796"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.nf_add" class="idref" href="#Ackermann.nf_add"><span class="id" title="lemma">nf_add</span></a> <a id="a:1798" class="idref" href="#a:1798"><span class="id" title="binder">a</span></a> <a id="b:1799" class="idref" href="#b:1799"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1798"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1799"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1798"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1799"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3add_eq0" class="idref" href="#Ackermann.T3add_eq0"><span class="id" title="lemma">T3add_eq0</span></a> <a id="m:1800" class="idref" href="#m:1800"><span class="id" title="binder">m</span></a> <a id="n:1801" class="idref" href="#n:1801"><span class="id" title="binder">n</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1800"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1801"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1800"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1801"><span class="id" title="variable">n</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_le1" class="idref" href="#Ackermann.add_le1"><span class="id" title="lemma">add_le1</span></a> <a id="a:1802" class="idref" href="#a:1802"><span class="id" title="binder">a</span></a> <a id="b:1803" class="idref" href="#b:1803"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1802"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1802"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1803"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_le2" class="idref" href="#Ackermann.add_le2"><span class="id" title="lemma">add_le2</span></a> <a id="a:1804" class="idref" href="#a:1804"><span class="id" title="binder">a</span></a> <a id="b:1805" class="idref" href="#b:1805"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1805"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1804"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1805"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_le1" class="idref" href="#Ackermann.sub_le1"><span class="id" title="lemma">sub_le1</span></a> <a id="a:1806" class="idref" href="#a:1806"><span class="id" title="binder">a</span></a> <a id="b:1807" class="idref" href="#b:1807"><span class="id" title="binder">b</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1806"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1806"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1807"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1806"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_pr" class="idref" href="#Ackermann.sub_pr"><span class="id" title="lemma">sub_pr</span></a> <a id="a:1808" class="idref" href="#a:1808"><span class="id" title="binder">a</span></a> <a id="b:1809" class="idref" href="#b:1809"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1809"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1808"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1809"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1808"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1809"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_inj" class="idref" href="#Ackermann.add_inj"><span class="id" title="lemma">add_inj</span></a> <a id="a:1810" class="idref" href="#a:1810"><span class="id" title="binder">a</span></a> <a id="b:1811" class="idref" href="#b:1811"><span class="id" title="binder">b</span></a> <a id="c:1812" class="idref" href="#c:1812"><span class="id" title="binder">c</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1811"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1812"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1810"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1811"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1810"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1812"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1811"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1812"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_pr1" class="idref" href="#Ackermann.sub_pr1"><span class="id" title="lemma">sub_pr1</span></a> <a id="a:1813" class="idref" href="#a:1813"><span class="id" title="binder">a</span></a> <a id="b:1814" class="idref" href="#b:1814"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1814"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1813"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1814"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1814"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1813"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1814"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1813"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.omega_minus_one" class="idref" href="#Ackermann.omega_minus_one"><span class="id" title="lemma">omega_minus_one</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_nz" class="idref" href="#Ackermann.sub_nz"><span class="id" title="lemma">sub_nz</span></a> <a id="a:1815" class="idref" href="#a:1815"><span class="id" title="binder">a</span></a> <a id="b:1816" class="idref" href="#b:1816"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1816"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1815"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1816"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1816"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1815"><span class="id" title="variable">a</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3addA" class="idref" href="#Ackermann.T3addA"><span class="id" title="lemma">T3addA</span></a> <a id="c1:1817" class="idref" href="#c1:1817"><span class="id" title="binder">c1</span></a> <a id="c2:1818" class="idref" href="#c2:1818"><span class="id" title="binder">c2</span></a> <a id="c3:1819" class="idref" href="#c3:1819"><span class="id" title="binder">c3</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c1:1817"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c2:1818"><span class="id" title="variable">c2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:1819"><span class="id" title="variable">c3</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c1:1817"><span class="id" title="variable">c1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c2:1818"><span class="id" title="variable">c2</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c3:1819"><span class="id" title="variable">c3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_add2l" class="idref" href="#Ackermann.T3le_add2l"><span class="id" title="lemma">T3le_add2l</span></a>  <a id="p:1820" class="idref" href="#p:1820"><span class="id" title="binder">p</span></a> <a id="m:1821" class="idref" href="#m:1821"><span class="id" title="binder">m</span></a> <a id="n:1822" class="idref" href="#n:1822"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1820"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1821"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1820"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1822"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1821"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1822"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_add2l" class="idref" href="#Ackermann.T3lt_add2l"><span class="id" title="lemma">T3lt_add2l</span></a>  <a id="p:1823" class="idref" href="#p:1823"><span class="id" title="binder">p</span></a> <a id="m:1824" class="idref" href="#m:1824"><span class="id" title="binder">m</span></a> <a id="n:1825" class="idref" href="#n:1825"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1823"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1824"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1823"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1825"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1824"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1825"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3lt_add2r" class="idref" href="#Ackermann.T3lt_add2r"><span class="id" title="lemma">T3lt_add2r</span></a>  <a id="p:1826" class="idref" href="#p:1826"><span class="id" title="binder">p</span></a> <a id="m:1827" class="idref" href="#m:1827"><span class="id" title="binder">m</span></a> <a id="n:1828" class="idref" href="#n:1828"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1827"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1826"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1828"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1826"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1827"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1828"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3le_add2r" class="idref" href="#Ackermann.T3le_add2r"><span class="id" title="lemma">T3le_add2r</span></a>  <a id="p:1829" class="idref" href="#p:1829"><span class="id" title="binder">p</span></a> <a id="m:1830" class="idref" href="#m:1830"><span class="id" title="binder">m</span></a> <a id="n:1831" class="idref" href="#n:1831"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1830"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#n:1831"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1830"><span class="id" title="variable">m</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1829"><span class="id" title="variable">p</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1831"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1829"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3eq_add2l" class="idref" href="#Ackermann.T3eq_add2l"><span class="id" title="lemma">T3eq_add2l</span></a>  <a id="p:1832" class="idref" href="#p:1832"><span class="id" title="binder">p</span></a> <a id="m:1833" class="idref" href="#m:1833"><span class="id" title="binder">m</span></a> <a id="n:1834" class="idref" href="#n:1834"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#p:1832"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1833"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#p:1832"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1834"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#m:1833"><span class="id" title="variable">m</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1834"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_le3" class="idref" href="#Ackermann.add_le3"><span class="id" title="lemma">add_le3</span></a> <a id="a:1835" class="idref" href="#a:1835"><span class="id" title="binder">a</span></a> <a id="b:1836" class="idref" href="#b:1836"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1835"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1835"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1836"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1836"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_le4" class="idref" href="#Ackermann.add_le4"><span class="id" title="lemma">add_le4</span></a> <a id="a:1837" class="idref" href="#a:1837"><span class="id" title="binder">a</span></a> <a id="b:1838" class="idref" href="#b:1838"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1838"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1837"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1837"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1838"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_pr1r" class="idref" href="#Ackermann.sub_pr1r"><span class="id" title="lemma">sub_pr1r</span></a> <a id="a:1839" class="idref" href="#a:1839"><span class="id" title="binder">a</span></a> <a id="b:1840" class="idref" href="#b:1840"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1839"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1839"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1840"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1839"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1840"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sub_pr1rCE" class="idref" href="#Ackermann.sub_pr1rCE"><span class="id" title="lemma">sub_pr1rCE</span></a> (<a id="a:1841" class="idref" href="#a:1841"><span class="id" title="binder">a</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a>) (<a id="b:1842" class="idref" href="#b:1842"><span class="id" title="binder">b</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1841"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1842"><span class="id" title="variable">b</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1842"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1841"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.T3addS" class="idref" href="#Ackermann.T3addS"><span class="id" title="lemma">T3addS</span></a> <a id="a:1843" class="idref" href="#a:1843"><span class="id" title="binder">a</span></a> <a id="b:1844" class="idref" href="#b:1844"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1843"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1844"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1843"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1844"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab463"></a><h2 class="section">limit</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Notation</span> <a id="Ackermann.Tf" class="idref" href="#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.limit_of" class="idref" href="#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a id="f:1845" class="idref" href="#f:1845"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a>) <a id="x:1846" class="idref" href="#x:1846"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:1847" class="idref" href="#n:1847"><span class="id" title="binder">n</span></a> <a id="m:1848" class="idref" href="#m:1848"><span class="id" title="binder">m</span></a>, (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1847"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1848"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1845"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1847"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1845"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1848"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:1849" class="idref" href="#n:1849"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1845"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1849"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1846"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:1850" class="idref" href="#y:1850"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1850"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1850"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1846"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:1851" class="idref" href="#n:1851"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1850"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1845"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1851"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.fincP" class="idref" href="#Ackermann.fincP"><span class="id" title="lemma">fincP</span></a> (<a id="f:1852" class="idref" href="#f:1852"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:1853" class="idref" href="#n:1853"><span class="id" title="binder">n</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1852"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1853"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1852"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1853"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:1854" class="idref" href="#n:1854"><span class="id" title="binder">n</span></a> <a id="m:1855" class="idref" href="#m:1855"><span class="id" title="binder">m</span></a>, (<a class="idref" href="gaia_hydras.nfwfgaia.html#n:1854"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1855"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1852"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1854"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1852"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#m:1855"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.limit12_hyp" class="idref" href="#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a id="a:1856" class="idref" href="#a:1856"><span class="id" title="binder">a</span></a> <a id="b:1857" class="idref" href="#b:1857"><span class="id" title="binder">b</span></a> <a id="c:1858" class="idref" href="#c:1858"><span class="id" title="binder">c</span></a>:=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1858"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">b1</span> <span class="id" title="var">c1</span> <span class="id" title="var">n1</span> <span class="id" title="var">d1</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n1</span> <span class="id" title="notation">==</span> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">d1</span> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1856"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1857"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">b1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1856"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1857"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1858"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi0" class="idref" href="#Ackermann.phi0"><span class="id" title="definition">phi0</span></a>:= <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi1" class="idref" href="#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> <a id="a:1860" class="idref" href="#a:1860"><span class="id" title="binder">a</span></a> (<a id="f:1861" class="idref" href="#f:1861"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a>) := <span class="id" title="keyword">fun</span> <a id="n:1862" class="idref" href="#n:1862"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1860"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1861"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1862"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi5" class="idref" href="#Ackermann.phi5"><span class="id" title="definition">phi5</span></a> (<a id="f:1863" class="idref" href="#f:1863"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a>) := <span class="id" title="keyword">fun</span> <a id="n:1864" class="idref" href="#n:1864"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1863"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1864"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi12a" class="idref" href="#Ackermann.phi12a"><span class="id" title="definition">phi12a</span></a> <a id="a:1865" class="idref" href="#a:1865"><span class="id" title="binder">a</span></a> <a id="b:1866" class="idref" href="#b:1866"><span class="id" title="binder">b</span></a> (<a id="f:1867" class="idref" href="#f:1867"><span class="id" title="binder">f</span></a>:<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.Tf"><span class="id" title="abbreviation">Tf</span></a>) := <span class="id" title="keyword">fun</span> <a id="n:1868" class="idref" href="#n:1868"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1865"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1866"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1867"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1868"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit1" class="idref" href="#Ackermann.limit1"><span class="id" title="lemma">limit1</span></a> <a id="a:1869" class="idref" href="#a:1869"><span class="id" title="binder">a</span></a> <a id="b:1870" class="idref" href="#b:1870"><span class="id" title="binder">b</span></a> <a id="f:1871" class="idref" href="#f:1871"><span class="id" title="binder">f</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1869"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1871"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1870"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1869"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1871"><span class="id" title="variable">f</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#a:1869"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1870"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit5" class="idref" href="#Ackermann.limit5"><span class="id" title="lemma">limit5</span></a> <a id="f:1872" class="idref" href="#f:1872"><span class="id" title="binder">f</span></a> <a id="x:1873" class="idref" href="#x:1873"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1872"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1873"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi5"><span class="id" title="definition">phi5</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1872"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1873"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12a" class="idref" href="#Ackermann.limit12a"><span class="id" title="lemma">limit12a</span></a> <a id="f:1876" class="idref" href="#f:1876"><span class="id" title="binder">f</span></a> <a id="a:1877" class="idref" href="#a:1877"><span class="id" title="binder">a</span></a> <a id="b:1878" class="idref" href="#b:1878"><span class="id" title="binder">b</span></a> <a id="c:1879" class="idref" href="#c:1879"><span class="id" title="binder">c</span></a>:   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1877"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1878"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1879"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1876"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1879"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12a"><span class="id" title="definition">phi12a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1877"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1878"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1876"><span class="id" title="variable">f</span></a>)<a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1877"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1878"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1879"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi3" class="idref" href="#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a id="x:1882" class="idref" href="#x:1882"><span class="id" title="binder">x</span></a> <a id="n:1883" class="idref" href="#n:1883"><span class="id" title="binder">n</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1883"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi3:1884"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1882"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1883"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1882"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1882"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phi3v" class="idref" href="#Ackermann.phi3v"><span class="id" title="lemma">phi3v</span></a> <a id="a:1886" class="idref" href="#a:1886"><span class="id" title="binder">a</span></a> <a id="b:1887" class="idref" href="#b:1887"><span class="id" title="binder">b</span></a> <a id="c:1888" class="idref" href="#c:1888"><span class="id" title="binder">c</span></a> <a id="k:1889" class="idref" href="#k:1889"><span class="id" title="binder">k</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:1886"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:1887"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:1888"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:1889"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:1886"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:1887"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:1888"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:1889"><span class="id" title="variable">k</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit3" class="idref" href="#Ackermann.limit3"><span class="id" title="lemma">limit3</span></a> <a id="x:1890" class="idref" href="#x:1890"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1890"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1890"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit2" class="idref" href="#Ackermann.limit2"><span class="id" title="lemma">limit2</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12b1" class="idref" href="#Ackermann.limit12b1"><span class="id" title="lemma">limit12b1</span></a> <a id="x:1891" class="idref" href="#x:1891"><span class="id" title="binder">x</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1891"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1891"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1891"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi4" class="idref" href="#Ackermann.phi4"><span class="id" title="definition">phi4</span></a> <a id="x:1892" class="idref" href="#x:1892"><span class="id" title="binder">x</span></a> <a id="n:1893" class="idref" href="#n:1893"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1893"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1892"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi4:1894"><span class="id" title="definition">phi4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1892"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1893"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi4:1894"><span class="id" title="definition">phi4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1892"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1893"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1892"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit4" class="idref" href="#Ackermann.limit4"><span class="id" title="lemma">limit4</span></a> <a id="x:1896" class="idref" href="#x:1896"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi4"><span class="id" title="definition">phi4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1896"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1896"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi8" class="idref" href="#Ackermann.phi8"><span class="id" title="definition">phi8</span></a> <a id="x:1899" class="idref" href="#x:1899"><span class="id" title="binder">x</span></a> <a id="y:1900" class="idref" href="#y:1900"><span class="id" title="binder">y</span></a> <a id="n:1901" class="idref" href="#n:1901"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1901"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1899"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi8:1902"><span class="id" title="definition">phi8</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1899"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1900"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1901"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi8:1902"><span class="id" title="definition">phi8</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1899"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1900"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1901"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1899"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1900"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit8" class="idref" href="#Ackermann.limit8"><span class="id" title="lemma">limit8</span></a> <a id="x:1904" class="idref" href="#x:1904"><span class="id" title="binder">x</span></a> <a id="y:1905" class="idref" href="#y:1905"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi8"><span class="id" title="definition">phi8</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1904"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1905"><span class="id" title="variable">y</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1904"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1905"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi12b2" class="idref" href="#Ackermann.phi12b2"><span class="id" title="definition">phi12b2</span></a> <a id="x:1908" class="idref" href="#x:1908"><span class="id" title="binder">x</span></a> <a id="y:1909" class="idref" href="#y:1909"><span class="id" title="binder">y</span></a> <a id="n:1910" class="idref" href="#n:1910"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1910"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1908"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b2:1911"><span class="id" title="definition">phi12b2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1908"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1909"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1910"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b2:1911"><span class="id" title="definition">phi12b2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1908"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1909"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1910"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1909"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12b2" class="idref" href="#Ackermann.limit12b2"><span class="id" title="lemma">limit12b2</span></a> <a id="x:1913" class="idref" href="#x:1913"><span class="id" title="binder">x</span></a> <a id="y:1914" class="idref" href="#y:1914"><span class="id" title="binder">y</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1913"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1914"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b2"><span class="id" title="definition">phi12b2</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1913"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1914"><span class="id" title="variable">y</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1913"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1914"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi6" class="idref" href="#Ackermann.phi6"><span class="id" title="definition">phi6</span></a> <a id="x:1917" class="idref" href="#x:1917"><span class="id" title="binder">x</span></a> <a id="y:1918" class="idref" href="#y:1918"><span class="id" title="binder">y</span></a> <a id="n:1919" class="idref" href="#n:1919"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1919"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1917"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1918"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi6:1920"><span class="id" title="definition">phi6</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1917"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1918"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1919"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1917"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1918"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi10" class="idref" href="#Ackermann.phi10"><span class="id" title="definition">phi10</span></a> <a id="x:1922" class="idref" href="#x:1922"><span class="id" title="binder">x</span></a> <a id="y:1923" class="idref" href="#y:1923"><span class="id" title="binder">y</span></a> <a id="z:1924" class="idref" href="#z:1924"><span class="id" title="binder">z</span></a> <a id="n:1925" class="idref" href="#n:1925"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1925"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1922"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1923"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi10:1926"><span class="id" title="definition">phi10</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1922"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1923"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1924"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1925"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1922"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1923"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:1924"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi12b4" class="idref" href="#Ackermann.phi12b4"><span class="id" title="definition">phi12b4</span></a> <a id="x:1928" class="idref" href="#x:1928"><span class="id" title="binder">x</span></a> <a id="y:1929" class="idref" href="#y:1929"><span class="id" title="binder">y</span></a> <a id="z:1930" class="idref" href="#z:1930"><span class="id" title="binder">z</span></a> <a id="n:1931" class="idref" href="#n:1931"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1931"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1928"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1929"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b4:1932"><span class="id" title="definition">phi12b4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1928"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1929"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1930"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1931"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1930"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit6" class="idref" href="#Ackermann.limit6"><span class="id" title="lemma">limit6</span></a> <a id="x:1934" class="idref" href="#x:1934"><span class="id" title="binder">x</span></a> <a id="y:1935" class="idref" href="#y:1935"><span class="id" title="binder">y</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi6"><span class="id" title="definition">phi6</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1934"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1935"><span class="id" title="variable">y</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1934"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1935"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit10" class="idref" href="#Ackermann.limit10"><span class="id" title="lemma">limit10</span></a> <a id="x:1938" class="idref" href="#x:1938"><span class="id" title="binder">x</span></a> <a id="y:1939" class="idref" href="#y:1939"><span class="id" title="binder">y</span></a> <a id="z:1940" class="idref" href="#z:1940"><span class="id" title="binder">z</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi10"><span class="id" title="definition">phi10</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1938"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1939"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1940"><span class="id" title="variable">z</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1938"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1939"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1940"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12b4" class="idref" href="#Ackermann.limit12b4"><span class="id" title="lemma">limit12b4</span></a> <a id="x:1943" class="idref" href="#x:1943"><span class="id" title="binder">x</span></a> <a id="y:1944" class="idref" href="#y:1944"><span class="id" title="binder">y</span></a> <a id="z:1945" class="idref" href="#z:1945"><span class="id" title="binder">z</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1943"><span class="id" title="variable">x</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1944"><span class="id" title="variable">y</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1945"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b4"><span class="id" title="definition">phi12b4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1943"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1944"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1945"><span class="id" title="variable">z</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1943"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1944"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:1945"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi7" class="idref" href="#Ackermann.phi7"><span class="id" title="definition">phi7</span></a> <a id="x:1948" class="idref" href="#x:1948"><span class="id" title="binder">x</span></a> <a id="y:1949" class="idref" href="#y:1949"><span class="id" title="binder">y</span></a> <a id="f:1950" class="idref" href="#f:1950"><span class="id" title="binder">f</span></a> <a id="n:1951" class="idref" href="#n:1951"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1951"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1948"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1950"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1951"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi7:1952"><span class="id" title="definition">phi7</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1948"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1949"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1950"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1951"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1949"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi9" class="idref" href="#Ackermann.phi9"><span class="id" title="definition">phi9</span></a> <a id="x:1954" class="idref" href="#x:1954"><span class="id" title="binder">x</span></a> <a id="y:1955" class="idref" href="#y:1955"><span class="id" title="binder">y</span></a> <a id="f:1956" class="idref" href="#f:1956"><span class="id" title="binder">f</span></a> <a id="n:1957" class="idref" href="#n:1957"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1957"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1956"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1957"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi9:1958"><span class="id" title="definition">phi9</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1954"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1955"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1956"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1957"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#phi9:1958"><span class="id" title="definition">phi9</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1954"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1955"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1956"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1957"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1954"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1955"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi11" class="idref" href="#Ackermann.phi11"><span class="id" title="definition">phi11</span></a> <a id="x:1960" class="idref" href="#x:1960"><span class="id" title="binder">x</span></a> <a id="y:1961" class="idref" href="#y:1961"><span class="id" title="binder">y</span></a> <a id="z:1962" class="idref" href="#z:1962"><span class="id" title="binder">z</span></a> <a id="f:1963" class="idref" href="#f:1963"><span class="id" title="binder">f</span></a> <a id="n:1964" class="idref" href="#n:1964"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1964"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1960"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1963"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1964"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi11:1965"><span class="id" title="definition">phi11</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1960"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1961"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1962"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1963"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1964"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1960"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1961"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:1962"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi12b3" class="idref" href="#Ackermann.phi12b3"><span class="id" title="definition">phi12b3</span></a> <a id="y:1967" class="idref" href="#y:1967"><span class="id" title="binder">y</span></a> <a id="f:1968" class="idref" href="#f:1968"><span class="id" title="binder">f</span></a> <a id="n:1969" class="idref" href="#n:1969"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1969"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1968"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1969"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b3:1970"><span class="id" title="definition">phi12b3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1967"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1968"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1969"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b3:1970"><span class="id" title="definition">phi12b3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1967"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1968"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1969"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1967"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phi12b5" class="idref" href="#Ackermann.phi12b5"><span class="id" title="definition">phi12b5</span></a> <a id="x:1972" class="idref" href="#x:1972"><span class="id" title="binder">x</span></a> <a id="z:1973" class="idref" href="#z:1973"><span class="id" title="binder">z</span></a> <a id="f:1974" class="idref" href="#f:1974"><span class="id" title="binder">f</span></a> <a id="n:1975" class="idref" href="#n:1975"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1975"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1972"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#f:1974"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1975"><span class="id" title="variable">n</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#phi12b5:1976"><span class="id" title="definition">phi12b5</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1972"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1973"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1974"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:1975"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1973"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit7" class="idref" href="#Ackermann.limit7"><span class="id" title="lemma">limit7</span></a> <a id="x:1978" class="idref" href="#x:1978"><span class="id" title="binder">x</span></a> <a id="y:1979" class="idref" href="#y:1979"><span class="id" title="binder">y</span></a> <a id="f:1980" class="idref" href="#f:1980"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1980"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1979"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> ( <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi7"><span class="id" title="definition">phi7</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1978"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1979"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1980"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1978"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:1979"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit9" class="idref" href="#Ackermann.limit9"><span class="id" title="lemma">limit9</span></a> <a id="x:1983" class="idref" href="#x:1983"><span class="id" title="binder">x</span></a> <a id="y:1984" class="idref" href="#y:1984"><span class="id" title="binder">y</span></a> <a id="f:1985" class="idref" href="#f:1985"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1985"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1983"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi9"><span class="id" title="definition">phi9</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1983"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1984"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1985"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1983"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1984"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit11" class="idref" href="#Ackermann.limit11"><span class="id" title="lemma">limit11</span></a> <a id="x:1988" class="idref" href="#x:1988"><span class="id" title="binder">x</span></a> <a id="y:1989" class="idref" href="#y:1989"><span class="id" title="binder">y</span></a> <a id="z:1990" class="idref" href="#z:1990"><span class="id" title="binder">z</span></a> <a id="f:1991" class="idref" href="#f:1991"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1991"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1989"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi11"><span class="id" title="definition">phi11</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1988"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1989"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1990"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1991"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1988"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1989"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:1990"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12b3" class="idref" href="#Ackermann.limit12b3"><span class="id" title="lemma">limit12b3</span></a> <a id="x:1994" class="idref" href="#x:1994"><span class="id" title="binder">x</span></a> <a id="y:1995" class="idref" href="#y:1995"><span class="id" title="binder">y</span></a> <a id="f:1996" class="idref" href="#f:1996"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1996"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1994"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1994"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1995"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b3"><span class="id" title="definition">phi12b3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1995"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:1996"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1994"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:1995"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.limit12b5" class="idref" href="#Ackermann.limit12b5"><span class="id" title="lemma">limit12b5</span></a> <a id="x:1999" class="idref" href="#x:1999"><span class="id" title="binder">x</span></a> <a id="y:2000" class="idref" href="#y:2000"><span class="id" title="binder">y</span></a> <a id="z:2001" class="idref" href="#z:2001"><span class="id" title="binder">z</span></a> <a id="f:2002" class="idref" href="#f:2002"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2002"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2000"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1999"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2000"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2001"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b5"><span class="id" title="definition">phi12b5</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:1999"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2001"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2002"><span class="id" title="variable">f</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:1999"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#y:2000"><span class="id" title="variable">y</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:2001"><span class="id" title="variable">z</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi_rec_psi" class="idref" href="#Ackermann.phi_rec_psi"><span class="id" title="definition">phi_rec_psi</span></a> <a id="f:2005" class="idref" href="#f:2005"><span class="id" title="binder">f</span></a> <a id="a:2006" class="idref" href="#a:2006"><span class="id" title="binder">a</span></a> <a id="b:2007" class="idref" href="#b:2007"><span class="id" title="binder">b</span></a> <a id="c:2008" class="idref" href="#c:2008"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi0"><span class="id" title="definition">phi0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi4"><span class="id" title="definition">phi4</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi5"><span class="id" title="definition">phi5</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi6"><span class="id" title="definition">phi6</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi7"><span class="id" title="definition">phi7</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi8"><span class="id" title="definition">phi8</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi9"><span class="id" title="definition">phi9</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi10"><span class="id" title="definition">phi10</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi11"><span class="id" title="definition">phi11</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>) <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp"><span class="id" title="definition">limit12_hyp</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b2"><span class="id" title="definition">phi12b2</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b3"><span class="id" title="definition">phi12b3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ"><span class="id" title="definition">T3is_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b4"><span class="id" title="definition">phi12b4</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred"><span class="id" title="definition">T3pred</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b5"><span class="id" title="definition">phi12b5</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi12a"><span class="id" title="definition">phi12a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2006"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2007"><span class="id" title="variable">b</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2005"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2008"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi_rec" class="idref" href="#Ackermann.phi_rec"><span class="id" title="definition">phi_rec</span></a> <a id="f:2009" class="idref" href="#f:2009"><span class="id" title="binder">f</span></a> (<a id="x:2010" class="idref" href="#x:2010"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2010"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">d</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="var">n</span> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec_psi"><span class="id" title="definition">phi_rec_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2009"><span class="id" title="variable">f</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec_psi"><span class="id" title="definition">phi_rec_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2009"><span class="id" title="variable">f</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2009"><span class="id" title="variable">f</span></a> <span class="id" title="var">d</span>) <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi0"><span class="id" title="definition">phi0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ackermann.phia" class="idref" href="#Ackermann.phia"><span class="id" title="definition">phia</span></a> <a id="k:2013" class="idref" href="#k:2013"><span class="id" title="binder">k</span></a> := <span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:2013"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">k</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec"><span class="id" title="definition">phi_rec</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#phia:2014"><span class="id" title="definition">phia</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#k:2013"><span class="id" title="variable">k</span></a>) <span class="id" title="keyword">else</span> (<span class="id" title="keyword">fun</span> <a id="x:2016" class="idref" href="#x:2016"><span class="id" title="binder">x</span></a> ⇒<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi0"><span class="id" title="definition">phi0</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.phi" class="idref" href="#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a id="x:2017" class="idref" href="#x:2017"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phia"><span class="id" title="definition">phia</span></a> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.size"><span class="id" title="definition">size</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2017"><span class="id" title="variable">x</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2017"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phiE" class="idref" href="#Ackermann.phiE"><span class="id" title="lemma">phiE</span></a> <a id="x:2018" class="idref" href="#x:2018"><span class="id" title="binder">x</span></a> : <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2018"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec"><span class="id" title="definition">phi_rec</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2018"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phiE_1" class="idref" href="#Ackermann.phiE_1"><span class="id" title="lemma">phiE_1</span></a> <a id="a:2021" class="idref" href="#a:2021"><span class="id" title="binder">a</span></a> <a id="b:2022" class="idref" href="#b:2022"><span class="id" title="binder">b</span></a> <a id="c:2023" class="idref" href="#c:2023"><span class="id" title="binder">c</span></a> <a id="n:2024" class="idref" href="#n:2024"><span class="id" title="binder">n</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2021"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2022"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2023"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2024"><span class="id" title="variable">n</span></a><span class="id" title="notation">.+1</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2021"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2022"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2023"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2024"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2021"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2022"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2023"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phiE_2" class="idref" href="#Ackermann.phiE_2"><span class="id" title="lemma">phiE_2</span></a> <a id="a:2025" class="idref" href="#a:2025"><span class="id" title="binder">a</span></a> <a id="b:2026" class="idref" href="#b:2026"><span class="id" title="binder">b</span></a> <a id="c:2027" class="idref" href="#c:2027"><span class="id" title="binder">c</span></a> <a id="n:2028" class="idref" href="#n:2028"><span class="id" title="binder">n</span></a> <a id="d:2029" class="idref" href="#d:2029"><span class="id" title="binder">d</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#d:2029"><span class="id" title="variable">d</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2025"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2026"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2027"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2028"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:2029"><span class="id" title="variable">d</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi1"><span class="id" title="definition">phi1</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2025"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2026"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2027"><span class="id" title="variable">c</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#n:2028"><span class="id" title="variable">n</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#d:2029"><span class="id" title="variable">d</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phiE_3" class="idref" href="#Ackermann.phiE_3"><span class="id" title="lemma">phiE_3</span></a> <a id="a:2030" class="idref" href="#a:2030"><span class="id" title="binder">a</span></a> <a id="b:2031" class="idref" href="#b:2031"><span class="id" title="binder">b</span></a> <a id="c:2032" class="idref" href="#c:2032"><span class="id" title="binder">c</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2030"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2031"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2032"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec_psi"><span class="id" title="definition">phi_rec_psi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2030"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2031"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2032"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.phiL" class="idref" href="#Ackermann.phiL"><span class="id" title="lemma">phiL</span></a> <a id="x:2033" class="idref" href="#x:2033"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2033"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2033"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2033"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2033"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.conc1" class="idref" href="#Ackermann.conc1"><span class="id" title="lemma">conc1</span></a> (<a id="x:2034" class="idref" href="#x:2034"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of"><span class="id" title="definition">limit_of</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.phi3"><span class="id" title="definition">phi3</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0"><span class="id" title="definition">epsilon0</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2034"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab464"></a><h2 class="section">additive principal</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.T3ap" class="idref" href="#Ackermann.T3ap"><span class="id" title="definition">T3ap</span></a> <a id="x:2035" class="idref" href="#x:2035"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2035"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="keyword">then</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><span class="id" title="notation">==</span>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">d</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr0" class="idref" href="#Ackermann.ap_pr0"><span class="id" title="lemma">ap_pr0</span></a> <a id="a:2037" class="idref" href="#a:2037"><span class="id" title="binder">a</span></a> <a id="b:2038" class="idref" href="#b:2038"><span class="id" title="binder">b</span></a> <a id="c:2039" class="idref" href="#c:2039"><span class="id" title="binder">c</span></a> (<a id="x:2040" class="idref" href="#x:2040"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2037"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2038"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2039"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>) <a id="u:2041" class="idref" href="#u:2041"><span class="id" title="binder">u</span></a> <a id="v:2042" class="idref" href="#v:2042"><span class="id" title="binder">v</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#u:2041"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2040"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:2042"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2040"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:2041"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:2042"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2040"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_limit" class="idref" href="#Ackermann.ap_limit"><span class="id" title="lemma">ap_limit</span></a> <a id="x:2045" class="idref" href="#x:2045"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap"><span class="id" title="definition">T3ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2045"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:2045"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2045"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr1" class="idref" href="#Ackermann.ap_pr1"><span class="id" title="lemma">ap_pr1</span></a> <a id="c:2046" class="idref" href="#c:2046"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:2047" class="idref" href="#a:2047"><span class="id" title="binder">a</span></a> <a id="b:2048" class="idref" href="#b:2048"><span class="id" title="binder">b</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2047"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2046"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2048"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2046"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2047"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2048"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2046"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2046"><span class="id" title="variable">c</span></a><span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap"><span class="id" title="definition">T3ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2046"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr2" class="idref" href="#Ackermann.ap_pr2"><span class="id" title="lemma">ap_pr2</span></a> <a id="c:2049" class="idref" href="#c:2049"><span class="id" title="binder">c</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:2050" class="idref" href="#a:2050"><span class="id" title="binder">a</span></a> <a id="b:2051" class="idref" href="#b:2051"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2050"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2051"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2050"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2051"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2050"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2051"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap"><span class="id" title="definition">T3ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2049"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr3" class="idref" href="#Ackermann.ap_pr3"><span class="id" title="lemma">ap_pr3</span></a> <a id="a:2053" class="idref" href="#a:2053"><span class="id" title="binder">a</span></a> <a id="b:2054" class="idref" href="#b:2054"><span class="id" title="binder">b</span></a> <a id="c:2055" class="idref" href="#c:2055"><span class="id" title="binder">c</span></a> <a id="y:2056" class="idref" href="#y:2056"><span class="id" title="binder">y</span></a> (<a id="x:2057" class="idref" href="#x:2057"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2053"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2054"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2055"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>): <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2056"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2057"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2056"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2057"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2057"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr4" class="idref" href="#Ackermann.ap_pr4"><span class="id" title="lemma">ap_pr4</span></a> <a id="x:2058" class="idref" href="#x:2058"><span class="id" title="binder">x</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:2059" class="idref" href="#b:2059"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2059"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2058"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2059"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2058"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2058"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#x:2058"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap"><span class="id" title="definition">T3ap</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2058"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.ap_pr2CE" class="idref" href="#Ackermann.ap_pr2CE"><span class="id" title="lemma">ap_pr2CE</span></a> (<a id="c:2060" class="idref" href="#c:2060"><span class="id" title="binder">c</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a> 1 <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="a:2061" class="idref" href="#a:2061"><span class="id" title="binder">a</span></a> <a id="b:2062" class="idref" href="#b:2062"><span class="id" title="binder">b</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2061"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2062"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2061"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2060"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2062"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2060"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2061"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd"><span class="id" title="notation">+</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2062"><span class="id" title="variable">b</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2060"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.psi_succ" class="idref" href="#Ackermann.psi_succ"><span class="id" title="definition">psi_succ</span></a> <a id="x:2068" class="idref" href="#x:2068"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2068"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="keyword">then</span> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span><span class="id" title="notation">==</span><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <span class="id" title="var">c</span><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2068"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.psi_succ_pr1" class="idref" href="#Ackermann.psi_succ_pr1"><span class="id" title="lemma">psi_succ_pr1</span></a> <a id="a:2070" class="idref" href="#a:2070"><span class="id" title="binder">a</span></a> <a id="b:2071" class="idref" href="#b:2071"><span class="id" title="binder">b</span></a> <a id="c:2072" class="idref" href="#c:2072"><span class="id" title="binder">c</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2070"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2071"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2072"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ"><span class="id" title="definition">psi_succ</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2070"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2071"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c:2072"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psi1" class="idref" href="#Ackermann.succ_psi1"><span class="id" title="lemma">succ_psi1</span></a> <a id="a:2073" class="idref" href="#a:2073"><span class="id" title="binder">a</span></a> <a id="b:2074" class="idref" href="#b:2074"><span class="id" title="binder">b</span></a> <a id="c:2075" class="idref" href="#c:2075"><span class="id" title="binder">c</span></a> (<a id="x:2076" class="idref" href="#x:2076"><span class="id" title="binder">x</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2073"><span class="id" title="variable">a</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#b:2074"><span class="id" title="variable">b</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c:2075"><span class="id" title="variable">c</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a:2073"><span class="id" title="variable">a</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b:2074"><span class="id" title="variable">b</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a':2077" class="idref" href="#a':2077"><span class="id" title="binder">a'</span></a> <a id="b':2078" class="idref" href="#b':2078"><span class="id" title="binder">b'</span></a> <a id="c':2079" class="idref" href="#c':2079"><span class="id" title="binder">c'</span></a>,  <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2076"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':2077"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':2078"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':2079"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ"><span class="id" title="definition">psi_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2076"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':2077"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':2078"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':2079"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psi2" class="idref" href="#Ackermann.succ_psi2"><span class="id" title="lemma">succ_psi2</span></a> <a id="u:2080" class="idref" href="#u:2080"><span class="id" title="binder">u</span></a> (<a id="x:2081" class="idref" href="#x:2081"><span class="id" title="binder">x</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:2080"><span class="id" title="variable">u</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="a':2082" class="idref" href="#a':2082"><span class="id" title="binder">a'</span></a> <a id="b':2083" class="idref" href="#b':2083"><span class="id" title="binder">b'</span></a> <a id="c':2084" class="idref" href="#c':2084"><span class="id" title="binder">c'</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c':2084"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2081"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':2082"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':2083"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':2084"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ"><span class="id" title="definition">psi_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2081"><span class="id" title="variable">x</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">)</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a>  <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#a':2082"><span class="id" title="variable">a'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#b':2083"><span class="id" title="variable">b'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c':2084"><span class="id" title="variable">c'</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_prCE" class="idref" href="#Ackermann.succ_prCE"><span class="id" title="lemma">succ_prCE</span></a>  (<a id="u:2085" class="idref" href="#u:2085"><span class="id" title="binder">u</span></a>:= <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.one"><span class="id" title="definition">one</span></a>) (<a id="v:2086" class="idref" href="#v:2086"><span class="id" title="binder">v</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a>):  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#u:2085"><span class="id" title="variable">u</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#v:2086"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#v:2086"><span class="id" title="variable">v</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#u:2085"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.succ_psiCE" class="idref" href="#Ackermann.succ_psiCE"><span class="id" title="lemma">succ_psiCE</span></a> (<a id="z:2087" class="idref" href="#z:2087"><span class="id" title="binder">z</span></a> := <a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">[</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad"><span class="id" title="definition">T3bad</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613"><span class="id" title="notation">]</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2087"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#z:2087"><span class="id" title="variable">z</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ"><span class="id" title="definition">psi_succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.omega"><span class="id" title="definition">omega</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~(</span></a><a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2087"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.sup_of" class="idref" href="#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> (<a id="f:2088" class="idref" href="#f:2088"><span class="id" title="binder">f</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3"><span class="id" title="inductive">T3</span></a>) <a id="x:2089" class="idref" href="#x:2089"><span class="id" title="binder">x</span></a> <a id="z:2090" class="idref" href="#z:2090"><span class="id" title="binder">z</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2090"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:2091" class="idref" href="#y:2091"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2091"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2091"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2089"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2088"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2091"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7"><span class="id" title="notation">≤</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2090"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="z':2092" class="idref" href="#z':2092"><span class="id" title="binder">z'</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':2092"><span class="id" title="variable">z'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':2092"><span class="id" title="variable">z'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2090"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:2093" class="idref" href="#y:2093"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2093"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2093"><span class="id" title="variable">y</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2089"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':2092"><span class="id" title="variable">z'</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2088"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2093"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Ackermann.normal" class="idref" href="#Ackermann.normal"><span class="id" title="definition">normal</span></a> <a id="f:2094" class="idref" href="#f:2094"><span class="id" title="binder">f</span></a>:=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <span class="id" title="keyword">∀</span> <a id="x:2095" class="idref" href="#x:2095"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2095"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2094"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2095"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:2096" class="idref" href="#x:2096"><span class="id" title="binder">x</span></a> <a id="y:2097" class="idref" href="#y:2097"><span class="id" title="binder">y</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2096"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2097"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2096"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2097"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2094"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2096"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2094"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#y:2097"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:2098" class="idref" href="#x:2098"><span class="id" title="binder">x</span></a>, <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2098"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2098"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2094"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2098"><span class="id" title="variable">x</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2094"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2098"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sup_unique" class="idref" href="#Ackermann.sup_unique"><span class="id" title="lemma">sup_unique</span></a> <a id="f:2099" class="idref" href="#f:2099"><span class="id" title="binder">f</span></a> <a id="x:2100" class="idref" href="#x:2100"><span class="id" title="binder">x</span></a> <a id="z:2101" class="idref" href="#z:2101"><span class="id" title="binder">z</span></a> <a id="z':2102" class="idref" href="#z':2102"><span class="id" title="binder">z'</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2099"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2100"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2101"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2099"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2100"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':2102"><span class="id" title="variable">z'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z:2101"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#z':2102"><span class="id" title="variable">z'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sup_Oalpha_zero" class="idref" href="#Ackermann.sup_Oalpha_zero"><span class="id" title="lemma">sup_Oalpha_zero</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sup_Oalpha_limit" class="idref" href="#Ackermann.sup_Oalpha_limit"><span class="id" title="lemma">sup_Oalpha_limit</span></a> <a id="x:2103" class="idref" href="#x:2103"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2103"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2103"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2103"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2103"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.sup_Oalpha_succ" class="idref" href="#Ackermann.sup_Oalpha_succ"><span class="id" title="lemma">sup_Oalpha_succ</span></a> <a id="x:2104" class="idref" href="#x:2104"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2104"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of"><span class="id" title="definition">sup_of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ"><span class="id" title="definition">T3succ</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2104"><span class="id" title="variable">x</span></a>) <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2104"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.normal_id" class="idref" href="#Ackermann.normal_id"><span class="id" title="lemma">normal_id</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.normal_limit" class="idref" href="#Ackermann.normal_limit"><span class="id" title="lemma">normal_limit</span></a> <a id="f:2105" class="idref" href="#f:2105"><span class="id" title="binder">f</span></a> <a id="x:2106" class="idref" href="#x:2106"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2105"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2106"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2106"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit"><span class="id" title="definition">T3limit</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2105"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#x:2106"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.normal_compose" class="idref" href="#Ackermann.normal_compose"><span class="id" title="lemma">normal_compose</span></a> <a id="f:2107" class="idref" href="#f:2107"><span class="id" title="binder">f</span></a> <a id="g:2108" class="idref" href="#g:2108"><span class="id" title="binder">g</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#f:2107"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:2108"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#f:2107"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#g:2108"><span class="id" title="variable">g</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ackermann.add_normal" class="idref" href="#Ackermann.add_normal"><span class="id" title="lemma">add_normal</span></a> <a id="a:2109" class="idref" href="#a:2109"><span class="id" title="binder">a</span></a>:  <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf"><span class="id" title="definition">T3nf</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2109"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann.T3add"><span class="id" title="definition">T3add</span></a> <a class="idref" href="gaia_hydras.nfwfgaia.html#a:2109"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.nfwfgaia.html#Ackermann"><span class="id" title="module">Ackermann</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/gaia_hydras.onType.html b/theories/html/gaia_hydras.onType.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/gaia_hydras.onType.html
@@ -0,0 +1,289 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>gaia_hydras.onType</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library gaia_hydras.onType</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Ordinal Notations (Experimental !!!!)  Warning:  This file is a draft !!!   We try to adapt to Gaia the ON class of ordinal notations 
+   (defined as a naïve (pre Stdpp) type class).
+
+<div class="paragraph"> </div>
+
+ An Ordinal  Notation is just a well-founded ordered type, with 
+    a trichotomic comparison 
+
+<div class="paragraph"> </div>
+
+    Notions of ordinal arithmetics should be defined in substructures
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_ssreflect</span> <span class="id" title="library">zify</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">fintype</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Inverse_Image</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab472"></a><h1 class="section">Preliminaries</h1>
+ restriction of a decidable relation 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="restrict" class="idref" href="#restrict"><span class="id" title="definition">restrict</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>} (<a id="p:2" class="idref" href="#p:2"><span class="id" title="binder">p</span></a>: <a class="idref" href="gaia_hydras.onType.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="r:3" class="idref" href="#r:3"><span class="id" title="binder">r</span></a>: <a class="idref" href="gaia_hydras.onType.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.onType.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>):=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a> <a id="y:5" class="idref" href="#y:5"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.onType.html#p:2"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.onType.html#x:4"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">(</span></a><a class="idref" href="gaia_hydras.onType.html#p:2"><span class="id" title="variable">p</span></a> <a class="idref" href="gaia_hydras.onType.html#y:5"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="gaia_hydras.onType.html#r:3"><span class="id" title="variable">r</span></a> <a class="idref" href="gaia_hydras.onType.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.onType.html#y:5"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab473"></a><h2 class="section">Complements on orderType</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="MoreOrderType" class="idref" href="#MoreOrderType"><span class="id" title="section">MoreOrderType</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="MoreOrderType.disp" class="idref" href="#MoreOrderType.disp"><span class="id" title="variable">disp</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="MoreOrderType.T" class="idref" href="#MoreOrderType.T"><span class="id" title="variable">T</span></a>: <span class="id" title="abbreviation">orderType</span> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.disp"><span class="id" title="variable">disp</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="limit_v2" class="idref" href="#limit_v2"><span class="id" title="definition">limit_v2</span></a> (<a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>)(<a id="x:9" class="idref" href="#x:9"><span class="id" title="binder">x</span></a>: <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), (<a class="idref" href="gaia_hydras.onType.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.onType.html#n:10"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#x:9"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">O</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:11" class="idref" href="#y:11"><span class="id" title="binder">y</span></a> : <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>,  (<a class="idref" href="gaia_hydras.onType.html#y:11"><span class="id" title="variable">y</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#x:9"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> (<a class="idref" href="gaia_hydras.onType.html#y:11"><span class="id" title="variable">y</span></a> <span class="id" title="notation">≤</span> <a class="idref" href="gaia_hydras.onType.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.onType.html#n:12"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">O</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="limit_of" class="idref" href="#limit_of"><span class="id" title="definition">limit_of</span></a> (<a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>) <a id="x:14" class="idref" href="#x:14"><span class="id" title="binder">x</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> <a id="m:16" class="idref" href="#m:16"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="gaia_hydras.onType.html#n:15"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#m:16"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="gaia_hydras.onType.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.onType.html#n:15"><span class="id" title="variable">n</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.onType.html#m:16"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">O</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="gaia_hydras.onType.html#limit_v2"><span class="id" title="definition">limit_v2</span></a> <a class="idref" href="gaia_hydras.onType.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="gaia_hydras.onType.html#x:14"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="is_successor_of" class="idref" href="#is_successor_of"><span class="id" title="definition">is_successor_of</span></a>  (<a id="x:17" class="idref" href="#x:17"><span class="id" title="binder">x</span></a> <a id="y:18" class="idref" href="#y:18"><span class="id" title="binder">y</span></a>: <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>):= <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="gaia_hydras.onType.html#x:17"><span class="id" title="variable">x</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#y:18"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="z:19" class="idref" href="#z:19"><span class="id" title="binder">z</span></a>,  (<a class="idref" href="gaia_hydras.onType.html#x:17"><span class="id" title="variable">x</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#z:19"><span class="id" title="variable">z</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> (<a class="idref" href="gaia_hydras.onType.html#z:19"><span class="id" title="variable">z</span></a> <span class="id" title="notation">&lt;</span>  <a class="idref" href="gaia_hydras.onType.html#y:18"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">O</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="MoreOrderType.Succ_no_limit" class="idref" href="#MoreOrderType.Succ_no_limit"><span class="id" title="section">Succ_no_limit</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="MoreOrderType.Succ_no_limit.x" class="idref" href="#MoreOrderType.Succ_no_limit.x"><span class="id" title="variable">x</span></a> <a id="MoreOrderType.Succ_no_limit.y" class="idref" href="#MoreOrderType.Succ_no_limit.y"><span class="id" title="variable">y</span></a>: <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>) (<a id="MoreOrderType.Succ_no_limit.s" class="idref" href="#MoreOrderType.Succ_no_limit.s"><span class="id" title="variable">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.T"><span class="id" title="variable">T</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="MoreOrderType.Succ_no_limit.Hsucc" class="idref" href="#MoreOrderType.Succ_no_limit.Hsucc"><span class="id" title="variable">Hsucc</span></a> : <a class="idref" href="gaia_hydras.onType.html#is_successor_of"><span class="id" title="definition">is_successor_of</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.x"><span class="id" title="variable">x</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.y"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="MoreOrderType.Succ_no_limit.Hlim" class="idref" href="#MoreOrderType.Succ_no_limit.Hlim"><span class="id" title="variable">Hlim</span></a> : <a class="idref" href="gaia_hydras.onType.html#limit_of"><span class="id" title="definition">limit_of</span></a> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.s"><span class="id" title="variable">s</span></a>  <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.y"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R0" class="idref" href="#R0"><span class="id" title="lemma">R0</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a> (<a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.x"><span class="id" title="variable">x</span></a> <span class="id" title="notation">≤</span> <span class="id" title="notation">(</span><a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.s"><span class="id" title="variable">s</span></a> <a class="idref" href="gaia_hydras.onType.html#n:25"><span class="id" title="variable">n</span></a><span class="id" title="notation">)</span>)%<span class="id" title="var">O</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="z:26" class="idref" href="#z:26"><span class="id" title="binder">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> (<a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.x"><span class="id" title="variable">x</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#z:26"><span class="id" title="variable">z</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">O</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="F" class="idref" href="#F"><span class="id" title="lemma">F</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit"><span class="id" title="section">Succ_no_limit</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#MoreOrderType"><span class="id" title="section">MoreOrderType</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Type for ordinal notations 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="ONDef" class="idref" href="#ONDef"><span class="id" title="module">ONDef</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Record</span> <a id="ONDef.mixin_of" class="idref" href="#ONDef.mixin_of"><span class="id" title="record">mixin_of</span></a> <a id="disp:27" class="idref" href="#disp:27"><span class="id" title="binder">disp</span></a> (<a id="T:28" class="idref" href="#T:28"><span class="id" title="binder">T</span></a>: <span class="id" title="abbreviation">orderType</span> <a class="idref" href="gaia_hydras.onType.html#disp:27"><span class="id" title="variable">disp</span></a>)  :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="ONDef.Mixin" class="idref" href="#ONDef.Mixin"><span class="id" title="constructor">Mixin</span></a> {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">_</span> : @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.onType.html#T:28"><span class="id" title="variable">T</span></a> <span class="id" title="notation">&lt;%</span><span class="id" title="notation">O</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="ONDef.Packing" class="idref" href="#ONDef.Packing"><span class="id" title="section">Packing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ONDef.Packing.disp" class="idref" href="#ONDef.Packing.disp"><span class="id" title="variable">disp</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Structure</span> <a id="ONDef.pack_type" class="idref" href="#ONDef.pack_type"><span class="id" title="record">pack_type</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="ONDef.Pack" class="idref" href="#ONDef.Pack"><span class="id" title="constructor">Pack</span></a>{ <a id="ONDef.type" class="idref" href="#ONDef.type"><span class="id" title="projection">type</span></a>: <span class="id" title="abbreviation">orderType</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.Packing.disp"><span class="id" title="variable">disp</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="gaia_hydras.onType.html#ONDef.mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="gaia_hydras.onType.html#type:32"><span class="id" title="method">type</span></a>}.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ONDef.Packing.cT" class="idref" href="#ONDef.Packing.cT"><span class="id" title="variable">cT</span></a>: <a class="idref" href="gaia_hydras.onType.html#ONDef.pack_type"><span class="id" title="record">pack_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ONDef.on_struct" class="idref" href="#ONDef.on_struct"><span class="id" title="definition">on_struct</span></a>: <a class="idref" href="gaia_hydras.onType.html#ONDef.mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.Packing.cT"><span class="id" title="variable">cT</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="gaia_hydras.onType.html#ONDef.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> := <a class="idref" href="gaia_hydras.onType.html#ONDef.Packing.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="gaia_hydras.onType.html#cT:34"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.Packing"><span class="id" title="section">Packing</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="ONDef.Exports" class="idref" href="#ONDef.Exports"><span class="id" title="module">Exports</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="ONDef.Exports.on" class="idref" href="#ONDef.Exports.on"><span class="id" title="abbreviation">on</span></a> := (<a class="idref" href="gaia_hydras.onType.html#ONDef.pack_type"><span class="id" title="record">pack_type</span></a> ).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="ONDef.Exports.ONMixin" class="idref" href="#ONDef.Exports.ONMixin"><span class="id" title="abbreviation">ONMixin</span></a> := <a class="idref" href="gaia_hydras.onType.html#ONDef.Mixin"><span class="id" title="constructor">Mixin</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="ONDef.Exports.ON" class="idref" href="#ONDef.Exports.ON"><span class="id" title="abbreviation">ON</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="gaia_hydras.onType.html#ONDef.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Coercion</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.type"><span class="id" title="projection">type</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.type"><span class="id" title="projection">:</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.type"><span class="id" title="projection">pack_type</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.type"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.type"><span class="id" title="projection">orderType</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="ONDef.Exports.Lemmas" class="idref" href="#ONDef.Exports.Lemmas"><span class="id" title="section">Lemmas</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ONDef.Exports.Lemmas.disp" class="idref" href="#ONDef.Exports.Lemmas.disp"><span class="id" title="variable">disp</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="ONDef.Exports.Lemmas.U" class="idref" href="#ONDef.Exports.Lemmas.U"><span class="id" title="variable">U</span></a>: @<a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.on"><span class="id" title="abbreviation">on</span></a>  <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.disp"><span class="id" title="variable">disp</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ONDef.Exports.tricho" class="idref" href="#ONDef.Exports.tricho"><span class="id" title="definition">tricho</span></a> (<a id="a:37" class="idref" href="#a:37"><span class="id" title="binder">a</span></a> <a id="b:38" class="idref" href="#b:38"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.U"><span class="id" title="variable">U</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.onType.html#a:37"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.onType.html#b:38"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="gaia_hydras.onType.html#a:37"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#b:38"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ONDef.Exports.trichoP" class="idref" href="#ONDef.Exports.trichoP"><span class="id" title="lemma">trichoP</span></a> (<a id="a:39" class="idref" href="#a:39"><span class="id" title="binder">a</span></a> <a id="b:40" class="idref" href="#b:40"><span class="id" title="binder">b</span></a>: <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.U"><span class="id" title="variable">U</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompareSpec"><span class="id" title="inductive">CompareSpec</span></a> (<a class="idref" href="gaia_hydras.onType.html#a:39"><span class="id" title="variable">a</span></a> <span class="id" title="notation">==</span> <a class="idref" href="gaia_hydras.onType.html#b:40"><span class="id" title="variable">b</span></a>) (<a class="idref" href="gaia_hydras.onType.html#a:39"><span class="id" title="variable">a</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#b:40"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">O</span> (<a class="idref" href="gaia_hydras.onType.html#b:40"><span class="id" title="variable">b</span></a> <span class="id" title="notation">&lt;</span> <a class="idref" href="gaia_hydras.onType.html#a:39"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">O</span> (<a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.tricho"><span class="id" title="definition">tricho</span></a> <a class="idref" href="gaia_hydras.onType.html#a:39"><span class="id" title="variable">a</span></a> <a class="idref" href="gaia_hydras.onType.html#b:40"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ONDef.Exports.wf" class="idref" href="#ONDef.Exports.wf"><span class="id" title="lemma">wf</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.U"><span class="id" title="variable">U</span></a> <span class="id" title="notation">&lt;%</span><span class="id" title="notation">O</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports.Lemmas"><span class="id" title="section">Lemmas</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports"><span class="id" title="module">Exports</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#ONDef"><span class="id" title="module">ONDef</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Export</span> <a class="idref" href="gaia_hydras.onType.html#ONDef.Exports"><span class="id" title="module">ONDef.Exports</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab474"></a><h1 class="section">First instances of ON</h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab475"></a><h2 class="section">The Ordinal notation for 'I_i</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wf_ltn" class="idref" href="#wf_ltn"><span class="id" title="lemma">wf_ltn</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<span class="id" title="keyword">fun</span> <a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">eta</span></a> <span class="id" title="definition">ltn</span> <a class="idref" href="gaia_hydras.onType.html#n:41"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">]</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="onFiniteDef" class="idref" href="#onFiniteDef"><span class="id" title="section">onFiniteDef</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="onFiniteDef.i" class="idref" href="#onFiniteDef.i"><span class="id" title="variable">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="I_i_wf" class="idref" href="#I_i_wf"><span class="id" title="lemma">I_i_wf</span></a>: @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <span class="id" title="notation">'</span><span class="id" title="notation">I_i</span> (<span class="id" title="notation">&lt;%</span><span class="id" title="notation">O</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="onFiniteMixin" class="idref" href="#onFiniteMixin"><span class="id" title="definition">onFiniteMixin</span></a> := <a class="idref" href="gaia_hydras.onType.html#ONMixin"><span class="id" title="abbreviation">ONMixin</span></a> <a class="idref" href="gaia_hydras.onType.html#I_i_wf"><span class="id" title="lemma">I_i_wf</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <a id="onFiniteType" class="idref" href="#onFiniteType"><span class="id" title="definition">onFiniteType</span></a> := <a class="idref" href="gaia_hydras.onType.html#ON"><span class="id" title="abbreviation">ON</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.onType.html#onFiniteMixin"><span class="id" title="definition">onFiniteMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#onFiniteDef"><span class="id" title="section">onFiniteDef</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Tests 
+Definition O12_33: onFiniteType 33. 
+by exists 12.
+Defined. 
+
+<div class="paragraph"> </div>
+
+Compute tricho O12_33 O12_33.
+
+<div class="paragraph"> </div>
+
+Goal (O12_33 &lt;= O12_33)
+<div class="paragraph"> </div>
+
+<a id="lab476"></a><h2 class="section">An ordinal notation for omega</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="onOmegaDef" class="idref" href="#onOmegaDef"><span class="id" title="section">onOmegaDef</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="omega_lt_wf" class="idref" href="#omega_lt_wf"><span class="id" title="lemma">omega_lt_wf</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <span class="id" title="definition">Order.NatOrder.orderType</span> <span class="id" title="notation">&lt;%</span><span class="id" title="notation">O</span>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="onOmegaMixin" class="idref" href="#onOmegaMixin"><span class="id" title="definition">onOmegaMixin</span></a> := <a class="idref" href="gaia_hydras.onType.html#ONMixin"><span class="id" title="abbreviation">ONMixin</span></a> <a class="idref" href="gaia_hydras.onType.html#omega_lt_wf"><span class="id" title="lemma">omega_lt_wf</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <a id="onOmegaType" class="idref" href="#onOmegaType"><span class="id" title="definition">onOmegaType</span></a> := <a class="idref" href="gaia_hydras.onType.html#ON"><span class="id" title="abbreviation">ON</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="gaia_hydras.onType.html#onOmegaMixin"><span class="id" title="definition">onOmegaMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#onOmegaDef"><span class="id" title="section">onOmegaDef</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="om12" class="idref" href="#om12"><span class="id" title="definition">om12</span></a> : <a class="idref" href="gaia_hydras.onType.html#onOmegaType"><span class="id" title="definition">onOmegaType</span></a> := 12.<br/>
+<span class="id" title="keyword">Example</span> <a id="om67" class="idref" href="#om67"><span class="id" title="definition">om67</span></a> : <a class="idref" href="gaia_hydras.onType.html#onOmegaType"><span class="id" title="definition">onOmegaType</span></a> := 67.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+To do :  Notation for epsilon0 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">gaia_hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="gaia_hydras.T1Bridge.html#"><span class="id" title="library">T1Bridge</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ONEpsilon0Def" class="idref" href="#ONEpsilon0Def"><span class="id" title="section">ONEpsilon0Def</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="gaia_hydras.onType.html#ONEpsilon0Def"><span class="id" title="section">ONEpsilon0Def</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.Deduction.html b/theories/html/hydras.Ackermann.Deduction.html
new file mode 100644
index 00000000..e4d5af87
--- /dev/null
+++ b/theories/html/hydras.Ackermann.Deduction.html
@@ -0,0 +1,82 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.Deduction</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.Deduction</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ FOL's deduction Lemma 
+
+<div class="paragraph"> </div>
+
+  Original file by Russel O'Connor 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Deduction_Theorem" class="idref" href="#Deduction_Theorem"><span class="id" title="section">Deduction_Theorem</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Deduction_Theorem.L" class="idref" href="#Deduction_Theorem.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.Formula" class="idref" href="#Deduction_Theorem.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.Formulas" class="idref" href="#Deduction_Theorem.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.System" class="idref" href="#Deduction_Theorem.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.Term" class="idref" href="#Deduction_Theorem.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.Terms" class="idref" href="#Deduction_Theorem.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.Prf" class="idref" href="#Deduction_Theorem.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.SysPrf" class="idref" href="#Deduction_Theorem.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Deduction_Theorem.list_incl" class="idref" href="#Deduction_Theorem.list_incl"><span class="id" title="variable">list_incl</span></a> (<a id="X:2" class="idref" href="#X:2"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="l:3" class="idref" href="#l:3"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formula"><span class="id" title="variable">Formula</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#l:3"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.Deduction.html#X:2"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#x:4"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="SysPrf_rephrase" class="idref" href="#SysPrf_rephrase"><span class="id" title="lemma">SysPrf_rephrase</span></a> <a id="X:5" class="idref" href="#X:5"><span class="id" title="binder">X</span></a> <a id="F:6" class="idref" href="#F:6"><span class="id" title="binder">F</span></a> : <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#X:5"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#F:6"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s:7" class="idref" href="#s:7"><span class="id" title="binder">s</span></a> (<span class="id" title="var">_</span> : <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#s:7"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#F:6"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.list_incl"><span class="id" title="variable">list_incl</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#X:5"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#s:7"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="DeductionTheorem" class="idref" href="#DeductionTheorem"><span class="id" title="lemma">DeductionTheorem</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="T:8" class="idref" href="#T:8"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.System"><span class="id" title="variable">System</span></a>) (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> <a id="g:10" class="idref" href="#g:10"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formula"><span class="id" title="variable">Formula</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="prf:11" class="idref" href="#prf:11"><span class="id" title="binder">prf</span></a> : <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.Deduction.html#T:8"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#g:10"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.Deduction.html#f:9"><span class="id" title="variable">f</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#T:8"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.Deduction.html#g:10"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#f:9"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.Deduction.html#Deduction_Theorem"><span class="id" title="section">Deduction_Theorem</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
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+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.LNN.html b/theories/html/hydras.Ackermann.LNN.html
new file mode 100644
index 00000000..3f328da1
--- /dev/null
+++ b/theories/html/hydras.Ackermann.LNN.html
@@ -0,0 +1,395 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.LNN</title>
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+
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+
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+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.LNN</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ LNN.v : Language of Natural Numbers (<span class="inlinecode"><span class="id" title="var">LNT</span></span>+ <span class="inlinecode">&lt;</span>) 
+
+<div class="paragraph"> </div>
+
+Original version by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>  <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab274"></a><h1 class="section">Instantiations of a few generic constructs</h1>
+
+
+<div class="paragraph"> </div>
+
+  <i>To do</i> perhaps these redefinitions should be deprecated, because they cause some issues 
+  in statements which mix <span class="inlinecode"><span class="id" title="var">LNN</span></span> and <span class="inlinecode"><span class="id" title="var">LNT</span></span> terms and formulas 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="definition">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="definition">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="System" class="idref" href="#System"><span class="id" title="definition">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Sentence" class="idref" href="#Sentence"><span class="id" title="definition">Sentence</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#Sentence"><span class="id" title="definition">Sentence</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Term" class="idref" href="#Term"><span class="id" title="definition">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="definition">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="SysPrf" class="idref" href="#SysPrf"><span class="id" title="definition">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab275"></a><h1 class="section">Notations (Experimental and unstable)</h1>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">Module</span> <a id="NNnotations" class="idref" href="#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NNnotations.Plus" class="idref" href="#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> (<a id="x:1" class="idref" href="#x:1"><span class="id" title="binder">x</span></a> <a id="y:2" class="idref" href="#y:2"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Plus_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:1"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:2"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NNnotations.Times" class="idref" href="#NNnotations.Times"><span class="id" title="definition">Times</span></a> (<a id="x:3" class="idref" href="#x:3"><span class="id" title="binder">x</span></a> <a id="y:4" class="idref" href="#y:4"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Times_"><span class="id" title="constructor">Times_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:3"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:4"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NNnotations.Succ" class="idref" href="#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NNnotations.Zero" class="idref" href="#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NNnotations.LT" class="idref" href="#NNnotations.LT"><span class="id" title="definition">LT</span></a> (<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a> <a id="y:7" class="idref" href="#y:7"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LT_"><span class="id" title="constructor">LT_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:6"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:7"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="NNnotations.S_" class="idref" href="#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="138f411abfc162998ee0d11413343f95" class="idref" href="#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a>: <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="018d848f24ca8d774a2830f1a5e4a70f" class="idref" href="#018d848f24ca8d774a2830f1a5e4a70f"><span class="id" title="notation">&quot;</span></a>n + p" := (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">∀</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#018d848f24ca8d774a2830f1a5e4a70f"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#99fc41bec617cc902ba1cfab8f026785"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="d5e10e8b47737e9c618b2250abdff48c" class="idref" href="#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Times"><span class="id" title="definition">Times</span></a>: <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="3eb208196255ec4fdf2292f5d7aaaf16" class="idref" href="#3eb208196255ec4fdf2292f5d7aaaf16"><span class="id" title="notation">&quot;</span></a>n * p" := (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Times"><span class="id" title="definition">Times</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="NNnotations.::fol_scope:x_'&lt;'_x" class="idref" href="#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>&lt;" := <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.LT"><span class="id" title="definition">LT</span></a>: <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN_eqdec" class="idref" href="#LNN_eqdec"><span class="id" title="lemma">LNN_eqdec</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#language_decidable"><span class="id" title="definition">language_decidable</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Free_Variables" class="idref" href="#Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarPlus" class="idref" href="#freeVarPlus"><span class="id" title="lemma">freeVarPlus</span></a> (<a id="x:8" class="idref" href="#x:8"><span class="id" title="binder">x</span></a> <a id="y:9" class="idref" href="#y:9"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#Plus"><span class="id" title="definition">Plus</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:9"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:9"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarTimes" class="idref" href="#freeVarTimes"><span class="id" title="lemma">freeVarTimes</span></a> (<a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a> <a id="y:11" class="idref" href="#y:11"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#Times"><span class="id" title="definition">Times</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:11"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:11"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSucc" class="idref" href="#freeVarSucc"><span class="id" title="lemma">freeVarSucc</span></a> (<a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:12"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:12"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarZero" class="idref" href="#freeVarZero"><span class="id" title="lemma">freeVarZero</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarLT" class="idref" href="#freeVarLT"><span class="id" title="lemma">freeVarLT</span></a> (<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a> <a id="y:14" class="idref" href="#y:14"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:14"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#y:14"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNN.html#Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab276"></a><h1 class="section">Basic and derived properties</h1>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a id="Logic" class="idref" href="#Logic"><span class="id" title="section">Logic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Axm" class="idref" href="#Axm"><span class="id" title="lemma">Axm</span></a> (<a id="T:15" class="idref" href="#T:15"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:16" class="idref" href="#f:16"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>): <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:15"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:15"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:16"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysExtend" class="idref" href="#sysExtend"><span class="id" title="lemma">sysExtend</span></a> (<a id="T:17" class="idref" href="#T:17"><span class="id" title="binder">T</span></a> <a id="U:18" class="idref" href="#U:18"><span class="id" title="binder">U</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:19" class="idref" href="#f:19"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:17"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#U:18"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:17"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:19"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#U:18"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:19"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysWeaken" class="idref" href="#sysWeaken"><span class="id" title="lemma">sysWeaken</span></a> (<a id="T:20" class="idref" href="#T:20"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> <a id="g:22" class="idref" href="#g:22"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:20"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:20"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:22"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNN.html#f:21"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impI" class="idref" href="#impI"><span class="id" title="lemma">impI</span></a> (<a id="T:23" class="idref" href="#T:23"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:23"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:25"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNN.html#f:24"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:23"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:24"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impE" class="idref" href="#impE"><span class="id" title="lemma">impE</span></a>  (<a id="T:26" class="idref" href="#T:26"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> <a id="g:28" class="idref" href="#g:28"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:26"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:28"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:27"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:26"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:28"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:26"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:27"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="contradiction" class="idref" href="#contradiction"><span class="id" title="lemma">contradiction</span></a> (<a id="T:29" class="idref" href="#T:29"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> <a id="g:31" class="idref" href="#g:31"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:29"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:29"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:30"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:29"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:31"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnE" class="idref" href="#nnE"><span class="id" title="lemma">nnE</span></a> (<a id="T:32" class="idref" href="#T:32"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:33" class="idref" href="#f:33"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:32"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">~~</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:33"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:32"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:33"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="noMiddle" class="idref" href="#noMiddle"><span class="id" title="lemma">noMiddle</span></a> (<a id="T:34" class="idref" href="#T:34"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:35" class="idref" href="#f:35"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):  <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:34"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:35"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:35"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnI" class="idref" href="#nnI"><span class="id" title="lemma">nnI</span></a>  (<a id="T:36" class="idref" href="#T:36"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:36"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:36"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:37"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp1" class="idref" href="#cp1"><span class="id" title="lemma">cp1</span></a> (<a id="T:38" class="idref" href="#T:38"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:39" class="idref" href="#f:39"><span class="id" title="binder">f</span></a> <a id="g:40" class="idref" href="#g:40"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:39"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:40"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:40"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:39"><span class="id" title="variable">f</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp2" class="idref" href="#cp2"><span class="id" title="lemma">cp2</span></a> (<a id="T:41" class="idref" href="#T:41"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a> <a id="g:43" class="idref" href="#g:43"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:41"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:43"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:42"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:41"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:43"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI1" class="idref" href="#orI1"><span class="id" title="lemma">orI1</span></a> (<a id="T:44" class="idref" href="#T:44"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> <a id="g:46" class="idref" href="#g:46"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:44"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:44"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:46"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI2" class="idref" href="#orI2"><span class="id" title="lemma">orI2</span></a>  (<a id="T:47" class="idref" href="#T:47"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> <a id="g:49" class="idref" href="#g:49"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:47"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:49"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:47"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:49"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orE" class="idref" href="#orE"><span class="id" title="lemma">orE</span></a> (<a id="T:50" class="idref" href="#T:50"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:51" class="idref" href="#f:51"><span class="id" title="binder">f</span></a> <a id="g:52" class="idref" href="#g:52"><span class="id" title="binder">g</span></a> <a id="h:53" class="idref" href="#h:53"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:50"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:51"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:52"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:50"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:51"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:53"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:50"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:52"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:53"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:50"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:53"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSys" class="idref" href="#orSys"><span class="id" title="lemma">orSys</span></a>  (<a id="T:54" class="idref" href="#T:54"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:55" class="idref" href="#f:55"><span class="id" title="binder">f</span></a> <a id="g:56" class="idref" href="#g:56"><span class="id" title="binder">g</span></a> <a id="h:57" class="idref" href="#h:57"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:54"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:55"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNN.html#h:57"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:54"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:56"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNN.html#h:57"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNN.html#T:54"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:56"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>) <a class="idref" href="hydras.Ackermann.LNN.html#h:57"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andI" class="idref" href="#andI"><span class="id" title="lemma">andI</span></a> (<a id="T:58" class="idref" href="#T:58"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:59" class="idref" href="#f:59"><span class="id" title="binder">f</span></a> <a id="g:60" class="idref" href="#g:60"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:58"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:59"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:58"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:60"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:58"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:59"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:60"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE1" class="idref" href="#andE1"><span class="id" title="lemma">andE1</span></a> (<a id="T:61" class="idref" href="#T:61"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:62" class="idref" href="#f:62"><span class="id" title="binder">f</span></a> <a id="g:63" class="idref" href="#g:63"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:61"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:62"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:63"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:61"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:62"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE2" class="idref" href="#andE2"><span class="id" title="lemma">andE2</span></a>  (<a id="T:64" class="idref" href="#T:64"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:65" class="idref" href="#f:65"><span class="id" title="binder">f</span></a> <a id="g:66" class="idref" href="#g:66"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:64"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:65"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:66"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:64"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:66"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffI" class="idref" href="#iffI"><span class="id" title="lemma">iffI</span></a> (<a id="T:67" class="idref" href="#T:67"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:68" class="idref" href="#f:68"><span class="id" title="binder">f</span></a> <a id="g:69" class="idref" href="#g:69"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:68"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:69"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:69"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:68"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:68"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:69"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE1" class="idref" href="#iffE1"><span class="id" title="lemma">iffE1</span></a> (<a id="T:70" class="idref" href="#T:70"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:71" class="idref" href="#f:71"><span class="id" title="binder">f</span></a> <a id="g:72" class="idref" href="#g:72"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:70"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:71"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:72"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:70"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:71"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:72"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE2" class="idref" href="#iffE2"><span class="id" title="lemma">iffE2</span></a> (<a id="T:73" class="idref" href="#T:73"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a> <a id="g:75" class="idref" href="#g:75"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:73"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:74"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:75"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:73"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:75"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:74"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallI" class="idref" href="#forallI"><span class="id" title="lemma">forallI</span></a> (<a id="T:76" class="idref" href="#T:76"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:77" class="idref" href="#f:77"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:78" class="idref" href="#v:78"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:78"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:76"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:76"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:78"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:77"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallE" class="idref" href="#forallE"><span class="id" title="lemma">forallE</span></a>  (<a id="T:79" class="idref" href="#T:79"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:80" class="idref" href="#f:80"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:81" class="idref" href="#v:81"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:82" class="idref" href="#t:82"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:79"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:81"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:80"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:79"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:80"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:81"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#t:82"><span class="id" title="variable">t</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallSimp" class="idref" href="#forallSimp"><span class="id" title="lemma">forallSimp</span></a> (<a id="T:83" class="idref" href="#T:83"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:84" class="idref" href="#f:84"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:85" class="idref" href="#v:85"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:83"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:85"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:84"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:83"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:84"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existI" class="idref" href="#existI"><span class="id" title="lemma">existI</span></a> (<a id="T:86" class="idref" href="#T:86"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:87" class="idref" href="#f:87"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:88" class="idref" href="#v:88"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:89" class="idref" href="#t:89"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:86"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:87"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:88"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#t:89"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:86"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:88"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:87"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existE" class="idref" href="#existE"><span class="id" title="lemma">existE</span></a> (<a id="T:90" class="idref" href="#T:90"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:91" class="idref" href="#f:91"><span class="id" title="binder">f</span></a> <a id="g:92" class="idref" href="#g:92"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:93" class="idref" href="#v:93"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:93"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:90"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:93"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:92"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:90"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:93"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:91"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:90"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:91"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:92"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:90"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:92"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSimp" class="idref" href="#existSimp"><span class="id" title="lemma">existSimp</span></a> (<a id="T:94" class="idref" href="#T:94"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:95" class="idref" href="#f:95"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:96" class="idref" href="#v:96"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:94"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:95"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:94"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:96"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:95"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSys" class="idref" href="#existSys"><span class="id" title="lemma">existSys</span></a> (<a id="T:97" class="idref" href="#T:97"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:98" class="idref" href="#f:98"><span class="id" title="binder">f</span></a> <a id="g:99" class="idref" href="#g:99"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:100" class="idref" href="#v:100"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:100"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:97"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:100"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:99"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">SetAdds</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:97"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:98"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNN.html#g:99"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">SetAdds</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:97"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:100"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:98"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">)</span></a>%<span class="id" title="var">fol</span>) <a class="idref" href="hydras.Ackermann.LNN.html#g:99"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="absurd1" class="idref" href="#absurd1"><span class="id" title="lemma">absurd1</span></a>  (<a id="T:101" class="idref" href="#T:101"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:102" class="idref" href="#f:102"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:101"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:102"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:102"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:101"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:102"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nImp" class="idref" href="#nImp"><span class="id" title="lemma">nImp</span></a> (<a id="T:103" class="idref" href="#T:103"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:104" class="idref" href="#f:104"><span class="id" title="binder">f</span></a> <a id="g:105" class="idref" href="#g:105"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:103"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:105"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:103"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:105"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nOr" class="idref" href="#nOr"><span class="id" title="lemma">nOr</span></a> (<a id="T:106" class="idref" href="#T:106"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:107" class="idref" href="#f:107"><span class="id" title="binder">f</span></a> <a id="g:108" class="idref" href="#g:108"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:106"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:108"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:106"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:108"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nAnd" class="idref" href="#nAnd"><span class="id" title="lemma">nAnd</span></a> (<a id="T:109" class="idref" href="#T:109"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:110" class="idref" href="#f:110"><span class="id" title="binder">f</span></a> <a id="g:111" class="idref" href="#g:111"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:109"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:110"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:111"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:109"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#f:110"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:111"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nForall" class="idref" href="#nForall"><span class="id" title="lemma">nForall</span></a> (<a id="T:112" class="idref" href="#T:112"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:113" class="idref" href="#f:113"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:114" class="idref" href="#v:114"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:114"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:113"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:114"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:113"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nExist" class="idref" href="#nExist"><span class="id" title="lemma">nExist</span></a> (<a id="T:115" class="idref" href="#T:115"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:116" class="idref" href="#f:116"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:117" class="idref" href="#v:117"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:115"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:117"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:116"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:115"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:117"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:116"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impRefl" class="idref" href="#impRefl"><span class="id" title="lemma">impRefl</span></a> (<a id="T:118" class="idref" href="#T:118"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:119" class="idref" href="#f:119"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):  <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:118"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:119"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:119"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impTrans" class="idref" href="#impTrans"><span class="id" title="lemma">impTrans</span></a> (<a id="T:120" class="idref" href="#T:120"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:121" class="idref" href="#f:121"><span class="id" title="binder">f</span></a> <a id="g:122" class="idref" href="#g:122"><span class="id" title="binder">g</span></a> <a id="h:123" class="idref" href="#h:123"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:120"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:121"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:122"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:120"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:122"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:123"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:120"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:121"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:123"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSym" class="idref" href="#orSym"><span class="id" title="lemma">orSym</span></a> (<a id="T:124" class="idref" href="#T:124"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:125" class="idref" href="#f:125"><span class="id" title="binder">f</span></a> <a id="g:126" class="idref" href="#g:126"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:124"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:125"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:126"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:124"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:126"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:125"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andSym" class="idref" href="#andSym"><span class="id" title="lemma">andSym</span></a> (<a id="T:127" class="idref" href="#T:127"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:128" class="idref" href="#f:128"><span class="id" title="binder">f</span></a> <a id="g:129" class="idref" href="#g:129"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:127"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:128"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:129"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:127"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:129"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:128"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffRefl" class="idref" href="#iffRefl"><span class="id" title="lemma">iffRefl</span></a> (<a id="T:130" class="idref" href="#T:130"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:131" class="idref" href="#f:131"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):  <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:130"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:131"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:131"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffSym" class="idref" href="#iffSym"><span class="id" title="lemma">iffSym</span></a>  (<a id="T:132" class="idref" href="#T:132"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:133" class="idref" href="#f:133"><span class="id" title="binder">f</span></a> <a id="g:134" class="idref" href="#g:134"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:132"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:133"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:134"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:132"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:134"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#f:133"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffTrans" class="idref" href="#iffTrans"><span class="id" title="lemma">iffTrans</span></a> (<a id="T:135" class="idref" href="#T:135"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:136" class="idref" href="#f:136"><span class="id" title="binder">f</span></a> <a id="g:137" class="idref" href="#g:137"><span class="id" title="binder">g</span></a> <a id="h:138" class="idref" href="#h:138"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:135"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:136"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#g:137"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:135"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#g:137"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:138"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:135"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#f:136"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#h:138"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqRefl" class="idref" href="#eqRefl"><span class="id" title="lemma">eqRefl</span></a> (<a id="T:139" class="idref" href="#T:139"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:140" class="idref" href="#a:140"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:139"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:140"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#a:140"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqSym" class="idref" href="#eqSym"><span class="id" title="lemma">eqSym</span></a> (<a id="T:141" class="idref" href="#T:141"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:142" class="idref" href="#a:142"><span class="id" title="binder">a</span></a> <a id="b:143" class="idref" href="#b:143"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:141"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:142"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:143"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:141"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#b:143"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#a:142"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqTrans" class="idref" href="#eqTrans"><span class="id" title="lemma">eqTrans</span></a> (<a id="T:144" class="idref" href="#T:144"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:145" class="idref" href="#a:145"><span class="id" title="binder">a</span></a> <a id="b:146" class="idref" href="#b:146"><span class="id" title="binder">b</span></a> <a id="c:147" class="idref" href="#c:147"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:144"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:145"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:146"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:144"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#b:146"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#c:147"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:144"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:145"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#c:147"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqPlus" class="idref" href="#eqPlus"><span class="id" title="lemma">eqPlus</span></a> (<a id="T:148" class="idref" href="#T:148"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:149" class="idref" href="#a:149"><span class="id" title="binder">a</span></a> <a id="b:150" class="idref" href="#b:150"><span class="id" title="binder">b</span></a> <a id="c:151" class="idref" href="#c:151"><span class="id" title="binder">c</span></a> <a id="d:152" class="idref" href="#d:152"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:148"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:149"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:150"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:148"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#c:151"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:152"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:148"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:149"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#c:151"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:150"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:152"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqTimes" class="idref" href="#eqTimes"><span class="id" title="lemma">eqTimes</span></a> (<a id="T:153" class="idref" href="#T:153"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:154" class="idref" href="#a:154"><span class="id" title="binder">a</span></a> <a id="b:155" class="idref" href="#b:155"><span class="id" title="binder">b</span></a> <a id="c:156" class="idref" href="#c:156"><span class="id" title="binder">c</span></a> <a id="d:157" class="idref" href="#d:157"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:153"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:154"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:155"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:153"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#c:156"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:157"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:153"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:154"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#c:156"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:155"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:157"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqSucc" class="idref" href="#eqSucc"><span class="id" title="lemma">eqSucc</span></a> (<a id="T:158" class="idref" href="#T:158"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:159" class="idref" href="#a:159"><span class="id" title="binder">a</span></a> <a id="b:160" class="idref" href="#b:160"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:158"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:159"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:160"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:158"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#a:159"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:160"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqLT" class="idref" href="#eqLT"><span class="id" title="lemma">eqLT</span></a> (<a id="T:161" class="idref" href="#T:161"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:162" class="idref" href="#a:162"><span class="id" title="binder">a</span></a> <a id="b:163" class="idref" href="#b:163"><span class="id" title="binder">b</span></a> <a id="c:164" class="idref" href="#c:164"><span class="id" title="binder">c</span></a> <a id="d:165" class="idref" href="#d:165"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:161"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:163"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:161"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#c:164"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:165"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#T:161"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#c:164"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#b:163"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d:165"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNN.html#Logic"><span class="id" title="section">Logic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="natToTerm" class="idref" href="#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a id="n:166" class="idref" href="#n:166"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN.html#n:166"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.LNN.html#Zero"><span class="id" title="definition">Zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.LNN.html#Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm:167"><span class="id" title="definition">natToTerm</span></a> <span class="id" title="var">m</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="n2t" class="idref" href="#n2t"><span class="id" title="abbreviation">n2t</span></a> <span class="id" title="var">i</span> := (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <span class="id" title="var">i</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedNatToTerm" class="idref" href="#closedNatToTerm"><span class="id" title="lemma">closedNatToTerm</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:169" class="idref" href="#a:169"><span class="id" title="binder">a</span></a> <a id="v:170" class="idref" href="#v:170"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#v:170"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#a:169"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.LNN2LNT</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.LNN2LNT</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+LNN2LNT: 
+
+<div class="paragraph"> </div>
+
+   Translation of <span class="inlinecode"><span class="id" title="var">LNN</span></span>-formulas and proofs into <span class="inlinecode"><span class="id" title="var">LNT</span></span> by replacement of 
+    <span class="inlinecode">(<span class="id" title="var">t</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">t'</span>)%<span class="id" title="var">fol</span></span> subformulas with <span class="inlinecode">(<span class="id" title="tactic">∃</span></span> <span class="inlinecode"><span class="id" title="var">v</span>,</span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">Succ</span></span> <span class="inlinecode"><span class="id" title="var">v</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">t'</span>)%<span class="id" title="var">nt</span></span>.
+
+<div class="paragraph"> </div>
+
+   Original file by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab295"></a><h1 class="section">Translation of terms</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="LNN2LNT_term" class="idref" href="#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> (<a id="t:1" class="idref" href="#t:1"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:1"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_terms:5"><span class="id" title="definition">LNN2LNT_terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="LNN2LNT_terms" class="idref" href="#LNN2LNT_terms"><span class="id" title="definition">LNN2LNT_terms</span></a> (<a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:3" class="idref" href="#ts:3"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:2"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#Terms"><span class="id" title="definition">Terms</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:2"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#ts:3"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n0</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.LNT.html#Terms"><span class="id" title="definition">Terms</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n0:7"><span class="id" title="variable">n0</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ @<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> <span class="id" title="var">ss</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term:4"><span class="id" title="definition">LNN2LNT_term</span></a> <span class="id" title="var">s</span>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_terms:5"><span class="id" title="definition">LNN2LNT_terms</span></a> <span class="id" title="var">m</span> <span class="id" title="var">ss</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab296"></a><h2 class="section">Inverse translation</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="LNT2LNN_term" class="idref" href="#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> (<a id="t:28" class="idref" href="#t:28"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:28"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_terms:32"><span class="id" title="definition">LNT2LNN_terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="LNT2LNN_terms" class="idref" href="#LNT2LNN_terms"><span class="id" title="definition">LNT2LNN_terms</span></a> (<a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:30" class="idref" href="#ts:30"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Terms"><span class="id" title="definition">Terms</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:29"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:29"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#ts:30"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n0</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n0:34"><span class="id" title="variable">n0</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ @<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> <span class="id" title="var">ss</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term:31"><span class="id" title="definition">LNT2LNN_term</span></a> <span class="id" title="var">s</span>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_terms:32"><span class="id" title="definition">LNT2LNN_terms</span></a> <span class="id" title="var">m</span> <span class="id" title="var">ss</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNN_natToTerm" class="idref" href="#LNT2LNN_natToTerm"><span class="id" title="lemma">LNT2LNN_natToTerm</span></a> (<a id="n:36" class="idref" href="#n:36"><span class="id" title="binder">n</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:36"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#n:36"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNT_term" class="idref" href="#LNT2LNT_term"><span class="id" title="lemma">LNT2LNT_term</span></a> (<a id="t:37" class="idref" href="#t:37"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:37"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:37"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab297"></a><h1 class="section">Translation of formulas</h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab298"></a><h2 class="section">Translation of <span class="inlinecode">(<span class="id" title="var">v</span>#0</span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">v</span>#1)%<span class="id" title="var">fol</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LTFormula" class="idref" href="#LTFormula"><span class="id" title="definition">LTFormula</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> 2<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">LNT.Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab299"></a><h2 class="section">Translation of <span class="inlinecode">(<span class="id" title="var">t</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">t'</span>)%<span class="id" title="var">fol</span></span></h2>
+
+<div class="paragraph"> </div>
+
+  The function <span class="inlinecode"><span class="id" title="var">translateLT</span></span> is defined by an interactive proof (omitted).
+     It is specified by the lemma <span class="inlinecode"><span class="id" title="var">translateLT1</span></span> 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="translateLT" class="idref" href="#translateLT"><span class="id" title="definition">translateLT</span></a> (<a id="ts:42" class="idref" href="#ts:42"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LT_"><span class="id" title="constructor">LT_</span></a>)) : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="translateLT1" class="idref" href="#translateLT1"><span class="id" title="lemma">translateLT1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="a0:44" class="idref" href="#a0:44"><span class="id" title="binder">a0</span></a> <a id="b0:45" class="idref" href="#b0:45"><span class="id" title="binder">b0</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#translateLT"><span class="id" title="definition">translateLT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:43"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a0:44"><span class="id" title="variable">a0</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b0:45"><span class="id" title="variable">b0</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LTFormula"><span class="id" title="definition">LTFormula</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="H:46" class="idref" href="#H:46"><span class="id" title="binder">H</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:43"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="H0:47" class="idref" href="#H0:47"><span class="id" title="binder">H0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">_</span> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a0:44"><span class="id" title="variable">a0</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="H1:48" class="idref" href="#H1:48"><span class="id" title="binder">H1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">_</span> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#H1:48"><span class="id" title="variable">H1</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#H0:47"><span class="id" title="variable">H0</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#H:46"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab300"></a><h2 class="section">Translation of any <span class="inlinecode"><span class="id" title="var">LNN</span></span>-formula</h2>
+
+
+<div class="paragraph"> </div>
+
+   The translation of any <span class="inlinecode"><span class="id" title="var">LNN</span></span>-formula is straigthforward, except for 
+   the special case of <span class="inlinecode"><span class="id" title="var">t_1</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">t_2</span></span> (handled by <span class="inlinecode"><span class="id" title="var">translateLT</span></span> )
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="LNN2LNT_formula" class="idref" href="#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a id="f:49" class="idref" href="#f:49"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:49"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| (<span class="id" title="var">t1</span> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <span class="id" title="var">t2</span>)%<span class="id" title="var">fol</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> <span class="id" title="var">t1</span> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> <span class="id" title="var">t2</span>)%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">r</span> <span class="id" title="keyword">as</span> <span class="id" title="var">l</span> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#l:52"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#LT_"><span class="id" title="constructor">LT_</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="ts:53" class="idref" href="#ts:53"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LT_"><span class="id" title="constructor">LT_</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.LNN2LNT.html#translateLT"><span class="id" title="definition">translateLT</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#ts:53"><span class="id" title="variable">ts</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;| (<span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">B</span>)%<span class="id" title="var">fol</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula:50"><span class="id" title="definition">LNN2LNT_formula</span></a> <span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula:50"><span class="id" title="definition">LNN2LNT_formula</span></a> <span class="id" title="var">B</span>)%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">fol</span> ⇒  (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula:50"><span class="id" title="definition">LNN2LNT_formula</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">fol</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula:50"><span class="id" title="definition">LNN2LNT_formula</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab301"></a><h3 class="section">Helpful rewriting lemmas</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_or" class="idref" href="#LNN2LNT_or"><span class="id" title="lemma">LNN2LNT_or</span></a> (<a id="a:54" class="idref" href="#a:54"><span class="id" title="binder">a</span></a> <a id="b:55" class="idref" href="#b:55"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:54"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:55"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:54"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:55"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_and" class="idref" href="#LNN2LNT_and"><span class="id" title="lemma">LNN2LNT_and</span></a> (<a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a> <a id="b:57" class="idref" href="#b:57"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:57"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:56"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:57"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_iff" class="idref" href="#LNN2LNT_iff"><span class="id" title="lemma">LNN2LNT_iff</span></a> (<a id="a:58" class="idref" href="#a:58"><span class="id" title="binder">a</span></a> <a id="b:59" class="idref" href="#b:59"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:58"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:59"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:58"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#b:59"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_exist" class="idref" href="#LNN2LNT_exist"><span class="id" title="lemma">LNN2LNT_exist</span></a> (<a id="v:60" class="idref" href="#v:60"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a>  <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:60"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:61"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:60"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#a:61"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_freeVarF" class="idref" href="#LNN2LNT_freeVarF"><span class="id" title="lemma">LNN2LNT_freeVarF</span></a> (<a id="f:62" class="idref" href="#f:62"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) (<a id="v:63" class="idref" href="#v:63"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:63"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:62"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:63"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:62"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_freeVarF1" class="idref" href="#LNN2LNT_freeVarF1"><span class="id" title="lemma">LNN2LNT_freeVarF1</span></a> (<a id="f:64" class="idref" href="#f:64"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) (<a id="v:65" class="idref" href="#v:65"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:65"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:64"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:65"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:64"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_freeVarF2" class="idref" href="#LNN2LNT_freeVarF2"><span class="id" title="lemma">LNN2LNT_freeVarF2</span></a> (<a id="f:66" class="idref" href="#f:66"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) (<a id="v:67" class="idref" href="#v:67"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:67"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:66"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:67"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:66"><span class="id" title="variable">f</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNN2LNT_subFormula" class="idref" href="#LNN2LNT_subFormula"><span class="id" title="lemma">LNN2LNT_subFormula</span></a> <br/>
+&nbsp;(<a id="T:68" class="idref" href="#T:68"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:69" class="idref" href="#f:69"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) (<a id="v:70" class="idref" href="#v:70"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:71" class="idref" href="#s:71"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#T:68"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:69"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:70"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:71"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:69"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:70"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term"><span class="id" title="definition">LNN2LNT_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:71"><span class="id" title="variable">s</span></a>))).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab302"></a><h2 class="section">Inverse translation</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="LNT2LNN_formula" class="idref" href="#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> (<a id="f:96" class="idref" href="#f:96"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:96"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| (<span class="id" title="var">t1</span> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <span class="id" title="var">t2</span>)%<span class="id" title="var">nt</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <span class="id" title="var">t1</span> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <span class="id" title="var">t2</span>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <span class="id" title="keyword">match</span> <span class="id" title="var">r</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| (<span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">B</span>)%<span class="id" title="var">nt</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula:97"><span class="id" title="definition">LNT2LNN_formula</span></a> <span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula:97"><span class="id" title="definition">LNT2LNN_formula</span></a> <span class="id" title="var">B</span>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a>  <span class="id" title="var">A</span>)%<span class="id" title="var">nt</span> ⇒   (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula:97"><span class="id" title="definition">LNT2LNN_formula</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">nt</span> ⇒ (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula:97"><span class="id" title="definition">LNT2LNN_formula</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab303"></a><h3 class="section">Commutation lemmas</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNT_formula" class="idref" href="#LNT2LNT_formula"><span class="id" title="lemma">LNT2LNT_formula</span></a> (<a id="f:100" class="idref" href="#f:100"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>): <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:100"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:100"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNN_subTerm" class="idref" href="#LNT2LNN_subTerm"><span class="id" title="lemma">LNT2LNN_subTerm</span></a>  (<a id="t:101" class="idref" href="#t:101"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) (<a id="v:102" class="idref" href="#v:102"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:103" class="idref" href="#s:103"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:101"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:102"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:103"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:101"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:102"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:103"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNN_freeVarT" class="idref" href="#LNT2LNN_freeVarT"><span class="id" title="lemma">LNT2LNN_freeVarT</span></a> ( <a id="t:108" class="idref" href="#t:108"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:108"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#t:108"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNN_freeVarF" class="idref" href="#LNT2LNN_freeVarF"><span class="id" title="lemma">LNT2LNN_freeVarF</span></a> (<a id="f:113" class="idref" href="#f:113"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:113"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:113"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT2LNN_subFormula" class="idref" href="#LNT2LNN_subFormula"><span class="id" title="lemma">LNT2LNN_subFormula</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:114" class="idref" href="#f:114"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:115" class="idref" href="#v:115"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:116" class="idref" href="#s:116"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:114"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:115"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:116"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula"><span class="id" title="definition">LNT2LNN_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:114"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#v:115"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term"><span class="id" title="definition">LNT2LNN_term</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#s:116"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab304"></a><h1 class="section">Proof translation</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+  If the translation of every axiom of a <span class="inlinecode"><span class="id" title="var">LNN</span></span>-system <span class="inlinecode"><span class="id" title="var">U</span></span> is provable in a closed 
+      <span class="inlinecode"><span class="id" title="var">LNT</span></span>-system <span class="inlinecode"><span class="id" title="var">V</span></span>, 
+      then the translation of any <span class="inlinecode"><span class="id" title="var">LNN</span></span>-formula in <span class="inlinecode"><span class="id" title="var">U</span></span> is provable in <span class="inlinecode"><span class="id" title="var">V</span></span>.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="translateProof" class="idref" href="#translateProof"><span class="id" title="lemma">translateProof</span></a> (<a id="U:136" class="idref" href="#U:136"><span class="id" title="binder">U</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) (<a id="V:137" class="idref" href="#V:137"><span class="id" title="binder">V</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#ClosedSystem"><span class="id" title="definition">ClosedSystem</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#V:137"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:138" class="idref" href="#f:138"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>) <a class="idref" href="hydras.Ackermann.LNN2LNT.html#U:136"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:138"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#V:137"><span class="id" title="variable">V</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:138"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:139" class="idref" href="#f:139"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#U:136"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:139"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#V:137"><span class="id" title="variable">V</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#f:139"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,450 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.LNT</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.LNT</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ LNT.v : Language of Number Theory 
+
+<div class="paragraph"> </div>
+
+Original version by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>  <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>  <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab285"></a><h1 class="section">Instantiations of a few generic constructs</h1>
+
+
+<div class="paragraph"> </div>
+
+  <i>To do</i> perhaps these redefinitions should be deprecated, because they cause some issues 
+  in statements which mix <span class="inlinecode"><span class="id" title="var">LNN</span></span> and <span class="inlinecode"><span class="id" title="var">LNT</span></span> terms and formulas 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="definition">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="definition">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="System" class="idref" href="#System"><span class="id" title="definition">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Sentence" class="idref" href="#Sentence"><span class="id" title="definition">Sentence</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#Sentence"><span class="id" title="definition">Sentence</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Term" class="idref" href="#Term"><span class="id" title="definition">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="definition">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="SysPrf" class="idref" href="#SysPrf"><span class="id" title="definition">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="LNT" class="idref" href="#LNT"><span class="id" title="module">LNT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNT.Plus" class="idref" href="#LNT.Plus"><span class="id" title="definition">Plus</span></a> (<a id="x:1" class="idref" href="#x:1"><span class="id" title="binder">x</span></a> <a id="y:2" class="idref" href="#y:2"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Plus_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:1"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:2"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNT.Times" class="idref" href="#LNT.Times"><span class="id" title="definition">Times</span></a> (<a id="x:3" class="idref" href="#x:3"><span class="id" title="binder">x</span></a> <a id="y:4" class="idref" href="#y:4"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Times_"><span class="id" title="constructor">Times_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:3"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:4"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNT.Succ" class="idref" href="#LNT.Succ"><span class="id" title="definition">Succ</span></a> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNT.Zero" class="idref" href="#LNT.Zero"><span class="id" title="definition">Zero</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNT.html#LNT"><span class="id" title="module">LNT</span></a>.<br/>
+<span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.LNT.html#LNT"><span class="id" title="module">LNT</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab286"></a><h1 class="section">Notations for LNT formulas: experimental and unstable</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nt_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="fed6c14e49eff7036666eb0f91269e41" class="idref" href="#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">&quot;</span></a>=" := (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">_</span>): <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="215352f85b2d50eebfec8d16206295aa" class="idref" href="#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">&quot;</span></a>∨" := (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a>): <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="281ad9f5c20247dcb66364deeca521cd" class="idref" href="#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">&quot;</span></a>∧" := (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a>):<span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="::nt_scope:x_'-&gt;'_x" class="idref" href="#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">&quot;</span></a>→" := (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a>): <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="798e0981258b01fbb1da8c3e84f3db7e" class="idref" href="#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">&quot;</span></a>~ A" := (@<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span>): <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nt_scope:x_'&lt;-&gt;'_x" class="idref" href="#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">&quot;</span></a>A &lt;-&gt; B" := (@<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span>): <span class="id" title="var">nt_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="fed6c14e49eff7036666eb0f91269e41" class="idref" href="#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">&quot;</span></a>t = u" := (@<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">_</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span>): <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nt_scope:x_'&lt;&gt;'_x" class="idref" href="#::nt_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">&quot;</span></a>t &lt;&gt; u" := (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <span class="id" title="var">u</span>)%<span class="id" title="var">nt</span> : <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="c1d299e5a32db34ea906aad92603b7d6" class="idref" href="#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">&quot;</span></a>'v#' i" := (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">i</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="var">format</span> "'v#' i", <span class="id" title="var">i</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) : <span class="id" title="var">nt_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="14879ad85c55a476ba57071753760440" class="idref" href="#14879ad85c55a476ba57071753760440"><span class="id" title="notation">&quot;</span></a>'exH' x .. y , p" := (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a>  <span class="id" title="var">x</span> .. (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p</span>) ..)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">nt_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="4bb2f2560044148bb04f513fa721bd27" class="idref" href="#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">&quot;</span></a>'allH' x .. y , p" := (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a>  <span class="id" title="var">x</span> .. (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p</span>) ..)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">nt_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="ed08876ea67a642780c9e14210615148" class="idref" href="#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.Ackermann.LNT.html#Plus"><span class="id" title="definition">Plus</span></a> :<span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="8823e980df46f52eb8bcd9d68ef4ed03" class="idref" href="#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.Ackermann.LNT.html#Times"><span class="id" title="definition">Times</span></a> :<span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="S_" class="idref" href="#S_"><span class="id" title="abbreviation">S_</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> (<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab287"></a><h2 class="section">Notations for printing computed formulas/terms with derived connectives</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '\/'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 85, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '/\'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 80, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '&lt;-&gt;'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 95, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '&lt;-&gt;''' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 95, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="c559e70a5c5fc9dc7c84ae2eb6b09bac" class="idref" href="#c559e70a5c5fc9dc7c84ae2eb6b09bac"><span class="id" title="notation">&quot;</span></a>x \/' y" := (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span>)%<span class="id" title="var">nt</span> : <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="d0a47be881e3150d2a9af078bfd6971b" class="idref" href="#d0a47be881e3150d2a9af078bfd6971b"><span class="id" title="notation">&quot;</span></a>x /\' y" := (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(~</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.LNT.html#c559e70a5c5fc9dc7c84ae2eb6b09bac"><span class="id" title="notation">\/'</span></a>  <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span> : <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nt_scope:x_'&lt;-&gt;'''''_x" class="idref" href="#::nt_scope:x_'&lt;-&gt;'''''_x"><span class="id" title="notation">&quot;</span></a>x &lt;-&gt;'' y" := (<a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">x</span><a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>:  <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nt_scope:x_'&lt;-&gt;'''_x" class="idref" href="#::nt_scope:x_'&lt;-&gt;'''_x"><span class="id" title="notation">&quot;</span></a>x &lt;-&gt;' y" := (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(~</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c559e70a5c5fc9dc7c84ae2eb6b09bac"><span class="id" title="notation">\/'</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">~(</span></a><span class="id" title="var">y</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">x</span><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">))</span></a>)%<span class="id" title="var">nt</span> : <span class="id" title="var">nt_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="exH'" class="idref" href="#exH'"><span class="id" title="abbreviation">exH'</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> := (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <span class="id" title="var">A</span>)<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab288"></a><h2 class="section">Examples</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Examples" class="idref" href="#Examples"><span class="id" title="section">Examples</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Examples.f" class="idref" href="#Examples.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>.<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 0 1 <a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#Examples.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> 0 1 <a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNT.html#Examples"><span class="id" title="section">Examples</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab289"></a><h1 class="section">Basic properties</h1>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LNT_eqdec" class="idref" href="#LNT_eqdec"><span class="id" title="lemma">LNT_eqdec</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#language_decidable"><span class="id" title="definition">language_decidable</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab290"></a><h2 class="section">List of free variables of a formula</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Free_Variables" class="idref" href="#Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarPlus" class="idref" href="#freeVarPlus"><span class="id" title="lemma">freeVarPlus</span></a> (<a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> <a id="y:8" class="idref" href="#y:8"><span class="id" title="binder">y</span></a>: <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#Plus"><span class="id" title="definition">Plus</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:8"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:8"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarTimes" class="idref" href="#freeVarTimes"><span class="id" title="lemma">freeVarTimes</span></a> (<a id="x:9" class="idref" href="#x:9"><span class="id" title="binder">x</span></a> <a id="y:10" class="idref" href="#y:10"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#Times"><span class="id" title="definition">Times</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:10"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#y:10"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSucc" class="idref" href="#freeVarSucc"><span class="id" title="lemma">freeVarSucc</span></a> (<a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:11"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#x:11"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarZero" class="idref" href="#freeVarZero"><span class="id" title="lemma">freeVarZero</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNT.html#Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab291"></a><h2 class="section">Basic and derived proof rules</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Logic" class="idref" href="#Logic"><span class="id" title="section">Logic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Axm" class="idref" href="#Axm"><span class="id" title="lemma">Axm</span></a>  (<a id="T:12" class="idref" href="#T:12"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :  <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:12"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:12"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:13"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysExtend" class="idref" href="#sysExtend"><span class="id" title="lemma">sysExtend</span></a> (<a id="T:14" class="idref" href="#T:14"><span class="id" title="binder">T</span></a> <a id="U:15" class="idref" href="#U:15"><span class="id" title="binder">U</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:16" class="idref" href="#f:16"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:14"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#U:15"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:14"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#U:15"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:16"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysWeaken" class="idref" href="#sysWeaken"><span class="id" title="lemma">sysWeaken</span></a> (<a id="T:17" class="idref" href="#T:17"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a> <a id="g:19" class="idref" href="#g:19"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:17"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:17"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:19"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNT.html#f:18"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impI" class="idref" href="#impI"><span class="id" title="lemma">impI</span></a> (<a id="T:20" class="idref" href="#T:20"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> <a id="g:22" class="idref" href="#g:22"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:20"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:22"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNT.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:20"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:22"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:21"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impE" class="idref" href="#impE"><span class="id" title="lemma">impE</span></a> (<a id="T:23" class="idref" href="#T:23"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:23"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:24"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:23"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:23"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:24"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="contradiction" class="idref" href="#contradiction"><span class="id" title="lemma">contradiction</span></a> (<a id="T:26" class="idref" href="#T:26"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> <a id="g:28" class="idref" href="#g:28"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:26"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:26"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:27"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:26"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:28"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnE" class="idref" href="#nnE"><span class="id" title="lemma">nnE</span></a> (<a id="T:29" class="idref" href="#T:29"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:29"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:30"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:29"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:30"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="noMiddle" class="idref" href="#noMiddle"><span class="id" title="lemma">noMiddle</span></a> (<a id="T:31" class="idref" href="#T:31"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:32" class="idref" href="#f:32"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>): <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:31"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:32"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:32"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnI" class="idref" href="#nnI"><span class="id" title="lemma">nnI</span></a> (<a id="T:33" class="idref" href="#T:33"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:34" class="idref" href="#f:34"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:33"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:34"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:33"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:34"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp1" class="idref" href="#cp1"><span class="id" title="lemma">cp1</span></a> (<a id="T:35" class="idref" href="#T:35"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:36" class="idref" href="#f:36"><span class="id" title="binder">f</span></a> <a id="g:37" class="idref" href="#g:37"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:35"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:36"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:37"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:35"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:37"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:36"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp2" class="idref" href="#cp2"><span class="id" title="lemma">cp2</span></a> (<a id="T:38" class="idref" href="#T:38"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:39" class="idref" href="#f:39"><span class="id" title="binder">f</span></a> <a id="g:40" class="idref" href="#g:40"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:40"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:39"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:39"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:40"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI1" class="idref" href="#orI1"><span class="id" title="lemma">orI1</span></a> (<a id="T:41" class="idref" href="#T:41"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a> <a id="g:43" class="idref" href="#g:43"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:41"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:41"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:43"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI2" class="idref" href="#orI2"><span class="id" title="lemma">orI2</span></a> (<a id="T:44" class="idref" href="#T:44"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> <a id="g:46" class="idref" href="#g:46"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:44"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:46"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:44"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:46"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orE" class="idref" href="#orE"><span class="id" title="lemma">orE</span></a> (<a id="T:47" class="idref" href="#T:47"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> <a id="g:49" class="idref" href="#g:49"><span class="id" title="binder">g</span></a> <a id="h:50" class="idref" href="#h:50"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:47"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:49"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:47"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:50"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:47"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:49"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:50"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:47"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:50"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSys" class="idref" href="#orSys"><span class="id" title="lemma">orSys</span></a> (<a id="T:51" class="idref" href="#T:51"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a> <a id="g:53" class="idref" href="#g:53"><span class="id" title="binder">g</span></a> <a id="h:54" class="idref" href="#h:54"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:51"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:52"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNT.html#h:54"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:51"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:53"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.LNT.html#h:54"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:51"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:53"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>) <a class="idref" href="hydras.Ackermann.LNT.html#h:54"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andI" class="idref" href="#andI"><span class="id" title="lemma">andI</span></a> (<a id="T:55" class="idref" href="#T:55"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:56" class="idref" href="#f:56"><span class="id" title="binder">f</span></a> <a id="g:57" class="idref" href="#g:57"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:55"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:56"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:55"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:57"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:55"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:56"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:57"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE1" class="idref" href="#andE1"><span class="id" title="lemma">andE1</span></a> (<a id="T:58" class="idref" href="#T:58"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:59" class="idref" href="#f:59"><span class="id" title="binder">f</span></a> <a id="g:60" class="idref" href="#g:60"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:58"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:59"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:60"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:58"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:59"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE2" class="idref" href="#andE2"><span class="id" title="lemma">andE2</span></a> (<a id="T:61" class="idref" href="#T:61"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:62" class="idref" href="#f:62"><span class="id" title="binder">f</span></a> <a id="g:63" class="idref" href="#g:63"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:61"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:62"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:63"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:61"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:63"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffI" class="idref" href="#iffI"><span class="id" title="lemma">iffI</span></a> (<a id="T:64" class="idref" href="#T:64"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:65" class="idref" href="#f:65"><span class="id" title="binder">f</span></a> <a id="g:66" class="idref" href="#g:66"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:64"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:65"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:66"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:64"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:66"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:65"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:64"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:65"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:66"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE1" class="idref" href="#iffE1"><span class="id" title="lemma">iffE1</span></a> (<a id="T:67" class="idref" href="#T:67"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:68" class="idref" href="#f:68"><span class="id" title="binder">f</span></a> <a id="g:69" class="idref" href="#g:69"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:68"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:69"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:68"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:69"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE2" class="idref" href="#iffE2"><span class="id" title="lemma">iffE2</span></a> (<a id="T:70" class="idref" href="#T:70"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:71" class="idref" href="#f:71"><span class="id" title="binder">f</span></a> <a id="g:72" class="idref" href="#g:72"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:70"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:71"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:72"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:70"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:72"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:71"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallI" class="idref" href="#forallI"><span class="id" title="lemma">forallI</span></a> (<a id="T:73" class="idref" href="#T:73"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:75" class="idref" href="#v:75"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:75"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:73"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:73"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:74"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:73"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:75"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:74"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallE" class="idref" href="#forallE"><span class="id" title="lemma">forallE</span></a> (<a id="T:76" class="idref" href="#T:76"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:77" class="idref" href="#f:77"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:78" class="idref" href="#v:78"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:79" class="idref" href="#t:79"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:78"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:77"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:78"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#t:79"><span class="id" title="variable">t</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallSimp" class="idref" href="#forallSimp"><span class="id" title="lemma">forallSimp</span></a> (<a id="T:80" class="idref" href="#T:80"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:81" class="idref" href="#f:81"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:82" class="idref" href="#v:82"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:80"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:82"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:81"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:80"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:81"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existI" class="idref" href="#existI"><span class="id" title="lemma">existI</span></a> (<a id="T:83" class="idref" href="#T:83"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:84" class="idref" href="#f:84"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:85" class="idref" href="#v:85"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:86" class="idref" href="#t:86"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:83"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:84"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:85"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#t:86"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:83"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:85"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:84"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existE" class="idref" href="#existE"><span class="id" title="lemma">existE</span></a> (<a id="T:87" class="idref" href="#T:87"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:88" class="idref" href="#f:88"><span class="id" title="binder">f</span></a> <a id="g:89" class="idref" href="#g:89"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:90" class="idref" href="#v:90"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:90"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:87"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:90"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:89"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:87"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:90"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:88"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:87"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:88"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:89"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:87"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:89"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSimp" class="idref" href="#existSimp"><span class="id" title="lemma">existSimp</span></a> (<a id="T:91" class="idref" href="#T:91"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:92" class="idref" href="#f:92"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:93" class="idref" href="#v:93"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:91"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:92"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:91"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:93"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:92"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSys" class="idref" href="#existSys"><span class="id" title="lemma">existSys</span></a> (<a id="T:94" class="idref" href="#T:94"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:95" class="idref" href="#f:95"><span class="id" title="binder">f</span></a> <a id="g:96" class="idref" href="#g:96"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:97" class="idref" href="#v:97"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:97"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:94"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:97"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:96"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:94"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:95"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.LNT.html#g:96"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.LNT.html#T:94"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:97"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:95"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>) <a class="idref" href="hydras.Ackermann.LNT.html#g:96"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="absurd1" class="idref" href="#absurd1"><span class="id" title="lemma">absurd1</span></a> (<a id="T:98" class="idref" href="#T:98"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:99" class="idref" href="#f:99"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:98"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:99"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:99"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:98"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:99"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nImp" class="idref" href="#nImp"><span class="id" title="lemma">nImp</span></a> (<a id="T:100" class="idref" href="#T:100"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:101" class="idref" href="#f:101"><span class="id" title="binder">f</span></a> <a id="g:102" class="idref" href="#g:102"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:100"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:101"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.LNT.html#g:102"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:100"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#f:101"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:102"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nOr" class="idref" href="#nOr"><span class="id" title="lemma">nOr</span></a> (<a id="T:103" class="idref" href="#T:103"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:104" class="idref" href="#f:104"><span class="id" title="binder">f</span></a> <a id="g:105" class="idref" href="#g:105"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:103"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.LNT.html#g:105"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:103"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:105"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nAnd" class="idref" href="#nAnd"><span class="id" title="lemma">nAnd</span></a> (<a id="T:106" class="idref" href="#T:106"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:107" class="idref" href="#f:107"><span class="id" title="binder">f</span></a> <a id="g:108" class="idref" href="#g:108"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:106"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:108"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:106"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:108"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nForall" class="idref" href="#nForall"><span class="id" title="lemma">nForall</span></a> (<a id="T:109" class="idref" href="#T:109"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:110" class="idref" href="#f:110"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:111" class="idref" href="#v:111"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:109"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:111"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:110"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:109"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:111"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:110"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nExist" class="idref" href="#nExist"><span class="id" title="lemma">nExist</span></a> (<a id="T:112" class="idref" href="#T:112"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:113" class="idref" href="#f:113"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:114" class="idref" href="#v:114"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:114"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:113"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:114"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:113"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impRefl" class="idref" href="#impRefl"><span class="id" title="lemma">impRefl</span></a> (<a id="T:115" class="idref" href="#T:115"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:116" class="idref" href="#f:116"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>): <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:115"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:116"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:116"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impTrans" class="idref" href="#impTrans"><span class="id" title="lemma">impTrans</span></a> (<a id="T:117" class="idref" href="#T:117"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:118" class="idref" href="#f:118"><span class="id" title="binder">f</span></a> <a id="g:119" class="idref" href="#g:119"><span class="id" title="binder">g</span></a> <a id="h:120" class="idref" href="#h:120"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:117"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:118"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:119"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:117"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:119"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:120"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:117"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:118"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:120"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSym" class="idref" href="#orSym"><span class="id" title="lemma">orSym</span></a> (<a id="T:121" class="idref" href="#T:121"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:122" class="idref" href="#f:122"><span class="id" title="binder">f</span></a> <a id="g:123" class="idref" href="#g:123"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:121"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:122"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:123"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:121"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:123"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:122"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andSym" class="idref" href="#andSym"><span class="id" title="lemma">andSym</span></a> (<a id="T:124" class="idref" href="#T:124"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:125" class="idref" href="#f:125"><span class="id" title="binder">f</span></a> <a id="g:126" class="idref" href="#g:126"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:124"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:125"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:126"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:124"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:126"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:125"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffRefl" class="idref" href="#iffRefl"><span class="id" title="lemma">iffRefl</span></a> (<a id="T:127" class="idref" href="#T:127"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:128" class="idref" href="#f:128"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:127"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:128"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:128"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffSym" class="idref" href="#iffSym"><span class="id" title="lemma">iffSym</span></a> (<a id="T:129" class="idref" href="#T:129"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:130" class="idref" href="#f:130"><span class="id" title="binder">f</span></a> <a id="g:131" class="idref" href="#g:131"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:129"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:130"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:131"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:129"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:131"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#f:130"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffTrans" class="idref" href="#iffTrans"><span class="id" title="lemma">iffTrans</span></a> (<a id="T:132" class="idref" href="#T:132"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="f:133" class="idref" href="#f:133"><span class="id" title="binder">f</span></a> <a id="g:134" class="idref" href="#g:134"><span class="id" title="binder">g</span></a> <a id="h:135" class="idref" href="#h:135"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:132"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:133"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#g:134"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:132"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#g:134"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:135"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:132"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#f:133"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#h:135"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqRefl" class="idref" href="#eqRefl"><span class="id" title="lemma">eqRefl</span></a>  (<a id="T:136" class="idref" href="#T:136"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:137" class="idref" href="#a:137"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:136"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:137"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#a:137"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqSym" class="idref" href="#eqSym"><span class="id" title="lemma">eqSym</span></a> (<a id="T:138" class="idref" href="#T:138"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:139" class="idref" href="#a:139"><span class="id" title="binder">a</span></a> <a id="b:140" class="idref" href="#b:140"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:138"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:139"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:140"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:138"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#b:140"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#a:139"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqTrans" class="idref" href="#eqTrans"><span class="id" title="lemma">eqTrans</span></a> (<a id="T:141" class="idref" href="#T:141"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:142" class="idref" href="#a:142"><span class="id" title="binder">a</span></a> <a id="b:143" class="idref" href="#b:143"><span class="id" title="binder">b</span></a> <a id="c:144" class="idref" href="#c:144"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:141"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:142"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:143"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:141"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#b:143"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c:144"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:141"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:142"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c:144"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqPlus" class="idref" href="#eqPlus"><span class="id" title="lemma">eqPlus</span></a> (<a id="T:145" class="idref" href="#T:145"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:146" class="idref" href="#a:146"><span class="id" title="binder">a</span></a> <a id="b:147" class="idref" href="#b:147"><span class="id" title="binder">b</span></a> <a id="c:148" class="idref" href="#c:148"><span class="id" title="binder">c</span></a> <a id="d:149" class="idref" href="#d:149"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:145"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:146"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:147"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:145"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#c:148"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#d:149"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:145"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:146"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c:148"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:147"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#d:149"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqTimes" class="idref" href="#eqTimes"><span class="id" title="lemma">eqTimes</span></a>  (<a id="T:150" class="idref" href="#T:150"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:151" class="idref" href="#a:151"><span class="id" title="binder">a</span></a> <a id="b:152" class="idref" href="#b:152"><span class="id" title="binder">b</span></a> <a id="c:153" class="idref" href="#c:153"><span class="id" title="binder">c</span></a> <a id="d:154" class="idref" href="#d:154"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:150"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:151"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:152"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:150"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#c:153"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#d:154"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:150"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:151"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c:153"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:152"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#d:154"><span class="id" title="variable">d</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqSucc" class="idref" href="#eqSucc"><span class="id" title="lemma">eqSucc</span></a> (<a id="T:155" class="idref" href="#T:155"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#System"><span class="id" title="definition">System</span></a>) (<a id="a:156" class="idref" href="#a:156"><span class="id" title="binder">a</span></a> <a id="b:157" class="idref" href="#b:157"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:155"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#a:156"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:157"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#T:155"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#a:156"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#b:157"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.LNT.html#Logic"><span class="id" title="section">Logic</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Conversion from <span class="inlinecode"><span class="id" title="var">nat</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="natToTerm" class="idref" href="#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a id="n:158" class="idref" href="#n:158"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.LNT.html#n:158"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.LNT.html#Zero"><span class="id" title="definition">Zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.LNT.html#S_"><span class="id" title="abbreviation">S_</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm:159"><span class="id" title="definition">natToTerm</span></a> <span class="id" title="var">m</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedNatToTerm" class="idref" href="#closedNatToTerm"><span class="id" title="lemma">closedNatToTerm</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:161" class="idref" href="#a:161"><span class="id" title="binder">a</span></a> <a id="v:162" class="idref" href="#v:162"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#v:162"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#a:161"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.Languages.html b/theories/html/hydras.Ackermann.Languages.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.Languages.html
@@ -0,0 +1,234 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.Languages</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.Languages</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Languages :  Definitions of <span class="inlinecode"><span class="id" title="var">LNT</span></span> and <span class="inlinecode"><span class="id" title="var">LNN</span></span>
+
+<div class="paragraph"> </div>
+
+   Original development by Russel O'Connor 
+<div class="paragraph"> </div>
+
+ TO do : reorganize Alectryon snippets 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab281"></a><h1 class="section">Language of Number Theory: <span class="inlinecode"><span class="id" title="var">LNT</span></span></h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="LNTFunction" class="idref" href="#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Plus_" class="idref" href="#Plus_"><span class="id" title="constructor">Plus_</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction:1"><span class="id" title="inductive">LNTFunction</span></a><br/>
+&nbsp;&nbsp;| <a id="Times_" class="idref" href="#Times_"><span class="id" title="constructor">Times_</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction:1"><span class="id" title="inductive">LNTFunction</span></a><br/>
+&nbsp;&nbsp;| <a id="Succ_" class="idref" href="#Succ_"><span class="id" title="constructor">Succ_</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction:1"><span class="id" title="inductive">LNTFunction</span></a><br/>
+&nbsp;&nbsp;| <a id="Zero_" class="idref" href="#Zero_"><span class="id" title="constructor">Zero_</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction:1"><span class="id" title="inductive">LNTFunction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="LNTRelation" class="idref" href="#LNTRelation"><span class="id" title="inductive">LNTRelation</span></a> : <span class="id" title="keyword">Set</span> :=.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNTFunctionArity" class="idref" href="#LNTFunctionArity"><span class="id" title="definition">LNTFunctionArity</span></a> (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.Languages.html#x:5"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Plus_</span></a> ⇒ 2<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Times_"><span class="id" title="constructor">Times_</span></a> ⇒ 2<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNTRelationR" class="idref" href="#LNTRelationR"><span class="id" title="definition">LNTRelationR</span></a> (<a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> :  <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelation"><span class="id" title="inductive">LNTRelation</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.Languages.html#x:7"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span> <span class="id" title="var">bot</span> ⇒ <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelation_rec"><span class="id" title="definition">LNTRelation_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <span class="id" title="var">bot</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNT" class="idref" href="#LNT"><span class="id" title="definition">LNT</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#language"><span class="id" title="constructor">language</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelation"><span class="id" title="inductive">LNTRelation</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelationR"><span class="id" title="definition">LNTRelationR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunctionArity"><span class="id" title="definition">LNTFunctionArity</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab282"></a><h1 class="section">Language of Natural Numbers: <span class="inlinecode"><span class="id" title="var">LNN</span></span></h1>
+
+     Its functions are also <span class="inlinecode"><span class="id" title="var">LNT</span></span>'s 
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="LNNRelation" class="idref" href="#LNNRelation"><span class="id" title="inductive">LNNRelation</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="LT_" class="idref" href="#LT_"><span class="id" title="constructor">LT_</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNNRelation:9"><span class="id" title="inductive">LNNRelation</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNNArityR" class="idref" href="#LNNArityR"><span class="id" title="definition">LNNArityR</span></a> (<a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNNRelation"><span class="id" title="inductive">LNNRelation</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.Languages.html#x:11"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Ackermann.Languages.html#LT_"><span class="id" title="constructor">LT_</span></a> ⇒ 2 <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNNArityF" class="idref" href="#LNNArityF"><span class="id" title="definition">LNNArityF</span></a> (<a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#LNTFunctionArity"><span class="id" title="definition">LNTFunctionArity</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:13"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LNN" class="idref" href="#LNN"><span class="id" title="definition">LNN</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#language"><span class="id" title="constructor">language</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNNRelation"><span class="id" title="inductive">LNNRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#LNNArityR"><span class="id" title="definition">LNNArityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNNArityF"><span class="id" title="definition">LNNArityF</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab283"></a><h1 class="section">Goedel encoding for LNT</h1>
+
+<div class="paragraph"> </div>
+
+ This function is also used as encoding for <span class="inlinecode"><span class="id" title="var">LNN</span></span>  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeLNTFunction" class="idref" href="#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> (<a id="f:14" class="idref" href="#f:14"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.Languages.html#f:14"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Plus_</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Times_"><span class="id" title="constructor">Times_</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> ⇒ 2<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> ⇒ 3<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeLNTRelation" class="idref" href="#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a> (<a id="R:16" class="idref" href="#R:16"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelation"><span class="id" title="inductive">LNTRelation</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.Languages.html#R:16"><span class="id" title="variable">R</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeLNTFunctionInj" class="idref" href="#codeLNTFunctionInj"><span class="id" title="lemma">codeLNTFunctionInj</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a> <a id="g:19" class="idref" href="#g:19"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTFunction"><span class="id" title="inductive">LNTFunction</span></a>, <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#g:19"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#g:19"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeLNTRelationInj" class="idref" href="#codeLNTRelationInj"><span class="id" title="lemma">codeLNTRelationInj</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="R:20" class="idref" href="#R:20"><span class="id" title="binder">R</span></a> <a id="S:21" class="idref" href="#S:21"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNTRelation"><span class="id" title="inductive">LNTRelation</span></a>, <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#R:20"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#S:21"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#R:20"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#S:21"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeArityLNTR" class="idref" href="#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> (<a id="r:22" class="idref" href="#r:22"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := 0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>]<span class="id" title="keyword">Instance</span> <a id="codeArityLNTRIsPR" class="idref" href="#codeArityLNTRIsPR"><span class="id" title="instance">codeArityLNTRIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNTRIsCorrect1" class="idref" href="#codeArityLNTRIsCorrect1"><span class="id" title="lemma">codeArityLNTRIsCorrect1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="r:23" class="idref" href="#r:23"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> (<a class="idref" href="hydras.Ackermann.Languages.html#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:23"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:23"><span class="id" title="variable">r</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNTRIsCorrect2" class="idref" href="#codeArityLNTRIsCorrect2"><span class="id" title="lemma">codeArityLNTRIsCorrect2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTR"><span class="id" title="definition">codeArityLNTR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="r:25" class="idref" href="#r:25"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:25"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:24"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeArityLNTF" class="idref" href="#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> (<a id="f:26" class="idref" href="#f:26"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:26"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:26"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:26"><span class="id" title="variable">f</span></a>)) (<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:26"><span class="id" title="variable">f</span></a>))) 0 1) 2) 3)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;3.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>]<span class="id" title="keyword">Instance</span> <a id="codeArityLNTFIsPR" class="idref" href="#codeArityLNTFIsPR"><span class="id" title="instance">codeArityLNTFIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNTFIsCorrect1" class="idref" href="#codeArityLNTFIsCorrect1"><span class="id" title="lemma">codeArityLNTFIsCorrect1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:47" class="idref" href="#f:47"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> (<a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:47"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:47"><span class="id" title="variable">f</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNNFIsCorrect1" class="idref" href="#codeArityLNNFIsCorrect1"><span class="id" title="lemma">codeArityLNNFIsCorrect1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> (<a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:48"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#f:48"><span class="id" title="variable">f</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNTFIsCorrect2" class="idref" href="#codeArityLNTFIsCorrect2"><span class="id" title="lemma">codeArityLNTFIsCorrect2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:49"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:50" class="idref" href="#f:50"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:49"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab284"></a><h1 class="section">Goedel encoding for LNN</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeLNNRelation" class="idref" href="#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a> (<a id="R:51" class="idref" href="#R:51"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNNRelation"><span class="id" title="inductive">LNNRelation</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeLNNRelationInj" class="idref" href="#codeLNNRelationInj"><span class="id" title="lemma">codeLNNRelationInj</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="R:52" class="idref" href="#R:52"><span class="id" title="binder">R</span></a> <a id="S:53" class="idref" href="#S:53"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.Languages.html#LNNRelation"><span class="id" title="inductive">LNNRelation</span></a>, <a class="idref" href="hydras.Ackermann.Languages.html#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#R:52"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#S:53"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#R:52"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#S:53"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeArityLNNR" class="idref" href="#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> (<a id="r:54" class="idref" href="#r:54"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:54"><span class="id" title="variable">r</span></a> 0 3.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>]<span class="id" title="keyword">Instance</span> <a id="codeArityLNNRIsPR" class="idref" href="#codeArityLNNRIsPR"><span class="id" title="instance">codeArityLNNRIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNNRIsCorrect1" class="idref" href="#codeArityLNNRIsCorrect1"><span class="id" title="lemma">codeArityLNNRIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="r:61" class="idref" href="#r:61"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> (<a class="idref" href="hydras.Ackermann.Languages.html#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:61"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:61"><span class="id" title="variable">r</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNNRIsCorrect2" class="idref" href="#codeArityLNNRIsCorrect2"><span class="id" title="lemma">codeArityLNNRIsCorrect2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNNR"><span class="id" title="definition">codeArityLNNR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="r:63" class="idref" href="#r:63"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#r:63"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:62"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeArityLNNFIsCorrect2" class="idref" href="#codeArityLNNFIsCorrect2"><span class="id" title="lemma">codeArityLNNFIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:64" class="idref" href="#n:64"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#codeArityLNTF"><span class="id" title="definition">codeArityLNTF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:64"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:65" class="idref" href="#f:65"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#f:65"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#n:64"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.ListExt.html b/theories/html/hydras.Ackermann.ListExt.html
new file mode 100644
index 00000000..31acf1a4
--- /dev/null
+++ b/theories/html/hydras.Ackermann.ListExt.html
@@ -0,0 +1,76 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.ListExt</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.ListExt</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ ListExt.v: 
+
+<div class="paragraph"> </div>
+
+   The most part of the original contents by Russel have been 
+   replaced by lemmas from <span class="inlinecode"><span class="id" title="var">Coq.List</span></span>. 
+   We decided to keep <span class="inlinecode"><span class="id" title="var">in_remove_neq</span></span> (originally <span class="inlinecode"><span class="id" title="var">In_list_remove2</span></span> 
+   since it appears in many composed tactics. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="List_Remove" class="idref" href="#List_Remove"><span class="id" title="section">List_Remove</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="List_Remove.A" class="idref" href="#List_Remove.A"><span class="id" title="variable">A</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="List_Remove.Aeq_dec" class="idref" href="#List_Remove.Aeq_dec"><span class="id" title="variable">Aeq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a:2" class="idref" href="#a:2"><span class="id" title="binder">a</span></a> <a id="b:3" class="idref" href="#b:3"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.ListExt.html#List_Remove.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.ListExt.html#a:2"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#b:3"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.ListExt.html#a:2"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#b:3"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="in_remove_neq" class="idref" href="#in_remove_neq"><span class="id" title="lemma">in_remove_neq</span></a>:<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.ListExt.html#List_Remove.A"><span class="id" title="variable">A</span></a>) (<a id="l:7" class="idref" href="#l:7"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#List_Remove.A"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#a:5"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#List_Remove.Aeq_dec"><span class="id" title="variable">Aeq_dec</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#b:6"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#l:7"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.ListExt.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#b:6"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.ListExt.html#List_Remove"><span class="id" title="section">List_Remove</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Old name in Russel's development 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use ListExt.in_remove_neq instead")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="In_list_remove2" class="idref" href="#In_list_remove2"><span class="id" title="abbreviation">In_list_remove2</span></a> := <a class="idref" href="hydras.Ackermann.ListExt.html#in_remove_neq"><span class="id" title="lemma">in_remove_neq</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.NN.html b/theories/html/hydras.Ackermann.NN.html
new file mode 100644
index 00000000..91dfc198
--- /dev/null
+++ b/theories/html/hydras.Ackermann.NN.html
@@ -0,0 +1,153 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.NN</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.NN</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ NN.v : Natural Numbers: Axioms and basic properties
+
+<div class="paragraph"> </div>
+
+Original version by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a>  <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+</div>
+
+<div class="doc">
+<a id="lab292"></a><h1 class="section">Axioms of <span class="inlinecode"><span class="id" title="var">NN</span></span></h1>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a id="NN" class="idref" href="#NN"><span class="id" title="section">NN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN1" class="idref" href="#NN1"><span class="id" title="definition">NN1</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN2" class="idref" href="#NN2"><span class="id" title="definition">NN2</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN3" class="idref" href="#NN3"><span class="id" title="definition">NN3</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN4" class="idref" href="#NN4"><span class="id" title="definition">NN4</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1))%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN5" class="idref" href="#NN5"><span class="id" title="definition">NN5</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN6" class="idref" href="#NN6"><span class="id" title="definition">NN6</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN7" class="idref" href="#NN7"><span class="id" title="definition">NN7</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN8" class="idref" href="#NN8"><span class="id" title="definition">NN8</span></a> := <br/>
+&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a>(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1) <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN9" class="idref" href="#NN9"><span class="id" title="definition">NN9</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="NN" class="idref" href="#NN"><span class="id" title="definition">NN</span></a> := <a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetEnum'_x_x_'..'_x"><span class="id" title="notation">SetEnum</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN1"><span class="id" title="definition">NN1</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN2"><span class="id" title="definition">NN2</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN3"><span class="id" title="definition">NN3</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN4"><span class="id" title="definition">NN4</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN5"><span class="id" title="definition">NN5</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN6"><span class="id" title="definition">NN6</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN7"><span class="id" title="definition">NN7</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN8"><span class="id" title="definition">NN8</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN9"><span class="id" title="definition">NN9</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab293"></a><h1 class="section">Properties of the system <span class="inlinecode"><span class="id" title="var">NN</span></span></h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedNN1" class="idref" href="#closedNN1"><span class="id" title="lemma">closedNN1</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#ClosedSystem"><span class="id" title="definition">ClosedSystem</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedNN" class="idref" href="#closedNN"><span class="id" title="lemma">closedNN</span></a> : <span class="id" title="keyword">∀</span> <a id="v:1" class="idref" href="#v:1"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.NN.html#v:1"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab294"></a><h2 class="section">Generic instantiation of axioms</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn1" class="idref" href="#nn1"><span class="id" title="lemma">nn1</span></a> (<a id="a:2" class="idref" href="#a:2"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:2"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn2" class="idref" href="#nn2"><span class="id" title="lemma">nn2</span></a> (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> <a id="b:4" class="idref" href="#b:4"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):  <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:4"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:4"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn3" class="idref" href="#nn3"><span class="id" title="lemma">nn3</span></a> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:11"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn4" class="idref" href="#nn4"><span class="id" title="lemma">nn4</span></a> (<a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:13"><span class="id" title="variable">b</span></a>))%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn5" class="idref" href="#nn5"><span class="id" title="lemma">nn5</span></a> ( <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn6" class="idref" href="#nn6"><span class="id" title="lemma">nn6</span></a> (<a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a> <a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:21"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn7" class="idref" href="#nn7"><span class="id" title="lemma">nn7</span></a> (<a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.NN.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn8" class="idref" href="#nn8"><span class="id" title="lemma">nn8</span></a> (<a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>) : <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:31"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nn9" class="idref" href="#nn9"><span class="id" title="lemma">nn9</span></a> (<a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a> <a id="b:39" class="idref" href="#b:39"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Term"><span class="id" title="definition">Term</span></a>):  <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.NN.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:39"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:39"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.NN.html#b:39"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NN.html#a:38"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="section">NN</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.NN2PA.html b/theories/html/hydras.Ackermann.NN2PA.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.NN2PA.html
@@ -0,0 +1,83 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.NN2PA</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.NN2PA</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ NN2PA.v : 
+
+<div class="paragraph"> </div>
+
+Original version by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>  <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>  <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>  <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>  <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.PAtheory.html#"><span class="id" title="library">PAtheory</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#"><span class="id" title="library">LNN2LNT</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="NN2PA" class="idref" href="#NN2PA"><span class="id" title="lemma">NN2PA</span></a> (<a id="f:1" class="idref" href="#f:1"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#f:1"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If  <span class="inlinecode"><span class="id" title="var">F</span>[<span class="id" title="var">x</span>\0]</span>, <span class="inlinecode"><span class="id" title="var">F</span>[<span class="id" title="var">x</span>\1]</span> ... <span class="inlinecode"><span class="id" title="var">F</span>[<span class="id" title="var">x</span>\<span class="id" title="var">m</span>-1]</span> are provable in PA,
+       then <span class="inlinecode"></span> <span class="inlinecode"><span class="id" title="var">v_x</span></span> <span class="inlinecode">&lt;'</span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">F</span></span> is also provable (where <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">&lt;'</span> <span class="inlinecode"><span class="id" title="var">b</span></span> is the translation of <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">b</span></span> into PA).
+
+<div class="paragraph"> </div>
+
+   More precisely: 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="PAboundedLT" class="idref" href="#PAboundedLT"><span class="id" title="lemma">PAboundedLT</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m:2" class="idref" href="#m:2"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="F:3" class="idref" href="#F:3"><span class="id" title="binder">F</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">LNT.Formula</span></a>) (<a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.NN2PA.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#m:2"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#F:3"><span class="id" title="variable">F</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#x:4"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#n:5"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.NN2PA.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">LNN.natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#m:2"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.NN2PA.html#F:3"><span class="id" title="variable">F</span></a>)%<span class="id" title="var">nt</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.NNtheory</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.NNtheory</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="natNE" class="idref" href="#natNE"><span class="id" title="lemma">natNE</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="b:2" class="idref" href="#b:2"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.NNtheory.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:2"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:2"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="natLE" class="idref" href="#natLE"><span class="id" title="lemma">natLE</span></a> (<a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.NNtheory.html#b:8"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:8"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="natLT" class="idref" href="#natLT"><span class="id" title="lemma">natLT</span></a> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.NNtheory.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:10"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="natPlus" class="idref" href="#natPlus"><span class="id" title="lemma">natPlus</span></a> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:12"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.NNtheory.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:12"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nat</span>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="natTimes" class="idref" href="#natTimes"><span class="id" title="lemma">natTimes</span></a> (<a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a> <a id="b:14" class="idref" href="#b:14"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a><a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:14"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> (<a class="idref" href="hydras.Ackermann.NNtheory.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#b:14"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nat</span>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundedLT" class="idref" href="#boundedLT"><span class="id" title="lemma">boundedLT</span></a>  (<a id="m:15" class="idref" href="#m:15"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="x:17" class="idref" href="#x:17"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.NNtheory.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#m:15"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#x:17"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#n:18"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.NNtheory.html#x:17"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#m:15"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#a:16"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnPlusNotNeeded" class="idref" href="#nnPlusNotNeeded"><span class="id" title="lemma">nnPlusNotNeeded</span></a> (<a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a><br/>
+&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#n:19"><span class="id" title="variable">n</span></a>))%<span class="id" title="var">fol</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.NewNotations.html b/theories/html/hydras.Ackermann.NewNotations.html
new file mode 100644
index 00000000..5c13840d
--- /dev/null
+++ b/theories/html/hydras.Ackermann.NewNotations.html
@@ -0,0 +1,44 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.NewNotations</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.NewNotations</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":::'SetAdds'_x_x_'..'_x" class="idref" href="#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">&quot;</span></a>'SetAdds' E0  x1 .. xn" := <br/>
+&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> (.. (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> <span class="id" title="var">E0</span> <span class="id" title="var">x1</span>) .. ) <span class="id" title="var">xn</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E0</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x1</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">xn</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":::'SetEnum'_x_x_'..'_x" class="idref" href="#:::'SetEnum'_x_x_'..'_x"><span class="id" title="notation">&quot;</span></a>'SetEnum' x0  x1 .. xn" := <br/>
+&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> (.. (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Singleton"><span class="id" title="inductive">Singleton</span></a> <span class="id" title="var">_</span> <span class="id" title="var">x0</span>) <span class="id" title="var">x1</span>) .. ) <span class="id" title="var">xn</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x0</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x1</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">xn</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.PA.html b/theories/html/hydras.Ackermann.PA.html
new file mode 100644
index 00000000..3d881dfb
--- /dev/null
+++ b/theories/html/hydras.Ackermann.PA.html
@@ -0,0 +1,133 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.PA</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.PA</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ PA : Peano Arithmetic 
+
+<div class="paragraph"> </div>
+
+       Original development by Russel O'Connor 
+       Bullets and abbreviations by Pierre Casteran 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Decidable.html#"><span class="id" title="library">Decidable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#"><span class="id" title="library">LNT</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="PA" class="idref" href="#PA"><span class="id" title="section">PA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA1" class="idref" href="#PA1"><span class="id" title="definition">PA1</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA2" class="idref" href="#PA2"><span class="id" title="definition">PA2</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA3" class="idref" href="#PA3"><span class="id" title="definition">PA3</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA4" class="idref" href="#PA4"><span class="id" title="definition">PA4</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1))%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA5" class="idref" href="#PA5"><span class="id" title="definition">PA5</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA6" class="idref" href="#PA6"><span class="id" title="definition">PA6</span></a> := (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> 1 0<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA7" class="idref" href="#PA7"><span class="id" title="definition">PA7</span></a> (<a id="f:1" class="idref" href="#f:1"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:2" class="idref" href="#v:2"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="f_0:3" class="idref" href="#f_0:3"><span class="id" title="binder">f_0</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:2"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>%<span class="id" title="var">nt</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="f_Sv:4" class="idref" href="#f_Sv:4"><span class="id" title="binder">f_Sv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:2"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.PA.html#v:2"><span class="id" title="variable">v</span></a>)%<span class="id" title="var">nt</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#close"><span class="id" title="definition">close</span></a> <span class="id" title="var">_</span>  (<a class="idref" href="hydras.Ackermann.PA.html#f_0:3"><span class="id" title="variable">f_0</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:2"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f_Sv:4"><span class="id" title="variable">f_Sv</span></a><a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:2"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:1"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="InductionSchema" class="idref" href="#InductionSchema"><span class="id" title="definition">InductionSchema</span></a> (<a id="f:5" class="idref" href="#f:5"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g:6" class="idref" href="#g:6"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="v:7" class="idref" href="#v:7"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA7"><span class="id" title="definition">PA7</span></a> <a class="idref" href="hydras.Ackermann.PA.html#g:6"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:7"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PA" class="idref" href="#PA"><span class="id" title="definition">PA</span></a> := <a class="idref" href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x"><span class="id" title="notation">SetAdds</span></a> <a class="idref" href="hydras.Ackermann.PA.html#InductionSchema"><span class="id" title="definition">InductionSchema</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA1"><span class="id" title="definition">PA1</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA2"><span class="id" title="definition">PA2</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA3"><span class="id" title="definition">PA3</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA4"><span class="id" title="definition">PA4</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA5"><span class="id" title="definition">PA5</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA6"><span class="id" title="definition">PA6</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="open" class="idref" href="#open"><span class="id" title="definition">open</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Formula_rec"><span class="id" title="definition">Formula_rec</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<span class="id" title="keyword">fun</span> <a id="t:8" class="idref" href="#t:8"><span class="id" title="binder">t</span></a> <a id="t0:9" class="idref" href="#t0:9"><span class="id" title="binder">t0</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a> ⇒ (<a class="idref" href="hydras.Ackermann.PA.html#t:8"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PA.html#t0:9"><span class="id" title="variable">t0</span></a>)%<span class="id" title="var">nt</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="r:10" class="idref" href="#r:10"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="ts:11" class="idref" href="#ts:11"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Terms"><span class="id" title="definition">Terms</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.PA.html#r:10"><span class="id" title="variable">r</span></a>)) ⇒  <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.PA.html#r:10"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.PA.html#ts:11"><span class="id" title="variable">ts</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) <span class="id" title="var">_</span> (<a id="f0:13" class="idref" href="#f0:13"><span class="id" title="binder">f0</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) <span class="id" title="var">_</span> ⇒ (<a class="idref" href="hydras.Ackermann.PA.html#f:12"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f0:13"><span class="id" title="variable">f0</span></a>)%<span class="id" title="var">nt</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="f:14" class="idref" href="#f:14"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) <span class="id" title="var">_</span> ⇒ (<a class="idref" href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:14"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <span class="id" title="var">_</span> (<a id="recf:16" class="idref" href="#recf:16"><span class="id" title="binder">recf</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.PA.html#recf:16"><span class="id" title="variable">recf</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="PAdec" class="idref" href="#PAdec"><span class="id" title="lemma">PAdec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:17" class="idref" href="#x:17"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Decidable.html#decidable"><span class="id" title="definition">decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="hydras.Ackermann.PA.html#x:17"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedPA1" class="idref" href="#closedPA1"><span class="id" title="lemma">closedPA1</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#ClosedSystem"><span class="id" title="definition">ClosedSystem</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="closedPA" class="idref" href="#closedPA"><span class="id" title="lemma">closedPA</span></a> : <span class="id" title="keyword">∀</span> <a id="v:24" class="idref" href="#v:24"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:24"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa1" class="idref" href="#pa1"><span class="id" title="lemma">pa1</span></a> (<a id="a:25" class="idref" href="#a:25"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.PA.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa2" class="idref" href="#pa2"><span class="id" title="lemma">pa2</span></a> (<a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.PA.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Ackermann.PA.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:27"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa3" class="idref" href="#pa3"><span class="id" title="lemma">pa3</span></a> (<a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PA.html#a:34"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa4" class="idref" href="#pa4"><span class="id" title="lemma">pa4</span></a> ( <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a> <a id="b:36" class="idref" href="#b:36"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) :  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:36"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:36"><span class="id" title="variable">b</span></a>))%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa5" class="idref" href="#pa5"><span class="id" title="lemma">pa5</span></a> (<a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>): <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pa6" class="idref" href="#pa6"><span class="id" title="lemma">pa6</span></a> (<a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> <a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>) :  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PA.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.PA.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.PA.html#a:44"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="induct" class="idref" href="#induct"><span class="id" title="lemma">induct</span></a> (<a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:53" class="idref" href="#v:53"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="f_0:54" class="idref" href="#f_0:54"><span class="id" title="binder">f_0</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:53"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <a id="f_Sv:55" class="idref" href="#f_Sv:55"><span class="id" title="binder">f_Sv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:53"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.PA.html#v:53"><span class="id" title="variable">v</span></a>))%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span>  <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f_0:54"><span class="id" title="variable">f_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:53"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f_Sv:55"><span class="id" title="variable">f_Sv</span></a>)%<span class="id" title="var">nt</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.PA.html#v:53"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PA.html#f:52"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">nt</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="section">PA</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.PAconsistent</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.PAconsistent</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+PAconsistent.v
+
+<div class="paragraph"> </div>
+
+    Original file by Russel O' Connor 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>  <a class="idref" href="hydras.Ackermann.PA.html#"><span class="id" title="library">PA</span></a> <a class="idref" href="hydras.Ackermann.model.html#"><span class="id" title="library">model</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="natModel" class="idref" href="#natModel"><span class="id" title="definition">natModel</span></a> : <a class="idref" href="hydras.Ackermann.model.html#Model"><span class="id" title="record">Model</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#model"><span class="id" title="constructor">model</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="f:1" class="idref" href="#f:1"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.PAconsistent.html#f:1"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.model.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.PAconsistent.html#f:2"><span class="id" title="variable">f</span></a>)) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Languages.Plus_</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="x:3" class="idref" href="#x:3"><span class="id" title="binder">x</span></a> <a id="y:4" class="idref" href="#y:4"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.PAconsistent.html#y:4"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.PAconsistent.html#x:3"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Times_"><span class="id" title="constructor">Languages.Times_</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.PAconsistent.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.PAconsistent.html#x:5"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Languages.Succ_</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Languages.Zero_</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="r:7" class="idref" href="#r:7"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.PAconsistent.html#r:7"><span class="id" title="variable">r</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">end</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="PAconsistent" class="idref" href="#PAconsistent"><span class="id" title="lemma">PAconsistent</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Consistent"><span class="id" title="definition">Consistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.PAtheory.html b/theories/html/hydras.Ackermann.PAtheory.html
new file mode 100644
index 00000000..8142abe8
--- /dev/null
+++ b/theories/html/hydras.Ackermann.PAtheory.html
@@ -0,0 +1,80 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.PAtheory</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.PAtheory</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ PAtheory.v   (Peano Arithmetic) 
+
+<div class="paragraph"> </div>
+
+     Original file by Russel O'Connor 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a> <a class="idref" href="hydras.Ackermann.LNN2LNT.html#"><span class="id" title="library">LNN2LNT</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.PA.html#"><span class="id" title="library">PA</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="paZeroOrSucc" class="idref" href="#paZeroOrSucc"><span class="id" title="lemma">paZeroOrSucc</span></a> (<a id="t:1" class="idref" href="#t:1"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>):<br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="nv:2" class="idref" href="#nv:2"><span class="id" title="binder">nv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#t:1"><span class="id" title="variable">t</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PAtheory.html#t:1"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">LNT.Zero</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#nv:2"><span class="id" title="variable">nv</span></a><a class="idref" href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#t:1"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">LNT.Succ</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.PAtheory.html#nv:2"><span class="id" title="variable">nv</span></a>))%<span class="id" title="var">nt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Opaque</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Transparent</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Opaque</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Transparent</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="paPlusSym" class="idref" href="#paPlusSym"><span class="id" title="lemma">paPlusSym</span></a> : <span class="id" title="keyword">∀</span> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Term"><span class="id" title="definition">Term</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.PAtheory.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.PAtheory.html#a:9"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">nt</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="NN72PA" class="idref" href="#NN72PA"><span class="id" title="lemma">NN72PA</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN7"><span class="id" title="definition">NN7</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="NN82PA" class="idref" href="#NN82PA"><span class="id" title="lemma">NN82PA</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN8"><span class="id" title="definition">NN8</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="NN92PA" class="idref" href="#NN92PA"><span class="id" title="lemma">NN92PA</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> (<a class="idref" href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula"><span class="id" title="definition">LNN2LNT_formula</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN9"><span class="id" title="definition">NN9</span></a>).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.cPair.html b/theories/html/hydras.Ackermann.cPair.html
new file mode 100644
index 00000000..c04c78c4
--- /dev/null
+++ b/theories/html/hydras.Ackermann.cPair.html
@@ -0,0 +1,396 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.cPair</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.cPair</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Cantor's pairing function 
+    (Originally  Russel O'Connors <span class="inlinecode"><span class="id" title="var">goedel</span></span> contrib  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a> <a class="idref" href="hydras.Prelude.ssrnat_extracts.html#"><span class="id" title="library">ssrnat_extracts</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab277"></a><h1 class="section">Bijection from <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> to <span class="inlinecode"><span class="id" title="var">nat</span></span></h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab278"></a><h2 class="section">Preliminary definitions</h2>
+
+<div class="paragraph"> </div>
+
+ Sum of all natural numbers upto <span class="inlinecode"><span class="id" title="var">n</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="sumToN" class="idref" href="#sumToN"><span class="id" title="definition">sumToN</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.cPair.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN:2"><span class="id" title="definition">sumToN</span></a> <span class="id" title="var">p</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sumToN1" class="idref" href="#sumToN1"><span class="id" title="lemma">sumToN1</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Ackermann.cPair.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:4"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sumToN2" class="idref" href="#sumToN2"><span class="id" title="lemma">sumToN2</span></a> <a id="b:5" class="idref" href="#b:5"><span class="id" title="binder">b</span></a> <a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a> :  <a class="idref" href="hydras.Ackermann.cPair.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:5"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:5"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="sumToNIsPR" class="idref" href="#sumToNIsPR"><span class="id" title="instance">sumToNIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab279"></a><h2 class="section">Definition and properties of <span class="inlinecode"><span class="id" title="var">cPair</span></span></h2>
+
+<div class="paragraph"> </div>
+
+  The <span class="inlinecode"><span class="id" title="var">cPair</span></span> function defined below is slightly different 
+     from the "usual" Cantor pairing function shown in  a big part 
+     of the litterature (and Coq's standard library). 
+      Since both versions are equivalent upto a swap of the 
+      arguments <span class="inlinecode"><span class="id" title="var">a</span></span> and <span class="inlinecode"><span class="id" title="var">b</span></span>, we still use  Russel O'Connors notations 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cPair" class="idref" href="#cPair"><span class="id" title="definition">cPair</span></a> (<a id="a:19" class="idref" href="#a:19"><span class="id" title="binder">a</span></a> <a id="b:20" class="idref" href="#b:20"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#a:19"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:20"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Usual definition (e.g. wikipedia) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="xPair" class="idref" href="#xPair"><span class="id" title="definition">xPair</span></a> <a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a> <a id="b:22" class="idref" href="#b:22"><span class="id" title="binder">b</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> ( <a class="idref" href="hydras.Ackermann.cPair.html#b:22"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:21"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="xPairDef" class="idref" href="#xPairDef"><span class="id" title="lemma">xPairDef</span></a> <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.cPair.html#xPair"><span class="id" title="definition">xPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:23"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cPairIsPR" class="idref" href="#cPairIsPR"><span class="id" title="instance">cPairIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="CPair_Injectivity" class="idref" href="#CPair_Injectivity"><span class="id" title="section">CPair_Injectivity</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="cPairInjHelp" class="idref" href="#cPairInjHelp"><span class="id" title="lemma">cPairInjHelp</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a> <a id="b:34" class="idref" href="#b:34"><span class="id" title="binder">b</span></a> <a id="c:35" class="idref" href="#c:35"><span class="id" title="binder">c</span></a> <a id="d:36" class="idref" href="#d:36"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:35"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:36"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:35"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:36"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairInj1" class="idref" href="#cPairInj1"><span class="id" title="lemma">cPairInj1</span></a> <a id="a:49" class="idref" href="#a:49"><span class="id" title="binder">a</span></a> <a id="b:50" class="idref" href="#b:50"><span class="id" title="binder">b</span></a> <a id="c:51" class="idref" href="#c:51"><span class="id" title="binder">c</span></a> <a id="d:52" class="idref" href="#d:52"><span class="id" title="binder">d</span></a>: <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:50"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:51"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:52"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:51"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairInj2" class="idref" href="#cPairInj2"><span class="id" title="lemma">cPairInj2</span></a>  <a id="a:53" class="idref" href="#a:53"><span class="id" title="binder">a</span></a> <a id="b:54" class="idref" href="#b:54"><span class="id" title="binder">b</span></a> <a id="c:55" class="idref" href="#c:55"><span class="id" title="binder">c</span></a> <a id="d:56" class="idref" href="#d:56"><span class="id" title="binder">d</span></a> : <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:53"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:54"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:55"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:56"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:54"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:56"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#CPair_Injectivity"><span class="id" title="section">CPair_Injectivity</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="CPair_projections" class="idref" href="#CPair_projections"><span class="id" title="section">CPair_projections</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="CPair_projections.searchXY" class="idref" href="#CPair_projections.searchXY"><span class="id" title="variable">searchXY</span></a> (<a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> (<span class="id" title="keyword">fun</span> <a id="a:58" class="idref" href="#a:58"><span class="id" title="binder">a</span></a> <a id="y:59" class="idref" href="#y:59"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:58"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#y:59"><span class="id" title="variable">y</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>)) <a class="idref" href="hydras.Ackermann.cPair.html#a:57"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="cPairPi1" class="idref" href="#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a id="a:60" class="idref" href="#a:60"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#CPair_projections.searchXY"><span class="id" title="variable">searchXY</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:60"><span class="id" title="variable">a</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="cPairPi2" class="idref" href="#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#CPair_projections.searchXY"><span class="id" title="variable">searchXY</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:61"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cPairProjectionsHelp" class="idref" href="#cPairProjectionsHelp"><span class="id" title="lemma">cPairProjectionsHelp</span></a> (<a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a> <a id="b:63" class="idref" href="#b:63"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#b:63"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:62"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#sumToN"><span class="id" title="definition">sumToN</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:62"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:63"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#CPair_projections.searchXY"><span class="id" title="variable">searchXY</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:63"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:62"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cPairProjections" class="idref" href="#cPairProjections"><span class="id" title="lemma">cPairProjections</span></a> <a id="a:80" class="idref" href="#a:80"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:80"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:80"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:80"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="searchXYIsPR" class="idref" href="#searchXYIsPR"><span class="id" title="instance">searchXYIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#CPair_projections.searchXY"><span class="id" title="variable">searchXY</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cPairPi1IsPR" class="idref" href="#cPairPi1IsPR"><span class="id" title="instance">cPairPi1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cPairProjections1" class="idref" href="#cPairProjections1"><span class="id" title="lemma">cPairProjections1</span></a> (<a id="a:103" class="idref" href="#a:103"><span class="id" title="binder">a</span></a> <a id="b:104" class="idref" href="#b:104"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:103"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:104"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:103"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cPairProjections2" class="idref" href="#cPairProjections2"><span class="id" title="lemma">cPairProjections2</span></a> (<a id="a:105" class="idref" href="#a:105"><span class="id" title="binder">a</span></a> <a id="b:106" class="idref" href="#b:106"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:105"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:106"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:106"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cPairPi2IsPR" class="idref" href="#cPairPi2IsPR"><span class="id" title="instance">cPairPi2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#CPair_projections"><span class="id" title="section">CPair_projections</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="CPair_Order" class="idref" href="#CPair_Order"><span class="id" title="section">CPair_Order</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLe1" class="idref" href="#cPairLe1"><span class="id" title="lemma">cPairLe1</span></a> (<a id="a:107" class="idref" href="#a:107"><span class="id" title="binder">a</span></a> <a id="b:108" class="idref" href="#b:108"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.cPair.html#a:107"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:107"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:108"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLe1A" class="idref" href="#cPairLe1A"><span class="id" title="lemma">cPairLe1A</span></a> (<a id="a:109" class="idref" href="#a:109"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:109"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:109"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLe2" class="idref" href="#cPairLe2"><span class="id" title="lemma">cPairLe2</span></a> (<a id="a:110" class="idref" href="#a:110"><span class="id" title="binder">a</span></a> <a id="b:111" class="idref" href="#b:111"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.cPair.html#b:111"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:110"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:111"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLe2A" class="idref" href="#cPairLe2A"><span class="id" title="lemma">cPairLe2A</span></a> (<a id="a:112" class="idref" href="#a:112"><span class="id" title="binder">a</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:112"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:112"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLe3" class="idref" href="#cPairLe3"><span class="id" title="lemma">cPairLe3</span></a> (<a id="a:113" class="idref" href="#a:113"><span class="id" title="binder">a</span></a> <a id="b:114" class="idref" href="#b:114"><span class="id" title="binder">b</span></a> <a id="c:115" class="idref" href="#c:115"><span class="id" title="binder">c</span></a> <a id="d:116" class="idref" href="#d:116"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#a:113"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:114"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:115"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:116"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:113"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:115"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:114"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#d:116"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLt1" class="idref" href="#cPairLt1"><span class="id" title="lemma">cPairLt1</span></a> (<a id="a:117" class="idref" href="#a:117"><span class="id" title="binder">a</span></a> <a id="b:118" class="idref" href="#b:118"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#a:117"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:117"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:118"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cPairLt2" class="idref" href="#cPairLt2"><span class="id" title="lemma">cPairLt2</span></a> (<a id="a:119" class="idref" href="#a:119"><span class="id" title="binder">a</span></a> <a id="b:120" class="idref" href="#b:120"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#b:120"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:119"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Ackermann.cPair.html#b:120"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#CPair_Order"><span class="id" title="section">CPair_Order</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.extraction.Extraction.html#"><span class="id" title="library">Extraction</span></a>.<br/>
+<span class="id" title="keyword">Section</span> <a id="code_nat_list" class="idref" href="#code_nat_list"><span class="id" title="section">code_nat_list</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="codeList" class="idref" href="#codeList"><span class="id" title="definition">codeList</span></a> (<a id="l:121" class="idref" href="#l:121"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.cPair.html#l:121"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList:122"><span class="id" title="definition">codeList</span></a> <span class="id" title="var">l'</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeListInj" class="idref" href="#codeListInj"><span class="id" title="lemma">codeListInj</span></a> (<a id="l:124" class="idref" href="#l:124"><span class="id" title="binder">l</span></a> <a id="m:125" class="idref" href="#m:125"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#l:124"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:125"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#l:124"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:125"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNth" class="idref" href="#codeNth"><span class="id" title="definition">codeNth</span></a> (<a id="n:126" class="idref" href="#n:126"><span class="id" title="binder">n</span></a> <a id="m:127" class="idref" href="#m:127"><span class="id" title="binder">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="X:129" class="idref" href="#X:129"><span class="id" title="binder">X</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#m:127"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="Hrecn:128" class="idref" href="#Hrecn:128"><span class="id" title="binder">Hrecn</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#Hrecn:128"><span class="id" title="variable">Hrecn</span></a>)) <a class="idref" href="hydras.Ackermann.cPair.html#n:126"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#X:129"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+drops <span class="inlinecode"><span class="id" title="var">n</span></span> first elements from <span class="inlinecode"><span class="id" title="var">l</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Let</span> <a id="code_nat_list.drop" class="idref" href="#code_nat_list.drop"><span class="id" title="variable">drop</span></a> (<a id="n:130" class="idref" href="#n:130"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">∀</span> (<a id="l:131" class="idref" href="#l:131"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNthCorrect" class="idref" href="#codeNthCorrect"><span class="id" title="lemma">codeNthCorrect</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:134" class="idref" href="#n:134"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:135" class="idref" href="#l:135"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:134"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#l:135"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:134"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#l:135"><span class="id" title="variable">l</span></a> 0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNthIsPR" class="idref" href="#codeNthIsPR"><span class="id" title="instance">codeNthIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#code_nat_list"><span class="id" title="section">code_nat_list</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Extraction</span> <span class="id" title="var">codeNth</span>.<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Strong_Recursion" class="idref" href="#Strong_Recursion"><span class="id" title="section">Strong_Recursion</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="evalStrongRecHelp" class="idref" href="#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> (<a id="n:164" class="idref" href="#n:164"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:165" class="idref" href="#f:165"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:165"><span class="id" title="variable">f</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:164"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt_lt_succ_r"><span class="id" title="lemma">Nat.lt_lt_succ_r</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt_succ_diag_r"><span class="id" title="lemma">Nat.lt_succ_diag_r</span></a>  <span class="id" title="var">_</span>))) <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> <span class="id" title="var">_</span>))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:166" class="idref" href="#a:166"><span class="id" title="binder">a</span></a> <a id="b:167" class="idref" href="#b:167"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:166"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:167"><span class="id" title="variable">b</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="evalStrongRec" class="idref" href="#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> (<a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:169" class="idref" href="#f:169"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:168"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:168"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:168"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="z:170" class="idref" href="#z:170"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#f:169"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#z:170"><span class="id" title="variable">z</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>) <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> <span class="id" title="var">_</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="z:171" class="idref" href="#z:171"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#z:171"><span class="id" title="variable">z</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-1'"><span class="id" title="notation">.-1</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span><br/>
+&nbsp;<a id="evalStrongRecIsPR" class="idref" href="#evalStrongRecIsPR"><span class="id" title="instance">evalStrongRecIsPR</span></a> (<a id="n:172" class="idref" href="#n:172"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:173" class="idref" href="#f:173"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:172"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <span class="id" title="var">_</span>  <a class="idref" href="hydras.Ackermann.cPair.html#f:173"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:172"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#f:173"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="computeEvalStrongRecHelp" class="idref" href="#computeEvalStrongRecHelp"><span class="id" title="lemma">computeEvalStrongRecHelp</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:188" class="idref" href="#n:188"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:189" class="idref" href="#f:189"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>) (<a id="c:190" class="idref" href="#c:190"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#f:189"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:190"><span class="id" title="variable">c</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#f:189"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:190"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a0:191" class="idref" href="#a0:191"><span class="id" title="binder">a0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#f:189"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:190"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a0:191"><span class="id" title="variable">a0</span></a>) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a0:191"><span class="id" title="variable">a0</span></a>) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:188"><span class="id" title="variable">n</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a1:192" class="idref" href="#a1:192"><span class="id" title="binder">a1</span></a> <a id="b0:193" class="idref" href="#b0:193"><span class="id" title="binder">b0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a1:192"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b0:193"><span class="id" title="variable">b0</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="listValues" class="idref" href="#listValues"><span class="id" title="definition">listValues</span></a>  (<a id="f:202" class="idref" href="#f:202"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="n:203" class="idref" href="#n:203"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.cPair.html#n:203"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:202"><span class="id" title="variable">f</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#listValues:204"><span class="id" title="definition">listValues</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#f:202"><span class="id" title="variable">f</span></a> <span class="id" title="var">m</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="evalStrongRecHelp1" class="idref" href="#evalStrongRecHelp1"><span class="id" title="lemma">evalStrongRecHelp1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:206" class="idref" href="#f:206"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="n:207" class="idref" href="#n:207"><span class="id" title="binder">n</span></a> <a id="m:208" class="idref" href="#m:208"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#m:208"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:207"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#n:207"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:208"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:206"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:207"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:206"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:208"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="evalStrongRecHelpParam" class="idref" href="#evalStrongRecHelpParam"><span class="id" title="lemma">evalStrongRecHelpParam</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:213" class="idref" href="#a:213"><span class="id" title="binder">a</span></a> <a id="n:214" class="idref" href="#n:214"><span class="id" title="binder">n</span></a> <a id="c:215" class="idref" href="#c:215"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:216" class="idref" href="#f:216"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:213"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#331a87f8f0f3c9336601c27f1e458dc3"><span class="id" title="notation">.+3</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:213"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:213"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Ackermann.cPair.html#f:216"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:214"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:215"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:213"><span class="id" title="variable">a</span></a> (<span class="id" title="keyword">fun</span> <a id="x:217" class="idref" href="#x:217"><span class="id" title="binder">x</span></a> <a id="y:218" class="idref" href="#y:218"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#f:216"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#x:217"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#y:218"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:215"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Ackermann.cPair.html#n:214"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="evalStrongRecHelp2" class="idref" href="#evalStrongRecHelp2"><span class="id" title="lemma">evalStrongRecHelp2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:243" class="idref" href="#a:243"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:244" class="idref" href="#f:244"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:243"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>) (<a id="n:245" class="idref" href="#n:245"><span class="id" title="binder">n</span></a> <a id="m:246" class="idref" href="#m:246"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#m:246"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:245"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <span class="id" title="var">_</span> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRecHelp"><span class="id" title="definition">evalStrongRecHelp</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:244"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:245"><span class="id" title="variable">n</span></a>) 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> <span class="id" title="var">_</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="b:247" class="idref" href="#b:247"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#n:245"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:246"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>) <a class="idref" href="hydras.Ackermann.cPair.html#b:247"><span class="id" title="variable">b</span></a>)) (<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.cPair.html#f:244"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#m:246"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span>  <a id="callIsPR" class="idref" href="#callIsPR"><span class="id" title="instance">callIsPR</span></a> (<a id="g:264" class="idref" href="#g:264"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="H:265" class="idref" href="#H:265"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#g:264"><span class="id" title="variable">g</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:266" class="idref" href="#a:266"><span class="id" title="binder">a</span></a> <a id="recs:267" class="idref" href="#recs:267"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#a:266"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#g:264"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:266"><span class="id" title="variable">a</span></a>)) <a class="idref" href="hydras.Ackermann.cPair.html#recs:267"><span class="id" title="variable">recs</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#Strong_Recursion"><span class="id" title="section">Strong_Recursion</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="div2IsPR" class="idref" href="#div2IsPR"><span class="id" title="instance">div2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cPairLemma1" class="idref" href="#cPairLemma1"><span class="id" title="lemma">cPairLemma1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:338" class="idref" href="#a:338"><span class="id" title="binder">a</span></a> <a id="b:339" class="idref" href="#b:339"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.cPair.html#a:338"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:339"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#a:338"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:339"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:338"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:338"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:339"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+abbreviations a la Lisp (to improve) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="LispAbbreviations" class="idref" href="#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.car" class="idref" href="#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.cdr" class="idref" href="#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a>  := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.caar" class="idref" href="#LispAbbreviations.caar"><span class="id" title="abbreviation">caar</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <span class="id" title="var">n</span>)).<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.cadr" class="idref" href="#LispAbbreviations.cadr"><span class="id" title="abbreviation">cadr</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">n</span>)).<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.caddr" class="idref" href="#LispAbbreviations.caddr"><span class="id" title="abbreviation">caddr</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">n</span>))).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.cddr" class="idref" href="#LispAbbreviations.cddr"><span class="id" title="abbreviation">cddr</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">n</span>)).<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.cdddr" class="idref" href="#LispAbbreviations.cdddr"><span class="id" title="abbreviation">cdddr</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">n</span>))).<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="LispAbbreviations.cddddr" class="idref" href="#LispAbbreviations.cddddr"><span class="id" title="abbreviation">cddddr</span></a> <span class="id" title="var">n</span> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">n</span>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab280"></a><h2 class="section">Triples</h2>
+
+   (moved from codeSubFormula.v)
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cTriple" class="idref" href="#cTriple"><span class="id" title="definition">cTriple</span></a> (<a id="a:340" class="idref" href="#a:340"><span class="id" title="binder">a</span></a> <a id="b:341" class="idref" href="#b:341"><span class="id" title="binder">b</span></a> <a id="c:342" class="idref" href="#c:342"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:340"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:341"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:342"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cTriplePi1" class="idref" href="#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a id="n:343" class="idref" href="#n:343"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:343"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cTriplePi2" class="idref" href="#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a id="n:344" class="idref" href="#n:344"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a>  (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:344"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cTriplePi3" class="idref" href="#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a> (<a id="n:345" class="idref" href="#n:345"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#n:345"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cTripleIsPR" class="idref" href="#cTripleIsPR"><span class="id" title="instance">cTripleIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cTriplePi1IsPR" class="idref" href="#cTriplePi1IsPR"><span class="id" title="instance">cTriplePi1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cTriplePi2IsPR" class="idref" href="#cTriplePi2IsPR"><span class="id" title="instance">cTriplePi2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="cTriplePi3IsPR" class="idref" href="#cTriplePi3IsPR"><span class="id" title="instance">cTriplePi3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cTripleProj1" class="idref" href="#cTripleProj1"><span class="id" title="lemma">cTripleProj1</span></a> (<a id="a:362" class="idref" href="#a:362"><span class="id" title="binder">a</span></a> <a id="b:363" class="idref" href="#b:363"><span class="id" title="binder">b</span></a> <a id="c:364" class="idref" href="#c:364"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :  <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:362"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:363"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:364"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:362"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cTripleProj2" class="idref" href="#cTripleProj2"><span class="id" title="lemma">cTripleProj2</span></a> (<a id="a:365" class="idref" href="#a:365"><span class="id" title="binder">a</span></a> <a id="b:366" class="idref" href="#b:366"><span class="id" title="binder">b</span></a> <a id="c:367" class="idref" href="#c:367"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:365"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:366"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:367"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:366"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cTripleProj3" class="idref" href="#cTripleProj3"><span class="id" title="lemma">cTripleProj3</span></a> (<a id="a:368" class="idref" href="#a:368"><span class="id" title="binder">a</span></a> <a id="b:369" class="idref" href="#b:369"><span class="id" title="binder">b</span></a> <a id="c:370" class="idref" href="#c:370"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:368"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#b:369"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:370"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#c:370"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cTripleProj" class="idref" href="#cTripleProj"><span class="id" title="lemma">cTripleProj</span></a> (<a id="a:371" class="idref" href="#a:371"><span class="id" title="binder">a</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:371"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:371"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:371"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#a:371"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.checkPrf</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.checkPrf</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+checkPrf.v
+
+<div class="paragraph"> </div>
+
+    Original file by Russel O'Connor 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#"><span class="id" title="library">codeFreeVar</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#"><span class="id" title="library">codeSubFormula</span></a>  <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>  <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>  <a class="idref" href="hydras.Ackermann.wellFormed.html#"><span class="id" title="library">wellFormed</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.prLogic.html#"><span class="id" title="library">prLogic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Check_Proof" class="idref" href="#Check_Proof"><span class="id" title="section">Check_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Check_Proof.L" class="idref" href="#Check_Proof.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Check_Proof.codeArityF" class="idref" href="#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Check_Proof.codeArityR" class="idref" href="#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityFIsPR:7" class="idref" href="#codeArityFIsPR:7"><span class="id" title="binder">codeArityFIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Check_Proof.codeArityFIsCorrect1" class="idref" href="#Check_Proof.codeArityFIsCorrect1"><span class="id" title="variable">codeArityFIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:8"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:8"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Check_Proof.codeArityFIsCorrect2" class="idref" href="#Check_Proof.codeArityFIsCorrect2"><span class="id" title="variable">codeArityFIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:11" class="idref" href="#f:11"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:11"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:10"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityRIsPR:13" class="idref" href="#codeArityRIsPR:13"><span class="id" title="binder">codeArityRIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Check_Proof.codeArityRIsCorrect1" class="idref" href="#Check_Proof.codeArityRIsCorrect1"><span class="id" title="variable">codeArityRIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="r:14" class="idref" href="#r:14"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:14"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:14"><span class="id" title="variable">r</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Check_Proof.codeArityRIsCorrect2" class="idref" href="#Check_Proof.codeArityRIsCorrect2"><span class="id" title="variable">codeArityRIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="r:17" class="idref" href="#r:17"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:17"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:16"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Check_Proof.codeFInj" class="idref" href="#Check_Proof.codeFInj"><span class="id" title="variable">codeFInj</span></a> : <span class="id" title="keyword">∀</span> <a id="f:19" class="idref" href="#f:19"><span class="id" title="binder">f</span></a> <a id="g:20" class="idref" href="#g:20"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:19"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#g:20"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:19"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#g:20"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="Check_Proof.codeRInj" class="idref" href="#Check_Proof.codeRInj"><span class="id" title="variable">codeRInj</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:22" class="idref" href="#R:22"><span class="id" title="binder">R</span></a> <a id="S:23" class="idref" href="#S:23"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#R:22"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#S:23"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#R:22"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#S:23"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.Term" class="idref" href="#Check_Proof.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.Terms" class="idref" href="#Check_Proof.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.Formula" class="idref" href="#Check_Proof.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.wellFormedTerm" class="idref" href="#Check_Proof.wellFormedTerm"><span class="id" title="variable">wellFormedTerm</span></a> := <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.wellFormedFormula" class="idref" href="#Check_Proof.wellFormedFormula"><span class="id" title="variable">wellFormedFormula</span></a> := <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedFormula"><span class="id" title="definition">wellFormedFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Check_Proof.Prf" class="idref" href="#Check_Proof.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The wellFormedness requirement isn't really neccesary,
+because any proof that used ``extra symbols'' could be
+turned into a proof that only used symbols from the
+axioms and the conclusion.
+
+<div class="paragraph"> </div>
+
+However making this assumption makes the proof easier 
+<div class="paragraph"> </div>
+
+  p is (cPair (formula) (proof of the Formula)) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfAXM" class="idref" href="#checkPrfAXM"><span class="id" title="definition">checkPrfAXM</span></a> (<a id="p:25" class="idref" href="#p:25"><span class="id" title="binder">p</span></a> <a id="recs:26" class="idref" href="#recs:26"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr"><span class="id" title="abbreviation">cddr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:25"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:25"><span class="id" title="variable">p</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:25"><span class="id" title="variable">p</span></a>) 0))) 0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfAXMIsPR" class="idref" href="#checkPrfAXMIsPR"><span class="id" title="instance">checkPrfAXMIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfAXM"><span class="id" title="definition">checkPrfAXM</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfMP" class="idref" href="#checkPrfMP"><span class="id" title="definition">checkPrfMP</span></a> (<a id="p:43" class="idref" href="#p:43"><span class="id" title="binder">p</span></a> <a id="recs:44" class="idref" href="#recs:44"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedFormula"><span class="id" title="variable">wellFormedFormula</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)))) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.caddr"><span class="id" title="abbreviation">caddr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:44"><span class="id" title="variable">recs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdddr"><span class="id" title="abbreviation">cdddr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:44"><span class="id" title="variable">recs</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">))</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.caddr"><span class="id" title="abbreviation">caddr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:44"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:43"><span class="id" title="variable">p</span></a>)))) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:44"><span class="id" title="variable">recs</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfMPIsPR" class="idref" href="#checkPrfMPIsPR"><span class="id" title="instance">checkPrfMPIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfMP"><span class="id" title="definition">checkPrfMP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfGEN" class="idref" href="#checkPrfGEN"><span class="id" title="definition">checkPrfGEN</span></a> (<a id="p:141" class="idref" href="#p:141"><span class="id" title="binder">p</span></a> <a id="recs:142" class="idref" href="#recs:142"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:142"><span class="id" title="variable">recs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormula"><span class="id" title="definition">codeFreeVarListFormula</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:142"><span class="id" title="variable">recs</span></a>)))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">))</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:141"><span class="id" title="variable">p</span></a>)))) <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:142"><span class="id" title="variable">recs</span></a>) 0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfGENIsPR" class="idref" href="#checkPrfGENIsPR"><span class="id" title="instance">checkPrfGENIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfGEN"><span class="id" title="definition">checkPrfGEN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfIMP1" class="idref" href="#checkPrfIMP1"><span class="id" title="definition">checkPrfIMP1</span></a> (<a id="p:231" class="idref" href="#p:231"><span class="id" title="binder">p</span></a> <a id="recs:232" class="idref" href="#recs:232"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:233" class="idref" href="#A:233"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:231"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="B:234" class="idref" href="#B:234"><span class="id" title="binder">B</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:231"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:233"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:234"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:233"><span class="id" title="variable">A</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:231"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfIMP1IsPR" class="idref" href="#checkPrfIMP1IsPR"><span class="id" title="instance">checkPrfIMP1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfIMP1"><span class="id" title="definition">checkPrfIMP1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfIMP2" class="idref" href="#checkPrfIMP2"><span class="id" title="definition">checkPrfIMP2</span></a> (<a id="p:263" class="idref" href="#p:263"><span class="id" title="binder">p</span></a> <a id="recs:264" class="idref" href="#recs:264"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:265" class="idref" href="#A:265"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:263"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="B:266" class="idref" href="#B:266"><span class="id" title="binder">B</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:263"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="C:267" class="idref" href="#C:267"><span class="id" title="binder">C</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:263"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:265"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:266"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#C:267"><span class="id" title="variable">C</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:265"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:266"><span class="id" title="variable">B</span></a>)) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:265"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#C:267"><span class="id" title="variable">C</span></a>))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:263"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfIMP2IsPR" class="idref" href="#checkPrfIMP2IsPR"><span class="id" title="instance">checkPrfIMP2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfIMP2"><span class="id" title="definition">checkPrfIMP2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfCP" class="idref" href="#checkPrfCP"><span class="id" title="definition">checkPrfCP</span></a> (<a id="p:316" class="idref" href="#p:316"><span class="id" title="binder">p</span></a> <a id="recs:317" class="idref" href="#recs:317"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:318" class="idref" href="#A:318"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:316"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="B:319" class="idref" href="#B:319"><span class="id" title="binder">B</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:316"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#A:318"><span class="id" title="variable">A</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#B:319"><span class="id" title="variable">B</span></a>))) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:319"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:318"><span class="id" title="variable">A</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:316"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfCPIsPR" class="idref" href="#checkPrfCPIsPR"><span class="id" title="instance">checkPrfCPIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfCP"><span class="id" title="definition">checkPrfCP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfFA1" class="idref" href="#checkPrfFA1"><span class="id" title="definition">checkPrfFA1</span></a> (<a id="p:352" class="idref" href="#p:352"><span class="id" title="binder">p</span></a> <a id="recs:353" class="idref" href="#recs:353"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:354" class="idref" href="#A:354"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:352"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:355" class="idref" href="#v:355"><span class="id" title="binder">v</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:352"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="t:356" class="idref" href="#t:356"><span class="id" title="binder">t</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:352"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedTerm"><span class="id" title="variable">wellFormedTerm</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#t:356"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:355"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:354"><span class="id" title="variable">A</span></a>)) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:354"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:355"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#t:356"><span class="id" title="variable">t</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:352"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfFA1IsPR" class="idref" href="#checkPrfFA1IsPR"><span class="id" title="instance">checkPrfFA1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA1"><span class="id" title="definition">checkPrfFA1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfFA2" class="idref" href="#checkPrfFA2"><span class="id" title="definition">checkPrfFA2</span></a> (<a id="p:405" class="idref" href="#p:405"><span class="id" title="binder">p</span></a> <a id="recs:406" class="idref" href="#recs:406"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:407" class="idref" href="#A:407"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:405"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:408" class="idref" href="#v:408"><span class="id" title="binder">v</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:405"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:408"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:407"><span class="id" title="variable">A</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:407"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:408"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:407"><span class="id" title="variable">A</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:405"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfFA2IsPR" class="idref" href="#checkPrfFA2IsPR"><span class="id" title="instance">checkPrfFA2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA2"><span class="id" title="definition">checkPrfFA2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfFA3" class="idref" href="#checkPrfFA3"><span class="id" title="definition">checkPrfFA3</span></a> (<a id="p:451" class="idref" href="#p:451"><span class="id" title="binder">p</span></a> <a id="recs:452" class="idref" href="#recs:452"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:453" class="idref" href="#A:453"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:451"><span class="id" title="variable">p</span></a>)) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="B:454" class="idref" href="#B:454"><span class="id" title="binder">B</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:451"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:455" class="idref" href="#v:455"><span class="id" title="binder">v</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:451"><span class="id" title="variable">p</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:455"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:453"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:454"><span class="id" title="variable">B</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:455"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:453"><span class="id" title="variable">A</span></a>)) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#v:455"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:454"><span class="id" title="variable">B</span></a>))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:451"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfFA3IsPR" class="idref" href="#checkPrfFA3IsPR"><span class="id" title="instance">checkPrfFA3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA3"><span class="id" title="definition">checkPrfFA3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfEQ1" class="idref" href="#checkPrfEQ1"><span class="id" title="definition">checkPrfEQ1</span></a> (<a id="p:516" class="idref" href="#p:516"><span class="id" title="binder">p</span></a> <a id="recs:517" class="idref" href="#recs:517"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:516"><span class="id" title="variable">p</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:516"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="checkPrfEQnIsPR" class="idref" href="#checkPrfEQnIsPR"><span class="id" title="lemma">checkPrfEQnIsPR</span></a> (<a id="n:518" class="idref" href="#n:518"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="p:519" class="idref" href="#p:519"><span class="id" title="binder">p</span></a> <a id="recs:520" class="idref" href="#recs:520"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:519"><span class="id" title="variable">p</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:518"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:519"><span class="id" title="variable">p</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfEQ1IsPR" class="idref" href="#checkPrfEQ1IsPR"><span class="id" title="instance">checkPrfEQ1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ1"><span class="id" title="definition">checkPrfEQ1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfEQ2" class="idref" href="#checkPrfEQ2"><span class="id" title="definition">checkPrfEQ2</span></a> (<a id="p:533" class="idref" href="#p:533"><span class="id" title="binder">p</span></a> <a id="recs:534" class="idref" href="#recs:534"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:533"><span class="id" title="variable">p</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:533"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfEQ2IsPR" class="idref" href="#checkPrfEQ2IsPR"><span class="id" title="instance">checkPrfEQ2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ2"><span class="id" title="definition">checkPrfEQ2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfEQ3" class="idref" href="#checkPrfEQ3"><span class="id" title="definition">checkPrfEQ3</span></a> (<a id="p:535" class="idref" href="#p:535"><span class="id" title="binder">p</span></a> <a id="recs:536" class="idref" href="#recs:536"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:535"><span class="id" title="variable">p</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:535"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfEQ3IsPR" class="idref" href="#checkPrfEQ3IsPR"><span class="id" title="instance">checkPrfEQ3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ3"><span class="id" title="definition">checkPrfEQ3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeAxmEqHelp" class="idref" href="#codeAxmEqHelp"><span class="id" title="definition">codeAxmEqHelp</span></a> (<a id="n:537" class="idref" href="#n:537"><span class="id" title="binder">n</span></a> <a id="f:538" class="idref" href="#f:538"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.Ackermann.checkPrf.html#f:538"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="m:539" class="idref" href="#m:539"><span class="id" title="binder">m</span></a> <a id="rec:540" class="idref" href="#rec:540"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.checkPrf.html#m:539"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:539"><span class="id" title="variable">m</span></a>)) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#m:539"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:539"><span class="id" title="variable">m</span></a>))))) <a class="idref" href="hydras.Ackermann.checkPrf.html#rec:540"><span class="id" title="variable">rec</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#n:537"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeAxmEqHelpIsPR" class="idref" href="#codeAxmEqHelpIsPR"><span class="id" title="instance">codeAxmEqHelpIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#codeAxmEqHelp"><span class="id" title="definition">codeAxmEqHelp</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNVars1" class="idref" href="#codeNVars1"><span class="id" title="definition">codeNVars1</span></a> (<a id="n:589" class="idref" href="#n:589"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="m:590" class="idref" href="#m:590"><span class="id" title="binder">m</span></a> <a id="rec:591" class="idref" href="#rec:591"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.checkPrf.html#m:590"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:590"><span class="id" title="variable">m</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#rec:591"><span class="id" title="variable">rec</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#n:589"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNVars1IsPR" class="idref" href="#codeNVars1IsPR"><span class="id" title="instance">codeNVars1IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars1"><span class="id" title="definition">codeNVars1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNVars2" class="idref" href="#codeNVars2"><span class="id" title="definition">codeNVars2</span></a> (<a id="n:618" class="idref" href="#n:618"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="m:619" class="idref" href="#m:619"><span class="id" title="binder">m</span></a> <a id="rec:620" class="idref" href="#rec:620"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#m:619"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:619"><span class="id" title="variable">m</span></a>))) <a class="idref" href="hydras.Ackermann.checkPrf.html#rec:620"><span class="id" title="variable">rec</span></a>)) <a class="idref" href="hydras.Ackermann.checkPrf.html#n:618"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNVars2IsPR" class="idref" href="#codeNVars2IsPR"><span class="id" title="instance">codeNVars2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars2"><span class="id" title="definition">codeNVars2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNVarsCorrect" class="idref" href="#codeNVarsCorrect"><span class="id" title="lemma">codeNVarsCorrect</span></a> (<a id="n:649" class="idref" href="#n:649"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars1"><span class="id" title="definition">codeNVars1</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:649"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#nVars"><span class="id" title="definition">nVars</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:649"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars2"><span class="id" title="definition">codeNVars2</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:649"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#nVars"><span class="id" title="definition">nVars</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:649"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfEQ4" class="idref" href="#checkPrfEQ4"><span class="id" title="definition">checkPrfEQ4</span></a> (<a id="p:650" class="idref" href="#p:650"><span class="id" title="binder">p</span></a> <a id="recs:651" class="idref" href="#recs:651"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="r:652" class="idref" href="#r:652"><span class="id" title="binder">r</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:650"><span class="id" title="variable">p</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="A:653" class="idref" href="#A:653"><span class="id" title="binder">A</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>)))) (<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars1"><span class="id" title="definition">codeNVars1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="B:654" class="idref" href="#B:654"><span class="id" title="binder">B</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>)))) (<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars2"><span class="id" title="definition">codeNVars2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#notZero"><span class="id" title="definition">notZero</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#codeAxmEqHelp"><span class="id" title="definition">codeAxmEqHelp</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#r:652"><span class="id" title="variable">r</span></a>)) (<a class="idref" href="hydras.Ackermann.code.html#codeIff"><span class="id" title="definition">codeIff</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#A:653"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#B:654"><span class="id" title="variable">B</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:650"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeOrIsPR" class="idref" href="#codeOrIsPR"><span class="id" title="instance">codeOrIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.code.html#codeOr"><span class="id" title="definition">codeOr</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeAndIsPR" class="idref" href="#codeAndIsPR"><span class="id" title="instance">codeAndIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.code.html#codeAnd"><span class="id" title="definition">codeAnd</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeIffIsPR" class="idref" href="#codeIffIsPR"><span class="id" title="instance">codeIffIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.code.html#codeIff"><span class="id" title="definition">codeIff</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfEQ4IsPR" class="idref" href="#checkPrfEQ4IsPR"><span class="id" title="instance">checkPrfEQ4IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ4"><span class="id" title="definition">checkPrfEQ4</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfEQ5" class="idref" href="#checkPrfEQ5"><span class="id" title="definition">checkPrfEQ5</span></a> (<a id="p:729" class="idref" href="#p:729"><span class="id" title="binder">p</span></a> <a id="recs:730" class="idref" href="#recs:730"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="f:731" class="idref" href="#f:731"><span class="id" title="binder">f</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:729"><span class="id" title="variable">p</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#notZero"><span class="id" title="definition">notZero</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#codeAxmEqHelp"><span class="id" title="definition">codeAxmEqHelp</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars1"><span class="id" title="definition">codeNVars1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.checkPrf.html#codeNVars2"><span class="id" title="definition">codeNVars2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:731"><span class="id" title="variable">f</span></a>))))))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:729"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfEQ5IsPR" class="idref" href="#checkPrfEQ5IsPR"><span class="id" title="instance">checkPrfEQ5IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ5"><span class="id" title="definition">checkPrfEQ5</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrfHelp" class="idref" href="#checkPrfHelp"><span class="id" title="definition">checkPrfHelp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="p:790" class="idref" href="#p:790"><span class="id" title="binder">p</span></a> <a id="recs:791" class="idref" href="#recs:791"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="type:792" class="idref" href="#type:792"><span class="id" title="binder">type</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>))))<br/>
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+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>)))))))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>))))))))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#type:792"><span class="id" title="variable">type</span></a>)))))))))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ5"><span class="id" title="definition">checkPrfEQ5</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ4"><span class="id" title="definition">checkPrfEQ4</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ3"><span class="id" title="definition">checkPrfEQ3</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ2"><span class="id" title="definition">checkPrfEQ2</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfEQ1"><span class="id" title="definition">checkPrfEQ1</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA3"><span class="id" title="definition">checkPrfFA3</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA2"><span class="id" title="definition">checkPrfFA2</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfFA1"><span class="id" title="definition">checkPrfFA1</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfCP"><span class="id" title="definition">checkPrfCP</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfIMP2"><span class="id" title="definition">checkPrfIMP2</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfIMP1"><span class="id" title="definition">checkPrfIMP1</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfGEN"><span class="id" title="definition">checkPrfGEN</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfMP"><span class="id" title="definition">checkPrfMP</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfAXM"><span class="id" title="definition">checkPrfAXM</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:790"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#recs:791"><span class="id" title="variable">recs</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfHelpIsPR" class="idref" href="#checkPrfHelpIsPR"><span class="id" title="instance">checkPrfHelpIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfHelp"><span class="id" title="definition">checkPrfHelp</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkPrf" class="idref" href="#checkPrf"><span class="id" title="definition">checkPrf</span></a> (<a id="f:821" class="idref" href="#f:821"><span class="id" title="binder">f</span></a> <a id="p:822" class="idref" href="#p:822"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedFormula"><span class="id" title="variable">wellFormedFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:821"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrfHelp"><span class="id" title="definition">checkPrfHelp</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:821"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:822"><span class="id" title="variable">p</span></a>)) 0.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkPrfIsPR" class="idref" href="#checkPrfIsPR"><span class="id" title="instance">checkPrfIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrf"><span class="id" title="definition">checkPrf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="checkPrfCorrect1" class="idref" href="#checkPrfCorrect1"><span class="id" title="lemma">checkPrfCorrect1</span></a> (<a id="l:835" class="idref" href="#l:835"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="f:836" class="idref" href="#f:836"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="p:837" class="idref" href="#p:837"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#l:835"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:836"><span class="id" title="variable">f</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrf"><span class="id" title="definition">checkPrf</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:836"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#l:835"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:836"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:837"><span class="id" title="variable">p</span></a>) <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#l:835"><span class="id" title="variable">l</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="checkPrfCorrect2" class="idref" href="#checkPrfCorrect2"><span class="id" title="lemma">checkPrfCorrect2</span></a> (<a id="n:844" class="idref" href="#n:844"><span class="id" title="binder">n</span></a> <a id="m:845" class="idref" href="#m:845"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.checkPrf.html#checkPrf"><span class="id" title="definition">checkPrf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:844"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:845"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:846" class="idref" href="#f:846"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula"><span class="id" title="variable">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:846"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#n:844"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l:847" class="idref" href="#l:847"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula"><span class="id" title="variable">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p:848" class="idref" href="#p:848"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#l:847"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:846"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#l:847"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#f:846"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#p:848"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.checkPrf.html#m:845"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.checkPrf.html#Check_Proof"><span class="id" title="section">Check_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> {<span class="id" title="var">L</span> <span class="id" title="var">cf</span> <span class="id" title="var">cr</span> <span class="id" title="var">cl</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.code</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.code</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab265"></a><h1 class="section">Encoding terms, formulas and proofs</h1>
+
+
+<div class="paragraph"> </div>
+
+Original script by Russel O'Connor 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_Term_Formula_Proof" class="idref" href="#Code_Term_Formula_Proof"><span class="id" title="section">Code_Term_Formula_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Code_Term_Formula_Proof.L" class="idref" href="#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Section</span> <a id="Code_Term_Formula_Proof.LcodeDef" class="idref" href="#Code_Term_Formula_Proof.LcodeDef"><span class="id" title="section">LcodeDef</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Code_Term_Formula_Proof.LcodeDef.cF" class="idref" href="#Code_Term_Formula_Proof.LcodeDef.cF"><span class="id" title="variable">cF</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Code_Term_Formula_Proof.LcodeDef.cR" class="idref" href="#Code_Term_Formula_Proof.LcodeDef.cR"><span class="id" title="variable">cR</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a id="Lcode" class="idref" href="#Lcode"><span class="id" title="record">Lcode</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{  <a id="codeFInj" class="idref" href="#codeFInj"><span class="id" title="projection">codeFInj</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:5" class="idref" href="#f:5"><span class="id" title="binder">f</span></a> <a id="g:6" class="idref" href="#g:6"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="hydras.Ackermann.code.html#g:6"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#g:6"><span class="id" title="variable">g</span></a>; <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="codeRInj" class="idref" href="#codeRInj"><span class="id" title="projection">codeRInj</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:8" class="idref" href="#R:8"><span class="id" title="binder">R</span></a> <a id="S:9" class="idref" href="#S:9"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cR"><span class="id" title="variable">cR</span></a> <a class="idref" href="hydras.Ackermann.code.html#R:8"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cR"><span class="id" title="variable">cR</span></a> <a class="idref" href="hydras.Ackermann.code.html#S:9"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#R:8"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#S:9"><span class="id" title="variable">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef"><span class="id" title="section">LcodeDef</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="codeF" class="idref" href="#codeF"><span class="id" title="definition">codeF</span></a> {<a id="cf:11" class="idref" href="#cf:11"><span class="id" title="binder">cf</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} {<a id="cr:12" class="idref" href="#cr:12"><span class="id" title="binder">cr</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="c:13" class="idref" href="#c:13"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.code.html#cf:11"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.code.html#cr:12"><span class="id" title="variable">cr</span></a>) := <a class="idref" href="hydras.Ackermann.code.html#cf:11"><span class="id" title="variable">cf</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="codeR" class="idref" href="#codeR"><span class="id" title="definition">codeR</span></a> {<a id="cf:14" class="idref" href="#cf:14"><span class="id" title="binder">cf</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} {<a id="cr:15" class="idref" href="#cr:15"><span class="id" title="binder">cr</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="c:16" class="idref" href="#c:16"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.code.html#cf:14"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.code.html#cr:15"><span class="id" title="variable">cr</span></a>) := <a class="idref" href="hydras.Ackermann.code.html#cr:15"><span class="id" title="variable">cr</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.Formula" class="idref" href="#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.Formulas" class="idref" href="#Code_Term_Formula_Proof.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.System" class="idref" href="#Code_Term_Formula_Proof.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.Term" class="idref" href="#Code_Term_Formula_Proof.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.Terms" class="idref" href="#Code_Term_Formula_Proof.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.Prf" class="idref" href="#Code_Term_Formula_Proof.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Term_Formula_Proof.SysPrf" class="idref" href="#Code_Term_Formula_Proof.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_Term_Formula_Proof.codeTermFormDef" class="idref" href="#Code_Term_Formula_Proof.codeTermFormDef"><span class="id" title="section">codeTermFormDef</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> `(<a id="cl:19" class="idref" href="#cl:19"><span class="id" title="binder">cl</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a id="cf:17" class="idref" href="#cf:17"><span class="id" title="binder">cf</span></a> <a id="cr:18" class="idref" href="#cr:18"><span class="id" title="binder">cr</span></a>).<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="codeTerm" class="idref" href="#codeTerm"><span class="id" title="definition">codeTerm</span></a> (<a id="t:20" class="idref" href="#t:20"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Term"><span class="id" title="variable">Term</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.code.html#t:20"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeF"><span class="id" title="definition">codeF</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef.cl"><span class="id" title="variable">cl</span></a> <span class="id" title="var">f</span>)) (<a class="idref" href="hydras.Ackermann.code.html#codeTerms:24"><span class="id" title="definition">codeTerms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="codeTerms" class="idref" href="#codeTerms"><span class="id" title="definition">codeTerms</span></a> (<a id="n:21" class="idref" href="#n:21"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:22" class="idref" href="#ts:22"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:21"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.code.html#ts:22"><span class="id" title="variable">ts</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">n</span> <span class="id" title="var">t</span> <span class="id" title="var">ss</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm:23"><span class="id" title="definition">codeTerm</span></a> <span class="id" title="var">t</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeTerms:24"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:21"><span class="id" title="variable">n</span></a> <span class="id" title="var">ss</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeTermInj" class="idref" href="#codeTermInj"><span class="id" title="lemma">codeTermInj</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:27" class="idref" href="#t:27"><span class="id" title="binder">t</span></a> <a id="s:28" class="idref" href="#s:28"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Term"><span class="id" title="variable">Term</span></a>, <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.code.html#t:27"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.code.html#s:28"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#t:27"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#s:28"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeTermsInj" class="idref" href="#codeTermsInj"><span class="id" title="lemma">codeTermsInj</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:35" class="idref" href="#n:35"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:36" class="idref" href="#ts:36"><span class="id" title="binder">ts</span></a> <a id="ss:37" class="idref" href="#ss:37"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:35"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.code.html#ts:36"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.code.html#ss:37"><span class="id" title="variable">ss</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#ts:36"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#ss:37"><span class="id" title="variable">ss</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="codeFormula" class="idref" href="#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a id="f:38" class="idref" href="#f:38"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.code.html#f:38"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <span class="id" title="var">t1</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <span class="id" title="var">t2</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">f1</span> <span class="id" title="var">f2</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula:39"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">f1</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula:39"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">f2</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">f1</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 (<a class="idref" href="hydras.Ackermann.code.html#codeFormula:39"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">f1</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f1</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula:39"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">f1</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">R</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (4 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeR"><span class="id" title="definition">codeR</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef.cl"><span class="id" title="variable">cl</span></a> <span class="id" title="var">R</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFormulaInj" class="idref" href="#codeFormulaInj"><span class="id" title="lemma">codeFormulaInj</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:41" class="idref" href="#f:41"><span class="id" title="binder">f</span></a> <a id="g:42" class="idref" href="#g:42"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#g:42"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#g:42"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="codePrf" class="idref" href="#codePrf"><span class="id" title="definition">codePrf</span></a> (<a id="Z:43" class="idref" href="#Z:43"><span class="id" title="binder">Z</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formulas"><span class="id" title="variable">Formulas</span></a>) (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="prf:45" class="idref" href="#prf:45"><span class="id" title="binder">prf</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.code.html#Z:43"><span class="id" title="variable">Z</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:44"><span class="id" title="variable">f</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">prf</span>} :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.code.html#prf:45"><span class="id" title="variable">prf</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#AXM"><span class="id" title="constructor">AXM</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#MP"><span class="id" title="constructor">MP</span></a> <span class="id" title="var">Axm1</span> <span class="id" title="var">Axm2</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">rec1</span> <span class="id" title="var">rec2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">B</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codePrf:46"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">rec1</span>)) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.code.html#codePrf:46"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">rec2</span>)))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#GEN"><span class="id" title="constructor">GEN</span></a> <span class="id" title="var">Axm</span> <span class="id" title="var">A</span> <span class="id" title="var">v</span> <span class="id" title="var">_</span> <span class="id" title="keyword">rec</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <span class="id" title="var">v</span> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.code.html#codePrf:46"><span class="id" title="definition">codePrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="keyword">rec</span>)))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#IMP1"><span class="id" title="constructor">IMP1</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">B</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#IMP2"><span class="id" title="constructor">IMP2</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 4 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">B</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">C</span>)))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#CP"><span class="id" title="constructor">CP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 5 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">B</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#FA1"><span class="id" title="constructor">FA1</span></a> <span class="id" title="var">A</span> <span class="id" title="var">v</span> <span class="id" title="var">t</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 6 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <span class="id" title="var">v</span> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <span class="id" title="var">t</span>)))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#FA2"><span class="id" title="constructor">FA2</span></a> <span class="id" title="var">A</span> <span class="id" title="var">v</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 7 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) <span class="id" title="var">v</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#FA3"><span class="id" title="constructor">FA3</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">v</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 8 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <span class="id" title="var">B</span>) <span class="id" title="var">v</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#EQ1"><span class="id" title="constructor">EQ1</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 9 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#EQ2"><span class="id" title="constructor">EQ2</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 10 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#EQ3"><span class="id" title="constructor">EQ3</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 11 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#EQ4"><span class="id" title="constructor">EQ4</span></a> <span class="id" title="var">r</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 12 (<a class="idref" href="hydras.Ackermann.code.html#codeR"><span class="id" title="definition">codeR</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef.cl"><span class="id" title="variable">cl</span></a> <span class="id" title="var">r</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.folProof.html#EQ5"><span class="id" title="constructor">EQ5</span></a> <span class="id" title="var">f</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 13 (<a class="idref" href="hydras.Ackermann.code.html#codeF"><span class="id" title="definition">codeF</span></a> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef.cl"><span class="id" title="variable">cl</span></a> <a class="idref" href="hydras.Ackermann.code.html#f:44"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codePrfInjAxm" class="idref" href="#codePrfInjAxm"><span class="id" title="lemma">codePrfInjAxm</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="b:49" class="idref" href="#b:49"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="A:50" class="idref" href="#A:50"><span class="id" title="binder">A</span></a> <a id="B:51" class="idref" href="#B:51"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formulas"><span class="id" title="variable">Formulas</span></a>) (<a id="p:52" class="idref" href="#p:52"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.code.html#A:50"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:48"><span class="id" title="variable">a</span></a>) (<a id="q:53" class="idref" href="#q:53"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Prf"><span class="id" title="variable">Prf</span></a> <a class="idref" href="hydras.Ackermann.code.html#B:51"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:49"><span class="id" title="variable">b</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="hydras.Ackermann.code.html#A:50"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#p:52"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> <a class="idref" href="hydras.Ackermann.code.html#B:51"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:49"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.code.html#q:53"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.code.html#A:50"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#B:51"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeImp" class="idref" href="#codeImp"><span class="id" title="definition">codeImp</span></a> (<a id="a:54" class="idref" href="#a:54"><span class="id" title="binder">a</span></a> <a id="b:55" class="idref" href="#b:55"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:54"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:55"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeImpCorrect" class="idref" href="#codeImpCorrect"><span class="id" title="lemma">codeImpCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a> <a id="b:57" class="idref" href="#b:57"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:56"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:57"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:57"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNot" class="idref" href="#codeNot"><span class="id" title="definition">codeNot</span></a> (<a id="a:58" class="idref" href="#a:58"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 <a class="idref" href="hydras.Ackermann.code.html#a:58"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNotCorrect" class="idref" href="#codeNotCorrect"><span class="id" title="lemma">codeNotCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:59" class="idref" href="#a:59"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:59"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:59"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeForall" class="idref" href="#codeForall"><span class="id" title="definition">codeForall</span></a> (<a id="n:60" class="idref" href="#n:60"><span class="id" title="binder">n</span></a> <a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:60"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:61"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeForallCorrect" class="idref" href="#codeForallCorrect"><span class="id" title="lemma">codeForallCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:63" class="idref" href="#a:63"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeForall"><span class="id" title="definition">codeForall</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:62"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:63"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.code.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:63"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeOr" class="idref" href="#codeOr"><span class="id" title="definition">codeOr</span></a> (<a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a> <a id="b:65" class="idref" href="#b:65"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:64"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Ackermann.code.html#b:65"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeOrCorrect" class="idref" href="#codeOrCorrect"><span class="id" title="lemma">codeOrCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:66" class="idref" href="#a:66"><span class="id" title="binder">a</span></a> <a id="b:67" class="idref" href="#b:67"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeOr"><span class="id" title="definition">codeOr</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:66"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:67"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:67"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeAnd" class="idref" href="#codeAnd"><span class="id" title="definition">codeAnd</span></a> (<a id="a:68" class="idref" href="#a:68"><span class="id" title="binder">a</span></a> <a id="b:69" class="idref" href="#b:69"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeOr"><span class="id" title="definition">codeOr</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:68"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:69"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeAndCorrect" class="idref" href="#codeAndCorrect"><span class="id" title="lemma">codeAndCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:70" class="idref" href="#a:70"><span class="id" title="binder">a</span></a> <a id="b:71" class="idref" href="#b:71"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeAnd"><span class="id" title="definition">codeAnd</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:70"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:71"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:70"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:71"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeIff" class="idref" href="#codeIff"><span class="id" title="definition">codeIff</span></a> (<a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> <a id="b:73" class="idref" href="#b:73"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Ackermann.code.html#codeAnd"><span class="id" title="definition">codeAnd</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:73"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:73"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:72"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeIffCorrect" class="idref" href="#codeIffCorrect"><span class="id" title="lemma">codeIffCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:74" class="idref" href="#a:74"><span class="id" title="binder">a</span></a> <a id="b:75" class="idref" href="#b:75"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeIff"><span class="id" title="definition">codeIff</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:74"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:75"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.code.html#a:74"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.code.html#b:75"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef"><span class="id" title="section">codeTermFormDef</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.code.html#Code_Term_Formula_Proof"><span class="id" title="section">Code_Term_Formula_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> {<span class="id" title="var">L</span>} {<span class="id" title="var">cf</span> <span class="id" title="var">cr</span> <span class="id" title="var">cl</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> {<span class="id" title="var">L</span>} {<span class="id" title="var">cf</span> <span class="id" title="var">cr</span> <span class="id" title="var">cl</span> <span class="id" title="var">n</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> {<span class="id" title="var">L</span>} {<span class="id" title="var">cf</span> <span class="id" title="var">cr</span> <span class="id" title="var">cl</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.code.html#codePrf"><span class="id" title="definition">codePrf</span></a> {<span class="id" title="var">L</span>} {<span class="id" title="var">cf</span> <span class="id" title="var">cr</span> <span class="id" title="var">cl</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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new file mode 100644
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@@ -0,0 +1,150 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.codeFreeVar</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.codeFreeVar</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ codeFreeVar.v
+
+<div class="paragraph"> </div>
+
+     Original content by Russel O'Connor 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>  <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.codeList.html#"><span class="id" title="library">codeList</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>  <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_Free_Vars" class="idref" href="#Code_Free_Vars"><span class="id" title="section">Code_Free_Vars</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Code_Free_Vars.L" class="idref" href="#Code_Free_Vars.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Free_Vars.Formula" class="idref" href="#Code_Free_Vars.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Free_Vars.Formulas" class="idref" href="#Code_Free_Vars.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Free_Vars.System" class="idref" href="#Code_Free_Vars.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Free_Vars.Term" class="idref" href="#Code_Free_Vars.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Free_Vars.Terms" class="idref" href="#Code_Free_Vars.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFreeVarTermTerms" class="idref" href="#codeFreeVarTermTerms"><span class="id" title="definition">codeFreeVarTermTerms</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="t:5" class="idref" href="#t:5"><span class="id" title="binder">t</span></a> <a id="recs:6" class="idref" href="#recs:6"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a>)) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:6"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a>) 0)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:6"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:5"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:6"><span class="id" title="variable">recs</span></a>))) 0)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFreeVarTerm" class="idref" href="#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> (<a id="t:7" class="idref" href="#t:7"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTerms"><span class="id" title="definition">codeFreeVarTermTerms</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:7"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFreeVarTerms" class="idref" href="#codeFreeVarTerms"><span class="id" title="definition">codeFreeVarTerms</span></a> (<a id="t:8" class="idref" href="#t:8"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTerms"><span class="id" title="definition">codeFreeVarTermTerms</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:8"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFreeVarTermCorrect" class="idref" href="#codeFreeVarTermCorrect"><span class="id" title="lemma">codeFreeVarTermCorrect</span></a> (<a id="t:9" class="idref" href="#t:9"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Term"><span class="id" title="variable">Term</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:9"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#t:9"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFreeVarTermsCorrect" class="idref" href="#codeFreeVarTermsCorrect"><span class="id" title="lemma">codeFreeVarTermsCorrect</span></a> (<a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:23" class="idref" href="#ts:23"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#n:22"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerms"><span class="id" title="definition">codeFreeVarTerms</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#ts:23"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#ts:23"><span class="id" title="variable">ts</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="codeFreeVarTermTermsIsPR" class="idref" href="#codeFreeVarTermTermsIsPR"><span class="id" title="instance">codeFreeVarTermTermsIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTerms"><span class="id" title="definition">codeFreeVarTermTerms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFreeVarTermIsPR" class="idref" href="#codeFreeVarTermIsPR"><span class="id" title="lemma">codeFreeVarTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="codeFreeVarTermsIsPR" class="idref" href="#codeFreeVarTermsIsPR"><span class="id" title="instance">codeFreeVarTermsIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerms"><span class="id" title="definition">codeFreeVarTerms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFreeVarFormula" class="idref" href="#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="f:130" class="idref" href="#f:130"><span class="id" title="binder">f</span></a> <a id="recs:131" class="idref" href="#recs:131"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerms"><span class="id" title="definition">codeFreeVarTerms</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeListRemove"><span class="id" title="definition">codeListRemove</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cadr"><span class="id" title="abbreviation">cadr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr"><span class="id" title="abbreviation">cddr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:131"><span class="id" title="variable">recs</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:131"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cadr"><span class="id" title="abbreviation">cadr</span></a>  <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:131"><span class="id" title="variable">recs</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr"><span class="id" title="abbreviation">cddr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:131"><span class="id" title="variable">recs</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cadr"><span class="id" title="abbreviation">cadr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr"><span class="id" title="abbreviation">cddr</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:130"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFreeVarFormulaCorrect" class="idref" href="#codeFreeVarFormulaCorrect"><span class="id" title="lemma">codeFreeVarFormulaCorrect</span></a> (<a id="f:132" class="idref" href="#f:132"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:132"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#f:132"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="codeFreeVarFormulaIsPR" class="idref" href="#codeFreeVarFormulaIsPR"><span class="id" title="instance">codeFreeVarFormulaIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeFreeVarListFormula" class="idref" href="#codeFreeVarListFormula"><span class="id" title="definition">codeFreeVarListFormula</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="l:293" class="idref" href="#l:293"><span class="id" title="binder">l</span></a> <a id="recs:294" class="idref" href="#recs:294"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:293"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:293"><span class="id" title="variable">l</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:293"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:293"><span class="id" title="variable">l</span></a>))) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#recs:294"><span class="id" title="variable">recs</span></a>)) 0).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeFreeVarListFormulaCorrect" class="idref" href="#codeFreeVarListFormulaCorrect"><span class="id" title="lemma">codeFreeVarListFormulaCorrect</span></a> (<a id="l:295" class="idref" href="#l:295"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormula"><span class="id" title="definition">codeFreeVarListFormula</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a>) <a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:295"><span class="id" title="variable">l</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#l:295"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="codeFreeVarListFormulaIsPR" class="idref" href="#codeFreeVarListFormulaIsPR"><span class="id" title="instance">codeFreeVarListFormulaIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormula"><span class="id" title="definition">codeFreeVarListFormula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars"><span class="id" title="section">Code_Free_Vars</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+++ b/theories/html/hydras.Ackermann.codeList.html
@@ -0,0 +1,141 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.codeList</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.codeList</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ codeList.v
+
+<div class="paragraph"> </div>
+
+     Original script by Russel O'Connor 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>  <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeLength" class="idref" href="#codeLength"><span class="id" title="definition">codeLength</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> <a id="Hrecs:2" class="idref" href="#Hrecs:2"><span class="id" title="binder">Hrecs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:1"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:1"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#Hrecs:2"><span class="id" title="variable">Hrecs</span></a>)) 0).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeLengthCorrect" class="idref" href="#codeLengthCorrect"><span class="id" title="lemma">codeLengthCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l:3" class="idref" href="#l:3"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.codeList.html#codeLength"><span class="id" title="definition">codeLength</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:3"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:3"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeLengthIsPR" class="idref" href="#codeLengthIsPR"><span class="id" title="instance">codeLengthIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeList.html#codeLength"><span class="id" title="definition">codeLength</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeApp" class="idref" href="#codeApp"><span class="id" title="definition">codeApp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:40" class="idref" href="#n:40"><span class="id" title="binder">n</span></a> <a id="Hrecs:41" class="idref" href="#Hrecs:41"><span class="id" title="binder">Hrecs</span></a> <a id="p1:42" class="idref" href="#p1:42"><span class="id" title="binder">p1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:40"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:40"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:40"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#Hrecs:41"><span class="id" title="variable">Hrecs</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#p1:42"><span class="id" title="variable">p1</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeAppCorrect" class="idref" href="#codeAppCorrect"><span class="id" title="lemma">codeAppCorrect</span></a> (<a id="l1:43" class="idref" href="#l1:43"><span class="id" title="binder">l1</span></a> <a id="l2:44" class="idref" href="#l2:44"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l1:43"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l2:44"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#l1:43"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l2:44"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeAppIsPR" class="idref" href="#codeAppIsPR"><span class="id" title="instance">codeAppIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeListRemove" class="idref" href="#codeListRemove"><span class="id" title="definition">codeListRemove</span></a> (<a id="a:105" class="idref" href="#a:105"><span class="id" title="binder">a</span></a> <a id="l:106" class="idref" href="#l:106"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a> <a id="Hrecs:108" class="idref" href="#Hrecs:108"><span class="id" title="binder">Hrecs</span></a> <a id="p1:109" class="idref" href="#p1:109"><span class="id" title="binder">p1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Ackermann.codeList.html#p1:109"><span class="id" title="variable">p1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#Hrecs:108"><span class="id" title="variable">Hrecs</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:107"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#Hrecs:108"><span class="id" title="variable">Hrecs</span></a>)))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="hydras.Ackermann.codeList.html#l:106"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:105"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeListRemoveCorrect" class="idref" href="#codeListRemoveCorrect"><span class="id" title="lemma">codeListRemoveCorrect</span></a> (<a id="a:110" class="idref" href="#a:110"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:111" class="idref" href="#l:111"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeList.html#codeListRemove"><span class="id" title="definition">codeListRemove</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:110"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:111"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:110"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:111"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeListRemoveIsPR" class="idref" href="#codeListRemoveIsPR"><span class="id" title="instance">codeListRemoveIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeList.html#codeListRemove"><span class="id" title="definition">codeListRemove</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeIn" class="idref" href="#codeIn"><span class="id" title="definition">codeIn</span></a> (<a id="a:210" class="idref" href="#a:210"><span class="id" title="binder">a</span></a> <a id="l:211" class="idref" href="#l:211"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:212" class="idref" href="#n:212"><span class="id" title="binder">n</span></a> <a id="Hrecs:213" class="idref" href="#Hrecs:213"><span class="id" title="binder">Hrecs</span></a> <a id="p1:214" class="idref" href="#p1:214"><span class="id" title="binder">p1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:212"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:212"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Ackermann.codeList.html#p1:214"><span class="id" title="variable">p1</span></a>) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#n:212"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#n:212"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#Hrecs:213"><span class="id" title="variable">Hrecs</span></a>)) 0) <a class="idref" href="hydras.Ackermann.codeList.html#l:211"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:210"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeInCorrect" class="idref" href="#codeInCorrect"><span class="id" title="lemma">codeInCorrect</span></a> (<a id="a:215" class="idref" href="#a:215"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:216" class="idref" href="#l:216"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:215"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:216"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#a:215"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:216"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ 1<br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ 0<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeInIsPR" class="idref" href="#codeInIsPR"><span class="id" title="instance">codeInIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNoDup" class="idref" href="#codeNoDup"><span class="id" title="definition">codeNoDup</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="l:295" class="idref" href="#l:295"><span class="id" title="binder">l</span></a> <a id="recs:296" class="idref" href="#recs:296"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#recs:296"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#recs:296"><span class="id" title="variable">recs</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:295"><span class="id" title="variable">l</span></a>))) <a class="idref" href="hydras.Ackermann.codeList.html#recs:296"><span class="id" title="variable">recs</span></a>)))) 0).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNoDupCorrect" class="idref" href="#codeNoDupCorrect"><span class="id" title="lemma">codeNoDupCorrect</span></a> (<a id="l:297" class="idref" href="#l:297"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeList.html#codeNoDup"><span class="id" title="definition">codeNoDup</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:297"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nodup"><span class="id" title="definition">List.nodup</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#l:297"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNoDupIsPR" class="idref" href="#codeNoDupIsPR"><span class="id" title="instance">codeNoDupIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeList.html#codeNoDup"><span class="id" title="definition">codeNoDup</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.codeNatToTerm.html b/theories/html/hydras.Ackermann.codeNatToTerm.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.codeNatToTerm.html
@@ -0,0 +1,66 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.codeNatToTerm</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.codeNatToTerm</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#"><span class="id" title="library">LNT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="natToTermLNN" class="idref" href="#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> := <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">LNN.natToTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="natToTermLNT" class="idref" href="#natToTermLNT"><span class="id" title="definition">natToTermLNT</span></a> := <a class="idref" href="hydras.Ackermann.LNT.html#natToTerm"><span class="id" title="definition">LNT.natToTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="codeNatToTerm" class="idref" href="#codeNatToTerm"><span class="id" title="definition">codeNatToTerm</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 4 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm:2"><span class="id" title="definition">codeNatToTerm</span></a> <span class="id" title="var">p</span>) 0))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="LcodeLNN" class="idref" href="#LcodeLNN"><span class="id" title="instance">LcodeLNN</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNNRelation"><span class="id" title="definition">codeLNNRelation</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="LcodeLNT" class="idref" href="#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTFunction"><span class="id" title="definition">codeLNTFunction</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#codeLNTRelation"><span class="id" title="definition">codeLNTRelation</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNatToTermCorrectLNN" class="idref" href="#codeNatToTermCorrectLNN"><span class="id" title="lemma">codeNatToTermCorrectLNN</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm"><span class="id" title="definition">codeNatToTerm</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN"><span class="id" title="definition">natToTermLNN</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#n:4"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNatToTermCorrectLNT" class="idref" href="#codeNatToTermCorrectLNT"><span class="id" title="lemma">codeNatToTermCorrectLNT</span></a> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm"><span class="id" title="definition">codeNatToTerm</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a>  (<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#natToTermLNT"><span class="id" title="definition">natToTermLNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#n:5"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNatToTermIsPR" class="idref" href="#codeNatToTermIsPR"><span class="id" title="instance">codeNatToTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm"><span class="id" title="definition">codeNatToTerm</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.codePA.html b/theories/html/hydras.Ackermann.codePA.html
new file mode 100644
index 00000000..495f27e9
--- /dev/null
+++ b/theories/html/hydras.Ackermann.codePA.html
@@ -0,0 +1,186 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.codePA</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.codePA</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ codePA.v
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="module">Bool</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>  <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a>  <a class="idref" href="hydras.Ackermann.codeList.html#"><span class="id" title="library">codeList</span></a>  <a class="idref" href="hydras.Ackermann.codeFreeVar.html#"><span class="id" title="library">codeFreeVar</span></a>  <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.prLogic.html#"><span class="id" title="library">prLogic</span></a>  <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#"><span class="id" title="library">codeNatToTerm</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="close" class="idref" href="#close"><span class="id" title="section">close</span></a>.<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="close.L" class="idref" href="#close.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a>: <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#close.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="close.Formula" class="idref" href="#close.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#close.L"><span class="id" title="variable">L</span></a> .<br/>
+<span class="id" title="keyword">Let</span> <a id="close.codeFormula" class="idref" href="#close.codeFormula"><span class="id" title="variable">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codePA.html#close.cL"><span class="id" title="variable">cL</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeCloseList" class="idref" href="#codeCloseList"><span class="id" title="definition">codeCloseList</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="l:5" class="idref" href="#l:5"><span class="id" title="binder">l</span></a> <a id="recs:6" class="idref" href="#recs:6"><span class="id" title="binder">recs</span></a> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#l:5"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#l:5"><span class="id" title="variable">l</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#l:5"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#l:5"><span class="id" title="variable">l</span></a>))) <a class="idref" href="hydras.Ackermann.codePA.html#recs:6"><span class="id" title="variable">recs</span></a>))) <a class="idref" href="hydras.Ackermann.codePA.html#f:7"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeCloseListCorrect" class="idref" href="#codeCloseListCorrect"><span class="id" title="lemma">codeCloseListCorrect</span></a> (<a id="l:8" class="idref" href="#l:8"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.codePA.html#close.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeCloseList"><span class="id" title="definition">codeCloseList</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#l:8"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.Ackermann.codePA.html#close.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:9"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#close.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#close.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#l:8"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:9"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeCloseListIsPR" class="idref" href="#codeCloseListIsPR"><span class="id" title="instance">codeCloseListIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codePA.html#codeCloseList"><span class="id" title="definition">codeCloseList</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeClose" class="idref" href="#codeClose"><span class="id" title="definition">codeClose</span></a> (<a id="f:64" class="idref" href="#f:64"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeCloseList"><span class="id" title="definition">codeCloseList</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#codeNoDup"><span class="id" title="definition">codeNoDup</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:64"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codePA.html#f:64"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeCloseCorrect" class="idref" href="#codeCloseCorrect"><span class="id" title="lemma">codeCloseCorrect</span></a> (<a id="f:65" class="idref" href="#f:65"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.codePA.html#close.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeClose"><span class="id" title="definition">codeClose</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#close.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:65"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#close.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#close"><span class="id" title="definition">close</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#close.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:65"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeCloseIsPR" class="idref" href="#codeCloseIsPR"><span class="id" title="instance">codeCloseIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codePA.html#codeClose"><span class="id" title="definition">codeClose</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.codePA.html#close"><span class="id" title="section">close</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.PA.html#"><span class="id" title="library">PA</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#"><span class="id" title="library">codeSubFormula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_PA" class="idref" href="#Code_PA"><span class="id" title="section">Code_PA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_PA.codeTerm" class="idref" href="#Code_PA.codeTerm"><span class="id" title="variable">codeTerm</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> (<span class="id" title="var">cl</span> :=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>).<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_PA.codeFormula" class="idref" href="#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span>:=<a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>).<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_PA.codeFormulaInj" class="idref" href="#Code_PA.codeFormulaInj"><span class="id" title="variable">codeFormulaInj</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormulaInj"><span class="id" title="lemma">codeFormulaInj</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNT"><span class="id" title="definition">LNT</span></a> <a class="idref" href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT"><span class="id" title="instance">LcodeLNT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeOpen" class="idref" href="#codeOpen"><span class="id" title="definition">codeOpen</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="f:70" class="idref" href="#f:70"><span class="id" title="binder">f</span></a> <a id="recs:71" class="idref" href="#recs:71"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>)))) <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.codePA.html#recs:71"><span class="id" title="variable">recs</span></a>)) <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codePA.html#f:70"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeOpenCorrect" class="idref" href="#codeOpenCorrect"><span class="id" title="lemma">codeOpenCorrect</span></a> (<a id="f:72" class="idref" href="#f:72"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeOpen"><span class="id" title="definition">codeOpen</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:72"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.PA.html#open"><span class="id" title="definition">open</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:72"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeOpenIsPR" class="idref" href="#codeOpenIsPR"><span class="id" title="instance">codeOpenIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codePA.html#codeOpen"><span class="id" title="definition">codeOpen</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeInductionSchema" class="idref" href="#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a> (<a id="f:139" class="idref" href="#f:139"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="n:140" class="idref" href="#n:140"><span class="id" title="binder">n</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddddr"><span class="id" title="abbreviation">cddddr</span></a>  (<a class="idref" href="hydras.Ackermann.codePA.html#codeOpen"><span class="id" title="definition">codeOpen</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:139"><span class="id" title="variable">f</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="g:141" class="idref" href="#g:141"><span class="id" title="binder">g</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cddddr"><span class="id" title="abbreviation">cddddr</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#codeOpen"><span class="id" title="definition">codeOpen</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:139"><span class="id" title="variable">f</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codePA.html#codeClose"><span class="id" title="definition">codeClose</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#g:141"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#n:140"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#LNT.Zero"><span class="id" title="definition">Zero</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeForall"><span class="id" title="definition">codeForall</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#n:140"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#g:141"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#g:141"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#n:140"><span class="id" title="variable">n</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeTerm"><span class="id" title="variable">codeTerm</span></a> (<a class="idref" href="hydras.Ackermann.LNT.html#LNT.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#n:140"><span class="id" title="variable">n</span></a>))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.code.html#codeForall"><span class="id" title="definition">codeForall</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#n:140"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#g:141"><span class="id" title="variable">g</span></a>)))) <a class="idref" href="hydras.Ackermann.codePA.html#f:139"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeInductionSchemaCorrect1" class="idref" href="#codeInductionSchemaCorrect1"><span class="id" title="lemma">codeInductionSchemaCorrect1</span></a> (<a id="f:142" class="idref" href="#f:142"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.PA.html#InductionSchema"><span class="id" title="definition">InductionSchema</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:142"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:142"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeInductionSchemaCorrect2" class="idref" href="#codeInductionSchemaCorrect2"><span class="id" title="lemma">codeInductionSchemaCorrect2</span></a> (<a id="f:143" class="idref" href="#f:143"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:143"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.PA.html#InductionSchema"><span class="id" title="definition">InductionSchema</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:143"><span class="id" title="variable">f</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Opaque</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Opaque</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeInductionSchemaCorrect3" class="idref" href="#codeInductionSchemaCorrect3"><span class="id" title="lemma">codeInductionSchemaCorrect3</span></a> (<a id="f:162" class="idref" href="#f:162"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.PA.html#InductionSchema"><span class="id" title="definition">InductionSchema</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:162"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:162"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="codeInductionSchemaIsPR" class="idref" href="#codeInductionSchemaIsPR"><span class="id" title="instance">codeInductionSchemaIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 1 <a class="idref" href="hydras.Ackermann.codePA.html#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codePA" class="idref" href="#codePA"><span class="id" title="definition">codePA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA6"><span class="id" title="definition">PA6</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA5"><span class="id" title="definition">PA5</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA4"><span class="id" title="definition">PA4</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA3"><span class="id" title="definition">PA3</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA2"><span class="id" title="definition">PA2</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA1"><span class="id" title="definition">PA1</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codeInductionSchema"><span class="id" title="definition">codeInductionSchema</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codePAcorrect1" class="idref" href="#codePAcorrect1"><span class="id" title="lemma">codePAcorrect1</span></a> (<a id="f:227" class="idref" href="#f:227"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codePA.html#codePA"><span class="id" title="definition">codePA</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:227"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:227"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codePAcorrect2" class="idref" href="#codePAcorrect2"><span class="id" title="lemma">codePAcorrect2</span></a> (<a id="f:228" class="idref" href="#f:228"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:228"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#codePA"><span class="id" title="definition">codePA</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:228"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codePAcorrect3" class="idref" href="#codePAcorrect3"><span class="id" title="lemma">codePAcorrect3</span></a> (<a id="f:231" class="idref" href="#f:231"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.Ackermann.LNT.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="hydras.Ackermann.PA.html#PA"><span class="id" title="definition">PA</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:231"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#codePA"><span class="id" title="definition">codePA</span></a> (<a class="idref" href="hydras.Ackermann.codePA.html#Code_PA.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codePA.html#f:231"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codePAIsPR" class="idref" href="#codePAIsPR"><span class="id" title="instance">codePAIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 1 <a class="idref" href="hydras.Ackermann.codePA.html#codePA"><span class="id" title="definition">codePA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.codePA.html#Code_PA"><span class="id" title="section">Code_PA</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.codeSubFormula.html b/theories/html/hydras.Ackermann.codeSubFormula.html
new file mode 100644
index 00000000..63023844
--- /dev/null
+++ b/theories/html/hydras.Ackermann.codeSubFormula.html
@@ -0,0 +1,535 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
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+<h1 class="libtitle">Library hydras.Ackermann.codeSubFormula</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+codeSubFormula:
+
+<div class="paragraph"> </div>
+
+    Original file by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span>  <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>  <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#"><span class="id" title="library">codeSubTerm</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeFreeVar.html#"><span class="id" title="library">codeFreeVar</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="module">Bool</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_Substitute_Formula" class="idref" href="#Code_Substitute_Formula"><span class="id" title="section">Code_Substitute_Formula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Code_Substitute_Formula.L" class="idref" href="#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="abbreviation">Formula</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="abbreviation">Formulas</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="System" class="idref" href="#System"><span class="id" title="abbreviation">System</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Term" class="idref" href="#Term"><span class="id" title="abbreviation">Term</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="abbreviation">Terms</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Formula.codeFormula" class="idref" href="#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> (<span class="id" title="var">cl</span> := <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.cL"><span class="id" title="variable">cL</span></a>).<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Formula.codeTerm" class="idref" href="#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> := <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a>  (<span class="id" title="var">cl</span> := <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.cL"><span class="id" title="variable">cL</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeNewVar" class="idref" href="#codeNewVar"><span class="id" title="definition">codeNewVar</span></a> (<a id="l:5" class="idref" href="#l:5"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> <a id="Hrecs:7" class="idref" href="#Hrecs:7"><span class="id" title="binder">Hrecs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:6"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:6"><span class="id" title="variable">n</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:6"><span class="id" title="variable">n</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Hrecs:7"><span class="id" title="variable">Hrecs</span></a>)) 0) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#l:5"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeNewVarCorrect" class="idref" href="#codeNewVarCorrect"><span class="id" title="lemma">codeNewVarCorrect</span></a> (<a id="l:8" class="idref" href="#l:8"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeNewVar"><span class="id" title="definition">codeNewVar</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#l:8"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#l:8"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNewVarIsPR" class="idref" href="#codeNewVarIsPR"><span class="id" title="instance">codeNewVarIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeNewVar"><span class="id" title="definition">codeNewVar</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="checkSubFormulaTrace" class="idref" href="#checkSubFormulaTrace"><span class="id" title="definition">checkSubFormulaTrace</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="trace:57" class="idref" href="#trace:57"><span class="id" title="binder">trace</span></a> <a id="recs:58" class="idref" href="#recs:58"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:59" class="idref" href="#v:59"><span class="id" title="binder">v</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="input:61" class="idref" href="#input:61"><span class="id" title="binder">input</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="output:62" class="idref" href="#output:62"><span class="id" title="binder">output</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="rest:63" class="idref" href="#rest:63"><span class="id" title="binder">rest</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi3"><span class="id" title="definition">cTriplePi3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="type:64" class="idref" href="#type:64"><span class="id" title="binder">type</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#type:64"><span class="id" title="variable">type</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#type:64"><span class="id" title="variable">type</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#type:64"><span class="id" title="variable">type</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#type:64"><span class="id" title="variable">type</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#type:64"><span class="id" title="variable">type</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerms"><span class="id" title="definition">codeSubTerms</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeIn"><span class="id" title="definition">codeIn</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)) (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">let</span> <a id="nv:65" class="idref" href="#nv:65"><span class="id" title="binder">nv</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeNewVar"><span class="id" title="definition">codeNewVar</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeList.html#codeApp"><span class="id" title="definition">codeApp</span></a> (<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm"><span class="id" title="definition">codeFreeVarTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula"><span class="id" title="definition">codeFreeVarFormula</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))))) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#nv:65"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#nv:65"><span class="id" title="variable">nv</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)) (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#trace:57"><span class="id" title="variable">trace</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rest:63"><span class="id" title="variable">rest</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:58"><span class="id" title="variable">recs</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#output:62"><span class="id" title="variable">output</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerm"><span class="id" title="definition">codeSubTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerm"><span class="id" title="definition">codeSubTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#input:61"><span class="id" title="variable">input</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:60"><span class="id" title="variable">s</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="makeTraceImp" class="idref" href="#makeTraceImp"><span class="id" title="definition">makeTraceImp</span></a> (<a id="f1:66" class="idref" href="#f1:66"><span class="id" title="binder">f1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="f1rec:67" class="idref" href="#f1rec:67"><span class="id" title="binder">f1rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f2:68" class="idref" href="#f2:68"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="f2rec:69" class="idref" href="#f2rec:69"><span class="id" title="binder">f2rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:70" class="idref" href="#p:70"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:71" class="idref" href="#v:71"><span class="id" title="binder">v</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:70"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="s:72" class="idref" href="#s:72"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:70"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:71"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:72"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f1:66"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:68"><span class="id" title="variable">f2</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f1:66"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:68"><span class="id" title="variable">f2</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:71"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:72"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f1rec:67"><span class="id" title="variable">f1rec</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:70"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2rec:69"><span class="id" title="variable">f2rec</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:70"><span class="id" title="variable">p</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="makeTraceNot" class="idref" href="#makeTraceNot"><span class="id" title="definition">makeTraceNot</span></a> <br/>
+&nbsp;&nbsp;(<a id="f:73" class="idref" href="#f:73"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="frec:74" class="idref" href="#frec:74"><span class="id" title="binder">frec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:75" class="idref" href="#p:75"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:76" class="idref" href="#v:76"><span class="id" title="binder">v</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:75"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="s:77" class="idref" href="#s:77"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:75"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:76"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:77"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:73"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:73"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:76"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:77"><span class="id" title="variable">s</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#frec:74"><span class="id" title="variable">frec</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:75"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="makeTraceForall" class="idref" href="#makeTraceForall"><span class="id" title="definition">makeTraceForall</span></a> (<a id="n:78" class="idref" href="#n:78"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:79" class="idref" href="#f:79"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="rec:81" class="idref" href="#rec:81"><span class="id" title="binder">rec</span></a> : <span class="id" title="keyword">∀</span> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:80"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:78"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:79"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:82" class="idref" href="#p:82"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="makeTrace" class="idref" href="#makeTrace"><span class="id" title="definition">makeTrace</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Formula_depth_rec2"><span class="id" title="definition">Formula_depth_rec2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="t:83" class="idref" href="#t:83"><span class="id" title="binder">t</span></a> <a id="t0:84" class="idref" href="#t0:84"><span class="id" title="binder">t0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="p:85" class="idref" href="#p:85"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:86" class="idref" href="#v:86"><span class="id" title="binder">v</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:85"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="s:87" class="idref" href="#s:87"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:85"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:86"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:87"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:83"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t0:84"><span class="id" title="variable">t0</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:83"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t0:84"><span class="id" title="variable">t0</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:86"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:87"><span class="id" title="variable">s</span></a>)) 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="r:88" class="idref" href="#r:88"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) <a id="t:89" class="idref" href="#t:89"><span class="id" title="binder">t</span></a> (<a id="p:90" class="idref" href="#p:90"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="v:91" class="idref" href="#v:91"><span class="id" title="binder">v</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:90"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="s:92" class="idref" href="#s:92"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:90"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:91"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:92"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#r:88"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:89"><span class="id" title="variable">t</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#r:88"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:89"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:91"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:92"><span class="id" title="variable">s</span></a>)) 0) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceImp"><span class="id" title="definition">makeTraceImp</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceNot"><span class="id" title="definition">makeTraceNot</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceForall"><span class="id" title="definition">makeTraceForall</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="makeTraceImpNice" class="idref" href="#makeTraceImpNice"><span class="id" title="lemma">makeTraceImpNice</span></a> (<a id="f2:93" class="idref" href="#f2:93"><span class="id" title="binder">f2</span></a> <a id="g:94" class="idref" href="#g:94"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="z1:95" class="idref" href="#z1:95"><span class="id" title="binder">z1</span></a> <a id="z2:96" class="idref" href="#z2:96"><span class="id" title="binder">z2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:97" class="idref" href="#q:97"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:95"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:97"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:96"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:97"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="z3:98" class="idref" href="#z3:98"><span class="id" title="binder">z3</span></a> <a id="z4:99" class="idref" href="#z4:99"><span class="id" title="binder">z4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:100" class="idref" href="#q:100"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z3:98"><span class="id" title="variable">z3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:100"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z4:99"><span class="id" title="variable">z4</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:100"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:101" class="idref" href="#q:101"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceImp"><span class="id" title="definition">makeTraceImp</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:93"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:95"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:94"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z3:98"><span class="id" title="variable">z3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:101"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceImp"><span class="id" title="definition">makeTraceImp</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:93"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:96"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:94"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z4:99"><span class="id" title="variable">z4</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:101"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="makeTraceNotNice" class="idref" href="#makeTraceNotNice"><span class="id" title="lemma">makeTraceNotNice</span></a> (<a id="f2:102" class="idref" href="#f2:102"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="z1:103" class="idref" href="#z1:103"><span class="id" title="binder">z1</span></a> <a id="z2:104" class="idref" href="#z2:104"><span class="id" title="binder">z2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:105" class="idref" href="#q:105"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:103"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:105"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:104"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:105"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="q:106" class="idref" href="#q:106"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceNot"><span class="id" title="definition">makeTraceNot</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:102"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:103"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:106"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceNot"><span class="id" title="definition">makeTraceNot</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:102"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:104"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:106"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="makeTraceForallNice" class="idref" href="#makeTraceForallNice"><span class="id" title="lemma">makeTraceForallNice</span></a> <br/>
+&nbsp;&nbsp;(<a id="v0:107" class="idref" href="#v0:107"><span class="id" title="binder">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:108" class="idref" href="#a:108"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="z1:110" class="idref" href="#z1:110"><span class="id" title="binder">z1</span></a> <a id="z2:111" class="idref" href="#z2:111"><span class="id" title="binder">z2</span></a> : <span class="id" title="keyword">∀</span> <a id="b:109" class="idref" href="#b:109"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:109"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:107"><span class="id" title="variable">v0</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:108"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="b:112" class="idref" href="#b:112"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="q:113" class="idref" href="#q:113"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:112"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:107"><span class="id" title="variable">v0</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:108"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="r:114" class="idref" href="#r:114"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>), <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:110"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:112"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:113"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#r:114"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:111"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:112"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:113"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#r:114"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:115" class="idref" href="#q:115"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceForall"><span class="id" title="definition">makeTraceForall</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:107"><span class="id" title="variable">v0</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:108"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z1:110"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:115"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTraceForall"><span class="id" title="definition">makeTraceForall</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:107"><span class="id" title="variable">v0</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:108"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#z2:111"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#q:115"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="makeTrace1" class="idref" href="#makeTrace1"><span class="id" title="lemma">makeTrace1</span></a> (<a id="f:116" class="idref" href="#f:116"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:117" class="idref" href="#v:117"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:118" class="idref" href="#s:118"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi1"><span class="id" title="definition">cTriplePi1</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTrace"><span class="id" title="definition">makeTrace</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:116"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:117"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:118"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:117"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:118"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:116"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="makeTrace2" class="idref" href="#makeTrace2"><span class="id" title="lemma">makeTrace2</span></a> (<a id="f:119" class="idref" href="#f:119"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:120" class="idref" href="#v:120"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:121" class="idref" href="#s:121"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cTriplePi2"><span class="id" title="definition">cTriplePi2</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTrace"><span class="id" title="definition">makeTrace</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:119"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:120"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:121"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:119"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:120"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:121"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="makeTraceCorrect" class="idref" href="#makeTraceCorrect"><span class="id" title="lemma">makeTraceCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:122" class="idref" href="#f:122"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:123" class="idref" href="#v:123"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:124" class="idref" href="#s:124"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#checkSubFormulaTrace"><span class="id" title="definition">checkSubFormulaTrace</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTrace"><span class="id" title="definition">makeTrace</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:122"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:123"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:124"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="checkTraceCorrect" class="idref" href="#checkTraceCorrect"><span class="id" title="lemma">checkTraceCorrect</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:131" class="idref" href="#f:131"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:132" class="idref" href="#v:132"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:133" class="idref" href="#s:133"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="n:134" class="idref" href="#n:134"><span class="id" title="binder">n</span></a> <a id="m:135" class="idref" href="#m:135"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#checkSubFormulaTrace"><span class="id" title="definition">checkSubFormulaTrace</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:132"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:133"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:131"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:134"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#m:135"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><br/>
+&nbsp;0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:131"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:132"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:133"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:134"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="switch5IsPR" class="idref" href="#switch5IsPR"><span class="id" title="lemma">switch5IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:144" class="idref" href="#f1:144"><span class="id" title="binder">f1</span></a> <a id="f2:145" class="idref" href="#f2:145"><span class="id" title="binder">f2</span></a> <a id="f3:146" class="idref" href="#f3:146"><span class="id" title="binder">f3</span></a> <a id="f4:147" class="idref" href="#f4:147"><span class="id" title="binder">f4</span></a> <a id="f5:148" class="idref" href="#f5:148"><span class="id" title="binder">f5</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:149" class="idref" href="#g:149"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f1:144"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:145"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f3:146"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f4:147"><span class="id" title="variable">f4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f5:148"><span class="id" title="variable">f5</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:149"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:150" class="idref" href="#n:150"><span class="id" title="binder">n</span></a> <a id="recs:151" class="idref" href="#recs:151"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:149"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:149"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:149"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#g:149"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a>)))) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f1:144"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:151"><span class="id" title="variable">recs</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f2:145"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:151"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f3:146"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:151"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f4:147"><span class="id" title="variable">f4</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:151"><span class="id" title="variable">recs</span></a>)) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f5:148"><span class="id" title="variable">f5</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:150"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:151"><span class="id" title="variable">recs</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="checkTraceIsPR" class="idref" href="#checkTraceIsPR"><span class="id" title="instance">checkTraceIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#checkSubFormulaTrace"><span class="id" title="definition">checkSubFormulaTrace</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ReplaceTermTermsTerm" class="idref" href="#ReplaceTermTermsTerm"><span class="id" title="definition">ReplaceTermTermsTerm</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="t:692" class="idref" href="#t:692"><span class="id" title="binder">t</span></a> <a id="recs:693" class="idref" href="#recs:693"><span class="id" title="binder">recs</span></a> <a id="s:694" class="idref" href="#s:694"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:693"><span class="id" title="variable">recs</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:694"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:693"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:692"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:693"><span class="id" title="variable">recs</span></a>)))) 0)).<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="ReplaceTermTermsTermIsPR" class="idref" href="#ReplaceTermTermsTermIsPR"><span class="id" title="instance">ReplaceTermTermsTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTerm"><span class="id" title="definition">ReplaceTermTermsTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ReplaceTermTerm" class="idref" href="#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a id="t:815" class="idref" href="#t:815"><span class="id" title="binder">t</span></a> <a id="s:816" class="idref" href="#s:816"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTerm"><span class="id" title="definition">ReplaceTermTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:815"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:816"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ReplaceTermsTerm" class="idref" href="#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> (<a id="t:817" class="idref" href="#t:817"><span class="id" title="binder">t</span></a> <a id="s:818" class="idref" href="#s:818"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTerm"><span class="id" title="definition">ReplaceTermTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:817"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:818"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ReplaceTermTermIsPR" class="idref" href="#ReplaceTermTermIsPR"><span class="id" title="instance">ReplaceTermTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ReplaceTermsTermIsPR" class="idref" href="#ReplaceTermsTermIsPR"><span class="id" title="instance">ReplaceTermsTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="ReplaceTermTermsTermMonotone" class="idref" href="#ReplaceTermTermsTermMonotone"><span class="id" title="lemma">ReplaceTermTermsTermMonotone</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:819" class="idref" href="#a:819"><span class="id" title="binder">a</span></a> <a id="s1:820" class="idref" href="#s1:820"><span class="id" title="binder">s1</span></a> <a id="s2:821" class="idref" href="#s2:821"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:820"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:821"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:819"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:820"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:819"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:821"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:819"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:820"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:819"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:821"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceTermTermMonotone" class="idref" href="#ReplaceTermTermMonotone"><span class="id" title="lemma">ReplaceTermTermMonotone</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:842" class="idref" href="#a:842"><span class="id" title="binder">a</span></a> <a id="s1:843" class="idref" href="#s1:843"><span class="id" title="binder">s1</span></a> <a id="s2:844" class="idref" href="#s2:844"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:843"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:844"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:842"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:843"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:842"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:844"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceTermsTermMonotone" class="idref" href="#ReplaceTermsTermMonotone"><span class="id" title="lemma">ReplaceTermsTermMonotone</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:845" class="idref" href="#a:845"><span class="id" title="binder">a</span></a> <a id="s1:846" class="idref" href="#s1:846"><span class="id" title="binder">s1</span></a> <a id="s2:847" class="idref" href="#s2:847"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:846"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:847"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:845"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:846"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:845"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:847"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxLemma" class="idref" href="#maxLemma"><span class="id" title="lemma">maxLemma</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:848" class="idref" href="#a:848"><span class="id" title="binder">a</span></a> <a id="b:849" class="idref" href="#b:849"><span class="id" title="binder">b</span></a> <a id="c:850" class="idref" href="#c:850"><span class="id" title="binder">c</span></a> <a id="d:851" class="idref" href="#d:851"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:848"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:849"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#c:850"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#d:851"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:848"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#c:850"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:849"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#d:851"><span class="id" title="variable">d</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxLemma2" class="idref" href="#maxLemma2"><span class="id" title="lemma">maxLemma2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:852" class="idref" href="#a:852"><span class="id" title="binder">a</span></a> <a id="b:853" class="idref" href="#b:853"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:852"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:852"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:853"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxLemma3" class="idref" href="#maxLemma3"><span class="id" title="lemma">maxLemma3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:854" class="idref" href="#a:854"><span class="id" title="binder">a</span></a> <a id="b:855" class="idref" href="#b:855"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:855"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:854"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:855"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxApp" class="idref" href="#maxApp"><span class="id" title="lemma">maxApp</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:856" class="idref" href="#a:856"><span class="id" title="binder">a</span></a> <a id="b:857" class="idref" href="#b:857"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:856"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:857"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:856"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:856"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:857"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:857"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundSubTerm" class="idref" href="#boundSubTerm"><span class="id" title="lemma">boundSubTerm</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:858" class="idref" href="#t:858"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:859" class="idref" href="#v:859"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:860" class="idref" href="#s:860"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:858"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:859"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:860"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:858"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:860"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:858"><span class="id" title="variable">t</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundSubTerms" class="idref" href="#boundSubTerms"><span class="id" title="lemma">boundSubTerms</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:881" class="idref" href="#n:881"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:882" class="idref" href="#ts:882"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:881"><span class="id" title="variable">n</span></a>) (<a id="v:883" class="idref" href="#v:883"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:884" class="idref" href="#s:884"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:882"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:883"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:884"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a>  <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:882"><span class="id" title="variable">ts</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:884"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:882"><span class="id" title="variable">ts</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceTermTermSub" class="idref" href="#ReplaceTermTermSub"><span class="id" title="lemma">ReplaceTermTermSub</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:897" class="idref" href="#t:897"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:898" class="idref" href="#v:898"><span class="id" title="binder">v</span></a> <a id="w:899" class="idref" href="#w:899"><span class="id" title="binder">w</span></a> <a id="s2:900" class="idref" href="#s2:900"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:897"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:898"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#w:899"><span class="id" title="variable">w</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:900"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:897"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:900"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceTermsTermSub" class="idref" href="#ReplaceTermsTermSub"><span class="id" title="lemma">ReplaceTermsTermSub</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:923" class="idref" href="#n:923"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:924" class="idref" href="#ts:924"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:923"><span class="id" title="variable">n</span></a>) (<a id="v:925" class="idref" href="#v:925"><span class="id" title="binder">v</span></a> <a id="w:926" class="idref" href="#w:926"><span class="id" title="binder">w</span></a> <a id="s2:927" class="idref" href="#s2:927"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:924"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:925"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#w:926"><span class="id" title="variable">w</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:927"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:924"><span class="id" title="variable">ts</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:927"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ReplaceFormulaTerm" class="idref" href="#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="f:940" class="idref" href="#f:940"><span class="id" title="binder">f</span></a> <a id="recs:941" class="idref" href="#recs:941"><span class="id" title="binder">recs</span></a> <a id="s:942" class="idref" href="#s:942"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm"><span class="id" title="definition">ReplaceTermsTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:942"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:942"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:941"><span class="id" title="variable">recs</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 2 (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:941"><span class="id" title="variable">recs</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:941"><span class="id" title="variable">recs</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#recs:941"><span class="id" title="variable">recs</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:942"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm"><span class="id" title="definition">ReplaceTermTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:940"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:942"><span class="id" title="variable">s</span></a>)))).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ReplaceFormulaTermIsPR" class="idref" href="#ReplaceFormulaTermIsPR"><span class="id" title="instance">ReplaceFormulaTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceFormulaTermMonotone" class="idref" href="#ReplaceFormulaTermMonotone"><span class="id" title="lemma">ReplaceFormulaTermMonotone</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:1209" class="idref" href="#a:1209"><span class="id" title="binder">a</span></a> <a id="s1:1210" class="idref" href="#s1:1210"><span class="id" title="binder">s1</span></a> <a id="s2:1211" class="idref" href="#s2:1211"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:1210"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:1211"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1209"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s1:1210"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1209"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:1211"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="varFormula" class="idref" href="#varFormula"><span class="id" title="definition">varFormula</span></a> (<a id="f:1232" class="idref" href="#f:1232"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1232"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">s</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula:1233"><span class="id" title="definition">varFormula</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula:1233"><span class="id" title="definition">varFormula</span></a> <span class="id" title="var">B</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula:1233"><span class="id" title="definition">varFormula</span></a> <span class="id" title="var">A</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> ⇒ <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula:1233"><span class="id" title="definition">varFormula</span></a> <span class="id" title="var">A</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ReplaceFormulaTermSub" class="idref" href="#ReplaceFormulaTermSub"><span class="id" title="lemma">ReplaceFormulaTermSub</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:1235" class="idref" href="#f:1235"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:1236" class="idref" href="#v:1236"><span class="id" title="binder">v</span></a> <a id="w:1237" class="idref" href="#w:1237"><span class="id" title="binder">w</span></a> <a id="s2:1238" class="idref" href="#s2:1238"><span class="id" title="binder">s2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1235"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1236"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#w:1237"><span class="id" title="variable">w</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:1238"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1235"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s2:1238"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="codeTermFreeVar" class="idref" href="#codeTermFreeVar"><span class="id" title="lemma">codeTermFreeVar</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="s:1251" class="idref" href="#s:1251"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1251"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1251"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxVarFreeVar" class="idref" href="#maxVarFreeVar"><span class="id" title="lemma">maxVarFreeVar</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:1256" class="idref" href="#f:1256"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1256"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1256"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxSubTerm" class="idref" href="#maxSubTerm"><span class="id" title="lemma">maxSubTerm</span></a> (<a id="t:1257" class="idref" href="#t:1257"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:1258" class="idref" href="#v:1258"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1259" class="idref" href="#s:1259"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:1257"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1258"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1259"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1259"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#t:1257"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="maxSubTerms" class="idref" href="#maxSubTerms"><span class="id" title="lemma">maxSubTerms</span></a> (<a id="n:1264" class="idref" href="#n:1264"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:1265" class="idref" href="#ts:1265"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#n:1264"><span class="id" title="variable">n</span></a>) (<a id="v:1266" class="idref" href="#v:1266"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1267" class="idref" href="#s:1267"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:1265"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1266"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1267"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1267"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#ts:1265"><span class="id" title="variable">ts</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode">3</span> <span class="inlinecode">^</span> <span class="inlinecode"><span class="id" title="var">n</span></span> 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="pow3" class="idref" href="#pow3"><span class="id" title="definition">pow3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 1 (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="rec:1268" class="idref" href="#rec:1268"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rec:1268"><span class="id" title="variable">rec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#rec:1268"><span class="id" title="variable">rec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rec:1268"><span class="id" title="variable">rec</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pow3IsPR" class="idref" href="#pow3IsPR"><span class="id" title="instance">pow3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pow3Monotone" class="idref" href="#pow3Monotone"><span class="id" title="lemma">pow3Monotone</span></a> : <span class="id" title="keyword">∀</span> <a id="a:1281" class="idref" href="#a:1281"><span class="id" title="binder">a</span></a> <a id="b:1282" class="idref" href="#b:1282"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1281"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1282"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1281"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1282"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pow3Min" class="idref" href="#pow3Min"><span class="id" title="lemma">pow3Min</span></a> : <span class="id" title="keyword">∀</span> <a id="a:1283" class="idref" href="#a:1283"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1283"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="mapListLemma" class="idref" href="#mapListLemma"><span class="id" title="lemma">mapListLemma</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:1284" class="idref" href="#l:1284"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#l:1284"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#l:1284"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="boundSubFormulaHelp2" class="idref" href="#boundSubFormulaHelp2"><span class="id" title="lemma">boundSubFormulaHelp2</span></a>  (<a id="a:1285" class="idref" href="#a:1285"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v0:1286" class="idref" href="#v0:1286"><span class="id" title="binder">v0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1287" class="idref" href="#s:1287"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:1286"><span class="id" title="variable">v0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1287"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1285"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:1286"><span class="id" title="variable">v0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1287"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1285"><span class="id" title="variable">a</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="boundSubFormulaHelp1" class="idref" href="#boundSubFormulaHelp1"><span class="id" title="lemma">boundSubFormulaHelp1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="b:1288" class="idref" href="#b:1288"><span class="id" title="binder">b</span></a> <a id="a:1289" class="idref" href="#a:1289"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v0:1290" class="idref" href="#v0:1290"><span class="id" title="binder">v0</span></a> <a id="v:1291" class="idref" href="#v:1291"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1292" class="idref" href="#s:1292"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1288"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1291"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:1290"><span class="id" title="variable">v0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1292"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1289"><span class="id" title="variable">a</span></a>))))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1288"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1288"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v0:1290"><span class="id" title="variable">v0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1292"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1291"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b:1288"><span class="id" title="variable">b</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a:1289"><span class="id" title="variable">a</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="boundSubFormulaHelp" class="idref" href="#boundSubFormulaHelp"><span class="id" title="lemma">boundSubFormulaHelp</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:1293" class="idref" href="#f:1293"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:1294" class="idref" href="#v:1294"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1295" class="idref" href="#s:1295"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1293"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1294"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1295"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1293"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1295"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1293"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1294"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1295"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1293"><span class="id" title="variable">f</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="boundComputation" class="idref" href="#boundComputation"><span class="id" title="definition">boundComputation</span></a> (<a id="d:1308" class="idref" href="#d:1308"><span class="id" title="binder">d</span></a> <a id="p1:1309" class="idref" href="#p1:1309"><span class="id" title="binder">p1</span></a> <a id="p2:1310" class="idref" href="#p2:1310"><span class="id" title="binder">p2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p1:1309"><span class="id" title="variable">p1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p2:1310"><span class="id" title="variable">p2</span></a> 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="rec:1311" class="idref" href="#rec:1311"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p1:1309"><span class="id" title="variable">p1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p2:1310"><span class="id" title="variable">p2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rec:1311"><span class="id" title="variable">rec</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#rec:1311"><span class="id" title="variable">rec</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#d:1308"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="boundComputationIsPR" class="idref" href="#boundComputationIsPR"><span class="id" title="instance">boundComputationIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#boundComputation"><span class="id" title="definition">boundComputation</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundComputationMonotone" class="idref" href="#boundComputationMonotone"><span class="id" title="lemma">boundComputationMonotone</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a1:1360" class="idref" href="#a1:1360"><span class="id" title="binder">a1</span></a> <a id="a2:1361" class="idref" href="#a2:1361"><span class="id" title="binder">a2</span></a> <a id="b1:1362" class="idref" href="#b1:1362"><span class="id" title="binder">b1</span></a> <a id="b2:1363" class="idref" href="#b2:1363"><span class="id" title="binder">b2</span></a> <a id="c1:1364" class="idref" href="#c1:1364"><span class="id" title="binder">c1</span></a> <a id="c2:1365" class="idref" href="#c2:1365"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#a1:1360"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a2:1361"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#b1:1362"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b2:1363"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#c1:1364"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#c2:1365"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#boundComputation"><span class="id" title="definition">boundComputation</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a1:1360"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b1:1362"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#c1:1364"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#boundComputation"><span class="id" title="definition">boundComputation</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#a2:1361"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#b2:1363"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#c2:1365"><span class="id" title="variable">c2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundMakeTrace" class="idref" href="#boundMakeTrace"><span class="id" title="lemma">boundMakeTrace</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:1366" class="idref" href="#f:1366"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>) (<a id="v:1367" class="idref" href="#v:1367"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1368" class="idref" href="#s:1368"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="C:1369" class="idref" href="#C:1369"><span class="id" title="binder">C</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1368"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1367"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1368"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#varFormula"><span class="id" title="definition">varFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a>))) <span class="id" title="tactic">in</span><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#makeTrace"><span class="id" title="definition">makeTrace</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1367"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1368"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#boundComputation"><span class="id" title="definition">boundComputation</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1369"><span class="id" title="variable">C</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1369"><span class="id" title="variable">C</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1369"><span class="id" title="variable">C</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1366"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1369"><span class="id" title="variable">C</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSubFormula" class="idref" href="#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> (<a id="f:1404" class="idref" href="#f:1404"><span class="id" title="binder">f</span></a> <a id="v:1405" class="idref" href="#v:1405"><span class="id" title="binder">v</span></a> <a id="s:1406" class="idref" href="#s:1406"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="C:1407" class="idref" href="#C:1407"><span class="id" title="binder">C</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#pow3"><span class="id" title="definition">pow3</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1405"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1406"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">))</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="p:1408" class="idref" href="#p:1408"><span class="id" title="binder">p</span></a> <a id="x:1409" class="idref" href="#x:1409"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> 0 (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#checkSubFormulaTrace"><span class="id" title="definition">checkSubFormulaTrace</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#p:1408"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#x:1409"><span class="id" title="variable">x</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1405"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1406"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#boundComputation"><span class="id" title="definition">boundComputation</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cTriple"><span class="id" title="definition">cTriple</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1407"><span class="id" title="variable">C</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1407"><span class="id" title="variable">C</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1407"><span class="id" title="variable">C</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm"><span class="id" title="definition">ReplaceFormulaTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1404"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#C:1407"><span class="id" title="variable">C</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSubFormulaCorrect" class="idref" href="#codeSubFormulaCorrect"><span class="id" title="lemma">codeSubFormulaCorrect</span></a> (<a id="f:1410" class="idref" href="#f:1410"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Formula"><span class="id" title="abbreviation">Formula</span></a>) (<a id="v:1411" class="idref" href="#v:1411"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:1412" class="idref" href="#s:1412"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Term"><span class="id" title="abbreviation">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1410"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1411"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm"><span class="id" title="variable">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1412"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula"><span class="id" title="variable">codeFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#f:1410"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#v:1411"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#s:1412"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeSubFormulaIsPR" class="idref" href="#codeSubFormulaIsPR"><span class="id" title="instance">codeSubFormulaIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.codeSubFormula.html#codeSubFormula"><span class="id" title="definition">codeSubFormula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula"><span class="id" title="section">Code_Substitute_Formula</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.codeSubTerm.html b/theories/html/hydras.Ackermann.codeSubTerm.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.codeSubTerm.html
@@ -0,0 +1,100 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.codeSubTerm</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.codeSubTerm</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span>  <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Code_Substitute_Term" class="idref" href="#Code_Substitute_Term"><span class="id" title="section">Code_Substitute_Term</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="Code_Substitute_Term.L" class="idref" href="#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Term.Formula" class="idref" href="#Code_Substitute_Term.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Term.Formulas" class="idref" href="#Code_Substitute_Term.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Term.System" class="idref" href="#Code_Substitute_Term.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Term.Term" class="idref" href="#Code_Substitute_Term.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Code_Substitute_Term.Terms" class="idref" href="#Code_Substitute_Term.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSubTermTerms" class="idref" href="#codeSubTermTerms"><span class="id" title="definition">codeSubTermTerms</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="t:5" class="idref" href="#t:5"><span class="id" title="binder">t</span></a> <a id="recs:6" class="idref" href="#recs:6"><span class="id" title="binder">recs</span></a> <a id="v:7" class="idref" href="#v:7"><span class="id" title="binder">v</span></a> <a id="s:8" class="idref" href="#s:8"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>)) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#recs:6"><span class="id" title="variable">recs</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:7"><span class="id" title="variable">v</span></a>) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:8"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#recs:6"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:5"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#recs:6"><span class="id" title="variable">recs</span></a>)))) 0)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSubTerm" class="idref" href="#codeSubTerm"><span class="id" title="definition">codeSubTerm</span></a> (<a id="t:9" class="idref" href="#t:9"><span class="id" title="binder">t</span></a> <a id="s:10" class="idref" href="#s:10"><span class="id" title="binder">s</span></a> <a id="v:11" class="idref" href="#v:11"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTermTerms"><span class="id" title="definition">codeSubTermTerms</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:9"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:10"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:11"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="codeSubTerms" class="idref" href="#codeSubTerms"><span class="id" title="definition">codeSubTerms</span></a> (<a id="t:12" class="idref" href="#t:12"><span class="id" title="binder">t</span></a> <a id="s:13" class="idref" href="#s:13"><span class="id" title="binder">s</span></a> <a id="v:14" class="idref" href="#v:14"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTermTerms"><span class="id" title="definition">codeSubTermTerms</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:13"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:14"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSubTermCorrect" class="idref" href="#codeSubTermCorrect"><span class="id" title="lemma">codeSubTermCorrect</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:15" class="idref" href="#t:15"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Term"><span class="id" title="variable">Term</span></a>) (<a id="v:16" class="idref" href="#v:16"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:17" class="idref" href="#s:17"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerm"><span class="id" title="definition">codeSubTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:15"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:16"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:17"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#t:15"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:16"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:17"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSubTermsCorrect" class="idref" href="#codeSubTermsCorrect"><span class="id" title="lemma">codeSubTermsCorrect</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:36" class="idref" href="#n:36"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:37" class="idref" href="#ts:37"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#n:36"><span class="id" title="variable">n</span></a>) (<a id="v:38" class="idref" href="#v:38"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:39" class="idref" href="#s:39"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerms"><span class="id" title="definition">codeSubTerms</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#ts:37"><span class="id" title="variable">ts</span></a>) <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:38"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:39"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#ts:37"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#v:38"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#s:39"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeSubTermTermsIsPR" class="idref" href="#codeSubTermTermsIsPR"><span class="id" title="instance">codeSubTermTermsIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTermTerms"><span class="id" title="definition">codeSubTermTerms</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeSubTermIsPR" class="idref" href="#codeSubTermIsPR"><span class="id" title="instance">codeSubTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerm"><span class="id" title="definition">codeSubTerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeSubTermsIsPR" class="idref" href="#codeSubTermsIsPR"><span class="id" title="lemma">codeSubTermsIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.codeSubTerm.html#codeSubTerms"><span class="id" title="definition">codeSubTerms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term"><span class="id" title="section">Code_Substitute_Term</span></a>.<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<title>hydras.Ackermann.expressible</title>
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+
+<h1 class="libtitle">Library hydras.Ackermann.expressible</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ expressible.v 
+
+<div class="paragraph"> </div>
+
+     Original file by Russel O'Connor
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="RepresentableExpressible" class="idref" href="#RepresentableExpressible"><span class="id" title="section">RepresentableExpressible</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="RepresentableExpressible.T" class="idref" href="#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="RepresentableExpressible.closedT1" class="idref" href="#RepresentableExpressible.closedT1"><span class="id" title="variable">closedT1</span></a>: (<a class="idref" href="hydras.Ackermann.folProp.html#ClosedSystem"><span class="id" title="definition">ClosedSystem</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="closedT" class="idref" href="#closedT"><span class="id" title="lemma">closedT</span></a> : <span class="id" title="keyword">∀</span> <a id="v:3" class="idref" href="#v:3"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="RepresentableHalf1" class="idref" href="#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.expressible.html#n:4"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 0) (<a id="A:8" class="idref" href="#A:8"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.expressible.html#A:8"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:7"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="A:10" class="idref" href="#A:10"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1:5"><span class="id" title="definition">RepresentableHalf1</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.expressible.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:11"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:10"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:11"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="RepresentableHalf2" class="idref" href="#RepresentableHalf2"><span class="id" title="definition">RepresentableHalf2</span></a> (<a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.expressible.html#n:12"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:15" class="idref" href="#f:15"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 0) (<a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:15"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:16"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:17" class="idref" href="#f:17"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:19" class="idref" href="#a:19"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf2:13"><span class="id" title="definition">RepresentableHalf2</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.expressible.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:19"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:18"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:19"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="RepresentableHalf1Alternate" class="idref" href="#RepresentableHalf1Alternate"><span class="id" title="lemma">RepresentableHalf1Alternate</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:20"><span class="id" title="variable">n</span></a>) (<a id="A:22" class="idref" href="#A:22"><span class="id" title="binder">A</span></a> <a id="B:23" class="idref" href="#B:23"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:22"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:23"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:23"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:22"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="RepresentableHalf2Alternate" class="idref" href="#RepresentableHalf2Alternate"><span class="id" title="lemma">RepresentableHalf2Alternate</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:25" class="idref" href="#f:25"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:24"><span class="id" title="variable">n</span></a>) (<a id="A:26" class="idref" href="#A:26"><span class="id" title="binder">A</span></a> <a id="B:27" class="idref" href="#B:27"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.expressible.html#A:26"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:27"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf2"><span class="id" title="definition">RepresentableHalf2</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:26"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf2"><span class="id" title="definition">RepresentableHalf2</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:27"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="RepresentableHelp" class="idref" href="#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> (<a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.expressible.html#n:28"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:31" class="idref" href="#f:31"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 0) (<a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.expressible.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:31"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:33" class="idref" href="#f:33"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="A:34" class="idref" href="#A:34"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp:29"><span class="id" title="definition">RepresentableHelp</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.expressible.html#f:33"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:35"><span class="id" title="variable">a</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:34"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:35"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="RepresentableHalfHelp" class="idref" href="#RepresentableHalfHelp"><span class="id" title="lemma">RepresentableHalfHelp</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:36" class="idref" href="#n:36"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:36"><span class="id" title="variable">n</span></a>) (<a id="A:38" class="idref" href="#A:38"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:36"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:38"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf2"><span class="id" title="definition">RepresentableHalf2</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:36"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:38"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:36"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:38"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Representable" class="idref" href="#Representable"><span class="id" title="definition">Representable</span></a> (<a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:40" class="idref" href="#f:40"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:39"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;(<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:42" class="idref" href="#v:42"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#v:42"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:41"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#v:42"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:39"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:40"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:41"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="RepresentableAlternate" class="idref" href="#RepresentableAlternate"><span class="id" title="lemma">RepresentableAlternate</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:43"><span class="id" title="variable">n</span></a>) (<a id="A:45" class="idref" href="#A:45"><span class="id" title="binder">A</span></a> <a id="B:46" class="idref" href="#B:46"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:45"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:46"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:43"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:45"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:43"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:46"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Representable_ext" class="idref" href="#Representable_ext"><span class="id" title="lemma">Representable_ext</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:47" class="idref" href="#n:47"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> <a id="g:49" class="idref" href="#g:49"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:47"><span class="id" title="variable">n</span></a>) (<a id="A:50" class="idref" href="#A:50"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:47"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#g:49"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:47"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:50"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHelp"><span class="id" title="definition">RepresentableHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:47"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#g:49"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:50"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="ExpressibleHelp" class="idref" href="#ExpressibleHelp"><span class="id" title="definition">ExpressibleHelp</span></a> (<a id="n:51" class="idref" href="#n:51"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:51"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.expressible.html#n:51"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="R:54" class="idref" href="#R:54"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 0) (<a id="A:55" class="idref" href="#A:55"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.expressible.html#R:54"><span class="id" title="variable">R</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> ⇒ <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:55"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> ⇒ <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> ( <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:55"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="R:57" class="idref" href="#R:57"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="A:58" class="idref" href="#A:58"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:59" class="idref" href="#a:59"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#ExpressibleHelp:52"><span class="id" title="definition">ExpressibleHelp</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.expressible.html#R:57"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:59"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:58"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#a:59"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Expressible" class="idref" href="#Expressible"><span class="id" title="definition">Expressible</span></a> (<a id="n:60" class="idref" href="#n:60"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:61" class="idref" href="#R:61"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:60"><span class="id" title="variable">n</span></a>) (<a id="A:62" class="idref" href="#A:62"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:63" class="idref" href="#v:63"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#v:63"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:62"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#v:63"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:60"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#v:63"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#ExpressibleHelp"><span class="id" title="definition">ExpressibleHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:60"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#R:61"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:62"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="expressibleAlternate" class="idref" href="#expressibleAlternate"><span class="id" title="lemma">expressibleAlternate</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:64" class="idref" href="#n:64"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:65" class="idref" href="#R:65"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:64"><span class="id" title="variable">n</span></a>) (<a id="A:66" class="idref" href="#A:66"><span class="id" title="binder">A</span></a> <a id="B:67" class="idref" href="#B:67"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:66"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:67"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#ExpressibleHelp"><span class="id" title="definition">ExpressibleHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:64"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#R:65"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:66"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#ExpressibleHelp"><span class="id" title="definition">ExpressibleHelp</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:64"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#R:65"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#B:67"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="RepresentableExpressible.nn1" class="idref" href="#RepresentableExpressible.nn1"><span class="id" title="variable">nn1</span></a>: <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible.T"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 0)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Representable2Expressible" class="idref" href="#Representable2Expressible"><span class="id" title="lemma">Representable2Expressible</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:69" class="idref" href="#n:69"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:70" class="idref" href="#R:70"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:69"><span class="id" title="variable">n</span></a>) (<a id="A:71" class="idref" href="#A:71"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#Representable"><span class="id" title="definition">Representable</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:69"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:69"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#R:70"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.Ackermann.expressible.html#A:71"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.expressible.html#Expressible"><span class="id" title="definition">Expressible</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#n:69"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#R:70"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.expressible.html#A:71"><span class="id" title="variable">A</span></a> 0 (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 1)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableExpressible"><span class="id" title="section">RepresentableExpressible</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.extEqualNat</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.extEqualNat</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+extEqualNat:
+
+<div class="paragraph"> </div>
+
+     Original script by Russel O'Connor 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="naryFunc" class="idref" href="#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc:2"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:1"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="naryRel" class="idref" href="#naryRel"><span class="id" title="definition">naryRel</span></a> (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:4"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel:5"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:4"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span>  <a id="extEqual" class="idref" href="#extEqual"><span class="id" title="definition">extEqual</span></a> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">∀</span>  (<a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a> <a id="b:9" class="idref" href="#b:9"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:7"><span class="id" title="variable">n</span></a>), <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:7"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <span class="id" title="keyword">fun</span> <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#b:13"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <span class="id" title="keyword">fun</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a> <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a> ⇒ <span class="id" title="keyword">∀</span> <a id="c:16" class="idref" href="#c:16"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual:10"><span class="id" title="definition">extEqual</span></a> <span class="id" title="var">p</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#c:16"><span class="id" title="variable">c</span></a>) (<a class="idref" href="hydras.Ackermann.extEqualNat.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#c:16"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="charFunction" class="idref" href="#charFunction"><span class="id" title="definition">charFunction</span></a> (<a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:17"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:17"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:19"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="R:20" class="idref" href="#R:20"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#R:20"><span class="id" title="variable">R</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <span class="id" title="keyword">fun</span> (<a id="R:22" class="idref" href="#R:22"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction:18"><span class="id" title="definition">charFunction</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#R:22"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:23"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualRefl" class="idref" href="#extEqualRefl"><span class="id" title="lemma">extEqualRefl</span></a> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> <a id="a:25" class="idref" href="#a:25"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:25"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualSym" class="idref" href="#extEqualSym"><span class="id" title="lemma">extEqualSym</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="b:28" class="idref" href="#b:28"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:26"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:27"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualTrans" class="idref" href="#extEqualTrans"><span class="id" title="lemma">extEqualTrans</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a> <a id="c:32" class="idref" href="#c:32"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:29"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:29"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:29"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#c:32"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#n:29"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#c:32"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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\ No newline at end of file
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.fol.html
@@ -0,0 +1,457 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.fol</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.fol</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+First Order Logic 
+
+<div class="paragraph"> </div>
+
+ Original author: Russell O'Connor 
+
+<div class="paragraph"> </div>
+
+This file is Public Domain 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab266"></a><h1 class="section">First Order Formulas over a language</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="Language" class="idref" href="#Language"><span class="id" title="record">Language</span></a> : <span class="id" title="keyword">Type</span> := <a id="language" class="idref" href="#language"><span class="id" title="constructor">language</span></a><br/>
+&nbsp;{ <a id="Relations" class="idref" href="#Relations"><span class="id" title="projection">Relations</span></a> : <span class="id" title="keyword">Set</span>; <br/>
+&nbsp;&nbsp;&nbsp;<a id="Functions" class="idref" href="#Functions"><span class="id" title="projection">Functions</span></a> : <span class="id" title="keyword">Set</span>; <br/>
+&nbsp;&nbsp;&nbsp;<a id="arityR" class="idref" href="#arityR"><span class="id" title="projection">arityR</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations:2"><span class="id" title="method">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;<a id="arityF" class="idref" href="#arityF"><span class="id" title="projection">arityF</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions:3"><span class="id" title="method">Functions</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="First_Order_Logic" class="idref" href="#First_Order_Logic"><span class="id" title="section">First_Order_Logic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="First_Order_Logic.L" class="idref" href="#First_Order_Logic.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Term" class="idref" href="#Term"><span class="id" title="inductive">Term</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="var" class="idref" href="#var"><span class="id" title="constructor">var</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term:7"><span class="id" title="inductive">Term</span></a><br/>
+&nbsp;&nbsp;| <a id="apply" class="idref" href="#apply"><span class="id" title="constructor">apply</span></a> : <span class="id" title="keyword">∀</span> <a id="f:11" class="idref" href="#f:11"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#Terms:8"><span class="id" title="inductive">Terms</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:11"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term:7"><span class="id" title="inductive">Term</span></a><br/>
+<span class="id" title="keyword">with</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="inductive">Terms</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Tnil" class="idref" href="#Tnil"><span class="id" title="constructor">Tnil</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms:8"><span class="id" title="inductive">Terms</span></a> 0<br/>
+&nbsp;&nbsp;| <a id="Tcons" class="idref" href="#Tcons"><span class="id" title="constructor">Tcons</span></a> : <span class="id" title="keyword">∀</span> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#Term:7"><span class="id" title="inductive">Term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms:8"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.fol.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms:8"><span class="id" title="inductive">Terms</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.fol.html#n:12"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="Term_Terms_ind" class="idref" href="#Term_Terms_ind"><span class="id" title="definition">Term_Terms_ind</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a id="Terms_Term_ind" class="idref" href="#Terms_Term_ind"><span class="id" title="definition">Terms_Term_ind</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="Term_Terms_rec" class="idref" href="#Term_Terms_rec"><span class="id" title="definition">Term_Terms_rec</span></a> := <span class="id" title="var">Minimality</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a id="Terms_Term_rec" class="idref" href="#Terms_Term_rec"><span class="id" title="definition">Terms_Term_rec</span></a> := <span class="id" title="var">Minimality</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Scheme</span> <a id="Term_Terms_rec_full" class="idref" href="#Term_Terms_rec_full"><span class="id" title="definition">Term_Terms_rec_full</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a>  <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a id="Terms_Term_rec_full" class="idref" href="#Terms_Term_rec_full"><span class="id" title="definition">Terms_Term_rec_full</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="inductive">Formula</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="equal" class="idref" href="#equal"><span class="id" title="constructor">equal</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a><br/>
+&nbsp;&nbsp;| <a id="atomic" class="idref" href="#atomic"><span class="id" title="constructor">atomic</span></a> : <span class="id" title="keyword">∀</span> <a id="r:15" class="idref" href="#r:15"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.fol.html#r:15"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a><br/>
+&nbsp;&nbsp;| <a id="impH" class="idref" href="#impH"><span class="id" title="constructor">impH</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a><br/>
+&nbsp;&nbsp;| <a id="notH" class="idref" href="#notH"><span class="id" title="constructor">notH</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a><br/>
+&nbsp;&nbsp;| <a id="forallH" class="idref" href="#forallH"><span class="id" title="constructor">forallH</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula:13"><span class="id" title="inductive">Formula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="definition">Formulas</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="System" class="idref" href="#System"><span class="id" title="definition">System</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="mem" class="idref" href="#mem"><span class="id" title="definition">mem</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">Ensembles.In</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab267"></a><h2 class="section">Extensions of the basic language of formulas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="orH" class="idref" href="#orH"><span class="id" title="definition">orH</span></a> (<a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a> <a id="B:17" class="idref" href="#B:17"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) := <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:16"><span class="id" title="variable">A</span></a>) <a class="idref" href="hydras.Ackermann.fol.html#B:17"><span class="id" title="variable">B</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="andH" class="idref" href="#andH"><span class="id" title="definition">andH</span></a> (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> <a id="B:19" class="idref" href="#B:19"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) := <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:18"><span class="id" title="variable">A</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#B:19"><span class="id" title="variable">B</span></a>)).<br/>
+<span class="id" title="keyword">Definition</span> <a id="iffH" class="idref" href="#iffH"><span class="id" title="definition">iffH</span></a> (<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a> <a id="B:21" class="idref" href="#B:21"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) := <a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#B:21"><span class="id" title="variable">B</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#B:21"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:20"><span class="id" title="variable">A</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="existH" class="idref" href="#existH"><span class="id" title="definition">existH</span></a> (<a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="A:23" class="idref" href="#A:23"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) := <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#x:22"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:23"><span class="id" title="variable">A</span></a>)).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ifThenElseH" class="idref" href="#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> (<a id="A:24" class="idref" href="#A:24"><span class="id" title="binder">A</span></a> <a id="B:25" class="idref" href="#B:25"><span class="id" title="binder">B</span></a> <a id="C:26" class="idref" href="#C:26"><span class="id" title="binder">C</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) := <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:24"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#B:25"><span class="id" title="variable">B</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:24"><span class="id" title="variable">A</span></a>) <a class="idref" href="hydras.Ackermann.fol.html#C:26"><span class="id" title="variable">C</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab268"></a><h2 class="section">Decidability of equality between terms, formulas, ...</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="First_Order_Logic.Formula_Decidability" class="idref" href="#First_Order_Logic.Formula_Decidability"><span class="id" title="section">Formula_Decidability</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="language_decidable" class="idref" href="#language_decidable"><span class="id" title="definition">language_decidable</span></a> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:27" class="idref" href="#x:27"><span class="id" title="binder">x</span></a> <a id="y:28" class="idref" href="#y:28"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:27"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:28"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:27"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:28"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a> <a id="y:30" class="idref" href="#y:30"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:29"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:30"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:29"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:30"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Hypothesis</span> <a id="First_Order_Logic.Formula_Decidability.language_eqdec" class="idref" href="#First_Order_Logic.Formula_Decidability.language_eqdec"><span class="id" title="variable">language_eqdec</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#language_decidable"><span class="id" title="definition">language_decidable</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="term_eqdec" class="idref" href="#term_eqdec"><span class="id" title="lemma">term_eqdec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:60" class="idref" href="#x:60"><span class="id" title="binder">x</span></a> <a id="y:61" class="idref" href="#y:61"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:60"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:61"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:60"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:61"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="terms_eqdec" class="idref" href="#terms_eqdec"><span class="id" title="lemma">terms_eqdec</span></a> <a id="n:102" class="idref" href="#n:102"><span class="id" title="binder">n</span></a>  (<a id="x:103" class="idref" href="#x:103"><span class="id" title="binder">x</span></a> <a id="y:104" class="idref" href="#y:104"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.fol.html#n:102"><span class="id" title="variable">n</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:103"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:104"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:103"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:104"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="formula_eqdec" class="idref" href="#formula_eqdec"><span class="id" title="lemma">formula_eqdec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:107" class="idref" href="#x:107"><span class="id" title="binder">x</span></a> <a id="y:108" class="idref" href="#y:108"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:108"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#y:108"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability"><span class="id" title="section">Formula_Decidability</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab269"></a><h1 class="section">Depth Induction</h1>
+
+
+<div class="paragraph"> </div>
+
+  Many functions on term and formulas are not structurally recursive (e.g. because of substitution of variables with terms).
+  In which case, we may use some notion of depth as a measure. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="First_Order_Logic.Formula_Depth_Induction" class="idref" href="#First_Order_Logic.Formula_Depth_Induction"><span class="id" title="section">Formula_Depth_Induction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="depth" class="idref" href="#depth"><span class="id" title="definition">depth</span></a> (<a id="A:137" class="idref" href="#A:137"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.fol.html#A:137"><span class="id" title="variable">A</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth:138"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:137"><span class="id" title="variable">A</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#depth:138"><span class="id" title="definition">depth</span></a> <span class="id" title="var">B</span>))<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth:138"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:137"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth:138"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:137"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt_depth" class="idref" href="#lt_depth"><span class="id" title="definition">lt_depth</span></a> (<a id="A:140" class="idref" href="#A:140"><span class="id" title="binder">A</span></a> <a id="B:141" class="idref" href="#B:141"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>) : <span class="id" title="keyword">Prop</span> := <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#A:140"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#B:141"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Formula_depth_ind" class="idref" href="#Formula_depth_ind"><span class="id" title="definition">Formula_depth_ind</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:427" class="idref" href="#P:427"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:428" class="idref" href="#a:428"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:429" class="idref" href="#b:429"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b:429"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a:428"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:427"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b:429"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:427"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a:428"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:430" class="idref" href="#a:430"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:427"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a:430"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Formula_depth_ind2" class="idref" href="#Formula_depth_ind2"><span class="id" title="lemma">Formula_depth_ind2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="P:435" class="idref" href="#P:435"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t:436" class="idref" href="#t:436"><span class="id" title="binder">t</span></a> <a id="t0:437" class="idref" href="#t0:437"><span class="id" title="binder">t0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="hydras.Ackermann.fol.html#t:436"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.fol.html#t0:437"><span class="id" title="variable">t0</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="r:438" class="idref" href="#r:438"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="t:439" class="idref" href="#t:439"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.fol.html#r:438"><span class="id" title="variable">r</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.fol.html#r:438"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.fol.html#t:439"><span class="id" title="variable">t</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:440" class="idref" href="#f:440"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:440"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="f0:441" class="idref" href="#f0:441"><span class="id" title="binder">f0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f0:441"><span class="id" title="variable">f0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:440"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f0:441"><span class="id" title="variable">f0</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:442" class="idref" href="#f:442"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:442"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:442"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="v:443" class="idref" href="#v:443"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:444" class="idref" href="#a:444"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:445" class="idref" href="#b:445"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b:445"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#v:443"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a:444"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b:445"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.fol.html#v:443"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a:444"><span class="id" title="variable">a</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:446" class="idref" href="#f:446"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a>, <a class="idref" href="hydras.Ackermann.fol.html#P:435"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#f:446"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Depth_Induction"><span class="id" title="section">Formula_Depth_Induction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#First_Order_Logic"><span class="id" title="section">First_Order_Logic</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab270"></a><h1 class="section">Implicit arguments and notations</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ In Russel O'Connor's work, the abstract syntax of first-order terms and formulas is available in three versions:
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> A generic one (for any Language <span class="inlinecode"><span class="id" title="var">L</span></span>)
+
+</li>
+<li> An instantiation for the language of number theory <span class="inlinecode"><span class="id" title="var">LNT</span></span>
+
+</li>
+<li> An instantiation for the language of natural numbers <span class="inlinecode"><span class="id" title="var">LNN</span></span>
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+In the current version, we propose to use three notation scopes: <span class="inlinecode"><span class="id" title="var">fol_scope</span></span>, <span class="inlinecode"><span class="id" title="var">nt_scope</span></span> and  <span class="inlinecode"><span class="id" title="var">nn_scope</span></span>, in order to make clear the relationship between the three sets of formulas. 
+
+<div class="paragraph"> </div>
+
+ <span class="inlinecode"><span class="id" title="var">fol_scope</span></span> is defined in this file, <span class="inlinecode"><span class="id" title="var">lnt_scope</span></span> in 
+<a href="./hydras.Ackermann.LNT.html">LNT.v</a>, and
+<span class="inlinecode"><span class="id" title="var">lnn_scope</span></span> in 
+<a href="./hydras.Ackermann.LNT.html">LNN.v</a>,
+
+<div class="paragraph"> </div>
+
+<a id="lab271"></a><h2 class="section">Implicit arguments</h2>
+
+The original code of this library contains some redefinitions like 
+
+<div class="paragraph"> </div>
+
+<pre>
+Definition Formula := Formula LNN.
+</pre>
+
+<div class="paragraph"> </div>
+
+We plan to use systematically implicit arguments and avoid such redefinitions, which make more complex formula and term displaying, e.g. in goals or results of computation.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> {<span class="id" title="var">L</span>}.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> {<span class="id" title="var">L</span>} {<span class="id" title="var">n</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="027d47fdf302b28789c559f47c303a6a" class="idref" href="#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">&quot;</span></a>'f[' A ']f'" := <span class="id" title="var">A</span> (<span class="id" title="var">A</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="146ef165103f02c940f12358e2d84d5c" class="idref" href="#146ef165103f02c940f12358e2d84d5c"><span class="id" title="notation">&quot;</span></a>x ∨ y" := (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 85, <span class="id" title="var">x</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="var">y</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="64a970f6f4c4d1914c59e379c64a481b" class="idref" href="#64a970f6f4c4d1914c59e379c64a481b"><span class="id" title="notation">&quot;</span></a>x ∧ y" := (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 80, <span class="id" title="var">x</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="var">y</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":fol::x_'-&gt;'_x" class="idref" href="#:fol::x_'-&gt;'_x"><span class="id" title="notation">&quot;</span></a>x -&gt; y" := (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":fol::x_'&lt;-&gt;'_x" class="idref" href="#:fol::x_'&lt;-&gt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt;-&gt; y" := (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="b9c468757fc602e0e6e4d7f82378fa40" class="idref" href="#b9c468757fc602e0e6e4d7f82378fa40"><span class="id" title="notation">&quot;</span></a>~ x" := (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">x</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 75, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="299c44a9fb7c7425e5e57147661f89e4" class="idref" href="#299c44a9fb7c7425e5e57147661f89e4"><span class="id" title="notation">&quot;</span></a>( x )" := <span class="id" title="var">x</span> (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="var">x</span>  <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="99fc41bec617cc902ba1cfab8f026785" class="idref" href="#99fc41bec617cc902ba1cfab8f026785"><span class="id" title="notation">&quot;</span></a>x = y" := (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":fol::x_'&lt;&gt;'_x" class="idref" href="#:fol::x_'&lt;&gt;'_x"><span class="id" title="notation">&quot;</span></a>x &lt;&gt; y" := (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="b8de2539629ab8cd49874e6c530636b7" class="idref" href="#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">&quot;</span></a>{ x }" := <span class="id" title="var">x</span> (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span>, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="80736841a977d43b8bf9e594dd2212d2" class="idref" href="#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">&quot;</span></a>'∀' i ',' f" := (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">i</span> <span class="id" title="var">f</span>) (<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="var">i</span> <span class="id" title="keyword">constr</span>, <span class="id" title="var">f</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="0e2109d6ebbe119a5bf3c8b38478024d" class="idref" href="#0e2109d6ebbe119a5bf3c8b38478024d"><span class="id" title="notation">&quot;</span></a>'∃' i , f" := (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a>  <span class="id" title="var">i</span> <span class="id" title="var">f</span>) <br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="var">i</span> <span class="id" title="keyword">constr</span>, <span class="id" title="var">f</span> <span class="id" title="var">custom</span> <span class="id" title="var">fol</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">∀</span></a> 3<a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0e2109d6ebbe119a5bf3c8b38478024d"><span class="id" title="notation">∃</span></a> 4<a class="idref" href="hydras.Ackermann.fol.html#0e2109d6ebbe119a5bf3c8b38478024d"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 3) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 4)<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#146ef165103f02c940f12358e2d84d5c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 3<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#99fc41bec617cc902ba1cfab8f026785"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 4<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b9c468757fc602e0e6e4d7f82378fa40"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a> <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 3) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 4) <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 3<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#:fol::x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 4<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b9c468757fc602e0e6e4d7f82378fa40"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#299c44a9fb7c7425e5e57147661f89e4"><span class="id" title="notation">(</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 3<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#99fc41bec617cc902ba1cfab8f026785"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 4<a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#299c44a9fb7c7425e5e57147661f89e4"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab272"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="var">fol_scope</span></span> notation scope</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="FolNotations" class="idref" href="#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">fol_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="3dd041be98fa842518e2776cd5aaa2a0" class="idref" href="#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">&quot;</span></a>=" := (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">_</span>): <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="95d02b143473b1bf1f30676a3f27357c" class="idref" href="#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">&quot;</span></a>∨" := (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a>): <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="4ba889964d74201d72d870be149fb4e5" class="idref" href="#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">&quot;</span></a>∧" := (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a>):<span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="FolNotations.::fol_scope:x_'-&gt;'_x" class="idref" href="#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">&quot;</span></a>→" := (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a>): <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="7648da2bbc0fe930b661dc530c384448" class="idref" href="#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">&quot;</span></a>~ A" := (@<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span>): <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="FolNotations.::fol_scope:x_'&lt;-&gt;'_x" class="idref" href="#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">&quot;</span></a>A &lt;-&gt; B" := (@<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span>): <span class="id" title="var">fol_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="a12ef5915927c1f20da57c369b8b6c71" class="idref" href="#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">&quot;</span></a>'v#' i" := (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">i</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="var">format</span> "'v#' i", <span class="id" title="var">i</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) : <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="0b83cac238622d86e6ca74792daf53c5" class="idref" href="#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">&quot;</span></a>'exH' x .. y , p" := (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a>  <span class="id" title="var">x</span> .. (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p</span>) ..)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="ab366141b54751aa5ea1077b9b863e32" class="idref" href="#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">&quot;</span></a>'allH' x .. y , p" := (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a>  <span class="id" title="var">x</span> .. (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p</span>) ..)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 200, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">fol_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="3dd041be98fa842518e2776cd5aaa2a0" class="idref" href="#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">&quot;</span></a>t = u" := (@<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">_</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span>): <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="FolNotations.::fol_scope:x_'&lt;&gt;'_x" class="idref" href="#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">&quot;</span></a>t &lt;&gt; u" := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <span class="id" title="var">u</span>)%<span class="id" title="var">fol</span> : <span class="id" title="var">fol_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+the following notations are used when  some computation 
+    expands a disjunction, conjuction, etc. 
+    in terms of implication and negation 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '\/'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 85, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '/\'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 80, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '&lt;-&gt;'' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 95, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '&lt;-&gt;''' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 95, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="e4a489994216e94595670248e88226a2" class="idref" href="#e4a489994216e94595670248e88226a2"><span class="id" title="notation">&quot;</span></a>x \/' y" := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span>)%<span class="id" title="var">fol</span> : <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="bb04b0324214688cf30460e944283f4c" class="idref" href="#bb04b0324214688cf30460e944283f4c"><span class="id" title="notation">&quot;</span></a>x /\' y" := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(~</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.fol.html#e4a489994216e94595670248e88226a2"><span class="id" title="notation">\/'</span></a>  <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span> : <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="FolNotations.::fol_scope:x_'&lt;-&gt;'''''_x" class="idref" href="#FolNotations.::fol_scope:x_'&lt;-&gt;'''''_x"><span class="id" title="notation">&quot;</span></a>x &lt;-&gt;'' y" := (<a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">x</span><a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>:  <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="FolNotations.::fol_scope:x_'&lt;-&gt;'''_x" class="idref" href="#FolNotations.::fol_scope:x_'&lt;-&gt;'''_x"><span class="id" title="notation">&quot;</span></a>x &lt;-&gt;' y" := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(~</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#e4a489994216e94595670248e88226a2"><span class="id" title="notation">\/'</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">x</span><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">))</span></a>)%<span class="id" title="var">fol</span> : <span class="id" title="var">fol_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="FolNotations.exH'" class="idref" href="#FolNotations.exH'"><span class="id" title="abbreviation">exH'</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">A</span>)<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>5)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab273"></a><h2 class="section">Examples</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="LExamples" class="idref" href="#LExamples"><span class="id" title="section">LExamples</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="LExamples.L" class="idref" href="#LExamples.L"><span class="id" title="variable">L</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Variables</span> <a id="LExamples.P" class="idref" href="#LExamples.P"><span class="id" title="variable">P</span></a> <a id="LExamples.Q" class="idref" href="#LExamples.Q"><span class="id" title="variable">Q</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.fol.html#LExamples.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="LExamples.ex1" class="idref" href="#LExamples.ex1"><span class="id" title="variable">ex1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.fol.html#LExamples.L"><span class="id" title="variable">L</span></a> :=  (<a class="idref" href="hydras.Ackermann.fol.html#LExamples.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#LExamples.Q"><span class="id" title="variable">Q</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="LExamples.ex2" class="idref" href="#LExamples.ex2"><span class="id" title="variable">ex2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.fol.html#LExamples.L"><span class="id" title="variable">L</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(~</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.fol.html#LExamples.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.fol.html#LExamples.Q"><span class="id" title="variable">Q</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="LExamples.ex3" class="idref" href="#LExamples.ex3"><span class="id" title="variable">ex3</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.fol.html#LExamples.L"><span class="id" title="variable">L</span></a>:= (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(~</span></a><a class="idref" href="hydras.Ackermann.fol.html#LExamples.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.fol.html#LExamples.Q"><span class="id" title="variable">Q</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#LExamples"><span class="id" title="section">LExamples</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Correctness" class="idref" href="#Correctness"><span class="id" title="section">Correctness</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Correctness.L" class="idref" href="#Correctness.L"><span class="id" title="variable">L</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> <a id="Correctness.P" class="idref" href="#Correctness.P"><span class="id" title="variable">P</span></a> <a id="Correctness.Q" class="idref" href="#Correctness.Q"><span class="id" title="variable">Q</span></a> <a id="Correctness.R" class="idref" href="#Correctness.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Correctness.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∨</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">Q</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">fol</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">\/'</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">Q</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">fol</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">∧</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Q</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">fol</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">/\'</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Q</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">fol</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.fol.html#Correctness"><span class="id" title="section">Correctness</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<title>hydras.Ackermann.folLogic</title>
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+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folLogic</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+folLogic.v:
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#"><span class="id" title="library">Deduction</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Logic_Rules" class="idref" href="#Logic_Rules"><span class="id" title="section">Logic_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Logic_Rules.L" class="idref" href="#Logic_Rules.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.Formula" class="idref" href="#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.Formulas" class="idref" href="#Logic_Rules.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.System" class="idref" href="#Logic_Rules.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.Term" class="idref" href="#Logic_Rules.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.Terms" class="idref" href="#Logic_Rules.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.Prf" class="idref" href="#Logic_Rules.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Logic_Rules.SysPrf" class="idref" href="#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> {<span class="id" title="var">U</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Axm" class="idref" href="#Axm"><span class="id" title="lemma">Axm</span></a> <a id="T:2" class="idref" href="#T:2"><span class="id" title="binder">T</span></a> <a id="f:3" class="idref" href="#f:3"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folLogic.html#T:2"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:2"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:3"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysExtend" class="idref" href="#sysExtend"><span class="id" title="lemma">sysExtend</span></a>  (<a id="T:4" class="idref" href="#T:4"><span class="id" title="binder">T</span></a> <a id="U:5" class="idref" href="#U:5"><span class="id" title="binder">U</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:6" class="idref" href="#f:6"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>): <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folLogic.html#T:4"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#U:5"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:4"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:6"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#U:5"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:6"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sysWeaken" class="idref" href="#sysWeaken"><span class="id" title="lemma">sysWeaken</span></a>  (<a id="T:7" class="idref" href="#T:7"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> <a id="g:9" class="idref" href="#g:9"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:7"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#T:7"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:9"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.folLogic.html#f:8"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impI" class="idref" href="#impI"><span class="id" title="lemma">impI</span></a> (<a id="T:10" class="idref" href="#T:10"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:11" class="idref" href="#f:11"><span class="id" title="binder">f</span></a> <a id="g:12" class="idref" href="#g:12"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#T:10"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:12"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.folLogic.html#f:11"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:10"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:12"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:11"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impE" class="idref" href="#impE"><span class="id" title="lemma">impE</span></a> (<a id="T:13" class="idref" href="#T:13"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:14" class="idref" href="#f:14"><span class="id" title="binder">f</span></a> <a id="g:15" class="idref" href="#g:15"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:13"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:15"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:14"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:13"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:15"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:13"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:14"><span class="id" title="variable">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="contradiction" class="idref" href="#contradiction"><span class="id" title="lemma">contradiction</span></a> (<a id="T:16" class="idref" href="#T:16"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:17" class="idref" href="#f:17"><span class="id" title="binder">f</span></a> <a id="g:18" class="idref" href="#g:18"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:16"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:16"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:17"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:16"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:18"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnE" class="idref" href="#nnE"><span class="id" title="lemma">nnE</span></a> (<a id="T:19" class="idref" href="#T:19"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:20" class="idref" href="#f:20"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>): <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:19"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:20"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:19"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:20"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nnI" class="idref" href="#nnI"><span class="id" title="lemma">nnI</span></a> (<a id="T:21" class="idref" href="#T:21"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:22" class="idref" href="#f:22"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>): <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:21"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:22"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:21"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:22"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ contraposition 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp1" class="idref" href="#cp1"><span class="id" title="lemma">cp1</span></a> (<a id="T:23" class="idref" href="#T:23"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> <a id="g:25" class="idref" href="#g:25"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:23"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:24"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:25"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:23"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:25"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:24"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cp2" class="idref" href="#cp2"><span class="id" title="lemma">cp2</span></a> (<a id="T:26" class="idref" href="#T:26"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> <a id="g:28" class="idref" href="#g:28"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:26"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:28"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:27"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:26"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#g:28"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI1" class="idref" href="#orI1"><span class="id" title="lemma">orI1</span></a> (<a id="T:29" class="idref" href="#T:29"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> <a id="g:31" class="idref" href="#g:31"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>): <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:29"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:29"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:31"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orI2" class="idref" href="#orI2"><span class="id" title="lemma">orI2</span></a> (<a id="T:32" class="idref" href="#T:32"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:33" class="idref" href="#f:33"><span class="id" title="binder">f</span></a> <a id="g:34" class="idref" href="#g:34"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>): <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:32"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:34"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:32"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:33"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:34"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="noMiddle" class="idref" href="#noMiddle"><span class="id" title="lemma">noMiddle</span></a>  (<a id="T:35" class="idref" href="#T:35"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:36" class="idref" href="#f:36"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):  <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:35"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:36"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:36"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+    
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orE" class="idref" href="#orE"><span class="id" title="lemma">orE</span></a> (<a id="T:37" class="idref" href="#T:37"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:38" class="idref" href="#f:38"><span class="id" title="binder">f</span></a> <a id="g:39" class="idref" href="#g:39"><span class="id" title="binder">g</span></a> <a id="h:40" class="idref" href="#h:40"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:37"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:39"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:37"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:40"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:37"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:39"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:40"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:37"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:40"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSys" class="idref" href="#orSys"><span class="id" title="lemma">orSys</span></a> (<a id="T:41" class="idref" href="#T:41"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a> <a id="g:43" class="idref" href="#g:43"><span class="id" title="binder">g</span></a> <a id="h:44" class="idref" href="#h:44"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#T:41"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:42"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folLogic.html#h:44"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:41"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:43"><span class="id" title="variable">g</span></a>) <a class="idref" href="hydras.Ackermann.folLogic.html#h:44"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#T:41"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:43"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>) <a class="idref" href="hydras.Ackermann.folLogic.html#h:44"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andI" class="idref" href="#andI"><span class="id" title="lemma">andI</span></a> (<a id="T:45" class="idref" href="#T:45"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:46" class="idref" href="#f:46"><span class="id" title="binder">f</span></a> <a id="g:47" class="idref" href="#g:47"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:45"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:45"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:47"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:45"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:47"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE1" class="idref" href="#andE1"><span class="id" title="lemma">andE1</span></a> (<a id="T:48" class="idref" href="#T:48"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:49" class="idref" href="#f:49"><span class="id" title="binder">f</span></a> <a id="g:50" class="idref" href="#g:50"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:48"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:50"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:48"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:49"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andE2" class="idref" href="#andE2"><span class="id" title="lemma">andE2</span></a>  (<a id="T:51" class="idref" href="#T:51"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a> <a id="g:53" class="idref" href="#g:53"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:51"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:53"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:51"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:53"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffI" class="idref" href="#iffI"><span class="id" title="lemma">iffI</span></a>  (<a id="T:54" class="idref" href="#T:54"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:55" class="idref" href="#f:55"><span class="id" title="binder">f</span></a> <a id="g:56" class="idref" href="#g:56"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:54"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:56"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:54"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:56"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:55"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:54"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:56"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE1" class="idref" href="#iffE1"><span class="id" title="lemma">iffE1</span></a> (<a id="T:57" class="idref" href="#T:57"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:58" class="idref" href="#f:58"><span class="id" title="binder">f</span></a> <a id="g:59" class="idref" href="#g:59"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:57"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:58"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:59"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:57"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:58"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:59"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffE2" class="idref" href="#iffE2"><span class="id" title="lemma">iffE2</span></a>  (<a id="T:60" class="idref" href="#T:60"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:61" class="idref" href="#f:61"><span class="id" title="binder">f</span></a> <a id="g:62" class="idref" href="#g:62"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:60"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:61"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:62"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:60"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:62"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:61"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallI" class="idref" href="#forallI"><span class="id" title="lemma">forallI</span></a> (<a id="T:63" class="idref" href="#T:63"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:64" class="idref" href="#f:64"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:65" class="idref" href="#v:65"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:65"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:63"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:63"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:64"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:63"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:65"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:64"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallE" class="idref" href="#forallE"><span class="id" title="lemma">forallE</span></a>  (<a id="T:66" class="idref" href="#T:66"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:67" class="idref" href="#f:67"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:68" class="idref" href="#v:68"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:69" class="idref" href="#t:69"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:66"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:68"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:67"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:66"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:67"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:68"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#t:69"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forallSimp" class="idref" href="#forallSimp"><span class="id" title="lemma">forallSimp</span></a> (<a id="T:70" class="idref" href="#T:70"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:71" class="idref" href="#f:71"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:72" class="idref" href="#v:72"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:70"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:72"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:71"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:70"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:71"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existI" class="idref" href="#existI"><span class="id" title="lemma">existI</span></a> (<a id="T:73" class="idref" href="#T:73"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:75" class="idref" href="#v:75"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:76" class="idref" href="#t:76"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:73"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:74"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:75"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#t:76"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:73"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:75"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:74"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existE" class="idref" href="#existE"><span class="id" title="lemma">existE</span></a> (<a id="T:77" class="idref" href="#T:77"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:78" class="idref" href="#f:78"><span class="id" title="binder">f</span></a> <a id="g:79" class="idref" href="#g:79"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:80" class="idref" href="#v:80"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:80"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:77"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:80"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:79"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:77"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:80"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:78"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:77"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:78"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:79"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:77"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:79"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSimp" class="idref" href="#existSimp"><span class="id" title="lemma">existSimp</span></a>  (<a id="T:81" class="idref" href="#T:81"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:82" class="idref" href="#f:82"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:83" class="idref" href="#v:83"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:81"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:82"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:81"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:83"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:82"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="existSys" class="idref" href="#existSys"><span class="id" title="lemma">existSys</span></a> (<a id="T:84" class="idref" href="#T:84"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:85" class="idref" href="#f:85"><span class="id" title="binder">f</span></a> <a id="g:86" class="idref" href="#g:86"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:87" class="idref" href="#v:87"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:87"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:84"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:87"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:86"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:84"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:85"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folLogic.html#g:86"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#T:84"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:87"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:85"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>) <a class="idref" href="hydras.Ackermann.folLogic.html#g:86"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Logic_Rules.Not_Rules" class="idref" href="#Logic_Rules.Not_Rules"><span class="id" title="section">Not_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="absurd1" class="idref" href="#absurd1"><span class="id" title="lemma">absurd1</span></a>  (<a id="T:88" class="idref" href="#T:88"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:89" class="idref" href="#f:89"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:88"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:89"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:89"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:88"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:89"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nImp" class="idref" href="#nImp"><span class="id" title="lemma">nImp</span></a>  (<a id="T:90" class="idref" href="#T:90"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:91" class="idref" href="#f:91"><span class="id" title="binder">f</span></a> <a id="g:92" class="idref" href="#g:92"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:90"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:91"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a>  <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#g:92"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:90"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:91"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:92"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nOr" class="idref" href="#nOr"><span class="id" title="lemma">nOr</span></a> (<a id="T:93" class="idref" href="#T:93"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:94" class="idref" href="#f:94"><span class="id" title="binder">f</span></a> <a id="g:95" class="idref" href="#g:95"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:94"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:95"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:94"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:95"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nAnd" class="idref" href="#nAnd"><span class="id" title="lemma">nAnd</span></a> (<a id="T:96" class="idref" href="#T:96"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:97" class="idref" href="#f:97"><span class="id" title="binder">f</span></a> <a id="g:98" class="idref" href="#g:98"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:96"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:97"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:98"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:96"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:97"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:98"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nForall" class="idref" href="#nForall"><span class="id" title="lemma">nForall</span></a>  (<a id="T:99" class="idref" href="#T:99"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:100" class="idref" href="#f:100"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:101" class="idref" href="#v:101"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:99"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:101"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:100"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:99"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a>  <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:101"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:100"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nExist" class="idref" href="#nExist"><span class="id" title="lemma">nExist</span></a> (<a id="T:102" class="idref" href="#T:102"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:103" class="idref" href="#f:103"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:104" class="idref" href="#v:104"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:102"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:104"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Ackermann.folLogic.html#f:103"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:102"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#v:104"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:103"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Not_Rules"><span class="id" title="section">Not_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Logic_Rules.Other_Rules" class="idref" href="#Logic_Rules.Other_Rules"><span class="id" title="section">Other_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impRefl" class="idref" href="#impRefl"><span class="id" title="lemma">impRefl</span></a> (<a id="T:105" class="idref" href="#T:105"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:106" class="idref" href="#f:106"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):  <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:105"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:106"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:106"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impTrans" class="idref" href="#impTrans"><span class="id" title="lemma">impTrans</span></a> (<a id="T:107" class="idref" href="#T:107"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:108" class="idref" href="#f:108"><span class="id" title="binder">f</span></a> <a id="g:109" class="idref" href="#g:109"><span class="id" title="binder">g</span></a> <a id="h:110" class="idref" href="#h:110"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:107"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:108"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:109"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:107"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:109"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:110"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:107"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:108"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:110"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orSym" class="idref" href="#orSym"><span class="id" title="lemma">orSym</span></a> (<a id="T:111" class="idref" href="#T:111"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:112" class="idref" href="#f:112"><span class="id" title="binder">f</span></a> <a id="g:113" class="idref" href="#g:113"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:111"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:112"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:113"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:111"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:113"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:112"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andSym" class="idref" href="#andSym"><span class="id" title="lemma">andSym</span></a> (<a id="T:114" class="idref" href="#T:114"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:115" class="idref" href="#f:115"><span class="id" title="binder">f</span></a> <a id="g:116" class="idref" href="#g:116"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:114"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:115"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:116"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:114"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:116"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:115"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffRefl" class="idref" href="#iffRefl"><span class="id" title="lemma">iffRefl</span></a> (<a id="T:117" class="idref" href="#T:117"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:118" class="idref" href="#f:118"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:117"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:118"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:118"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffSym" class="idref" href="#iffSym"><span class="id" title="lemma">iffSym</span></a> (<a id="T:119" class="idref" href="#T:119"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:120" class="idref" href="#f:120"><span class="id" title="binder">f</span></a> <a id="g:121" class="idref" href="#g:121"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:119"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:120"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="hydras.Ackermann.folLogic.html#g:121"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:119"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:121"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:120"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffTrans" class="idref" href="#iffTrans"><span class="id" title="lemma">iffTrans</span></a> (<a id="T:122" class="idref" href="#T:122"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:123" class="idref" href="#f:123"><span class="id" title="binder">f</span></a> <a id="g:124" class="idref" href="#g:124"><span class="id" title="binder">g</span></a> <a id="h:125" class="idref" href="#h:125"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:122"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:123"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#g:124"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:122"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#g:124"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:125"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:122"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic.html#f:123"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#h:125"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Other_Rules"><span class="id" title="section">Other_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="openClosed" class="idref" href="#openClosed"><span class="id" title="lemma">openClosed</span></a> (<a id="T:126" class="idref" href="#T:126"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="f:127" class="idref" href="#f:127"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:126"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#close"><span class="id" title="definition">close</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:127"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#T:126"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#f:127"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folLogic.html#Logic_Rules"><span class="id" title="section">Logic_Rules</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.folLogic2.html b/theories/html/hydras.Ackermann.folLogic2.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.folLogic2.html
@@ -0,0 +1,107 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.folLogic2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folLogic2</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+folLogic2.v
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="More_Logic_Rules" class="idref" href="#More_Logic_Rules"><span class="id" title="section">More_Logic_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="More_Logic_Rules.L" class="idref" href="#More_Logic_Rules.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.Formula" class="idref" href="#More_Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.Formulas" class="idref" href="#More_Logic_Rules.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.System" class="idref" href="#More_Logic_Rules.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.Term" class="idref" href="#More_Logic_Rules.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.Terms" class="idref" href="#More_Logic_Rules.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.Prf" class="idref" href="#More_Logic_Rules.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="More_Logic_Rules.SysPrf" class="idref" href="#More_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rebindForall" class="idref" href="#rebindForall"><span class="id" title="lemma">rebindForall</span></a> (<a id="T:2" class="idref" href="#T:2"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> <a id="b:4" class="idref" href="#b:4"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:5" class="idref" href="#f:5"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#b:4"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#f:5"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#T:2"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:3"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#f:5"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#b:4"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.folLogic2.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.Ackermann.folLogic2.html#b:4"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">)</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rebindExist" class="idref" href="#rebindExist"><span class="id" title="lemma">rebindExist</span></a> (<a id="T:6" class="idref" href="#T:6"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.System"><span class="id" title="variable">System</span></a>) (<a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#b:8"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#f:9"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#T:6"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#f:9"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#b:8"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.folLogic2.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#a:7"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#b:8"><span class="id" title="variable">b</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubTerm" class="idref" href="#subSubTerm"><span class="id" title="lemma">subSubTerm</span></a> (<a id="t:10" class="idref" href="#t:10"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term"><span class="id" title="variable">Term</span></a>) (<a id="v1:11" class="idref" href="#v1:11"><span class="id" title="binder">v1</span></a> <a id="v2:12" class="idref" href="#v2:12"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:13" class="idref" href="#s1:13"><span class="id" title="binder">s1</span></a> <a id="s2:14" class="idref" href="#s2:14"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#v1:11"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:12"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:11"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:14"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#t:10"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:11"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:13"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:12"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:14"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#t:10"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:12"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:14"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:11"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:13"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:12"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:14"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubTerms" class="idref" href="#subSubTerms"><span class="id" title="lemma">subSubTerms</span></a> (<a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:20" class="idref" href="#ts:20"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#n:19"><span class="id" title="variable">n</span></a>) (<a id="v1:21" class="idref" href="#v1:21"><span class="id" title="binder">v1</span></a> <a id="v2:22" class="idref" href="#v2:22"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:23" class="idref" href="#s1:23"><span class="id" title="binder">s1</span></a> <a id="s2:24" class="idref" href="#s2:24"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#v1:21"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:22"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:21"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:24"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#ts:20"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:21"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:23"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:22"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:24"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#ts:20"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:22"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:24"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:21"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:23"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:22"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:24"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubFormula" class="idref" href="#subSubFormula"><span class="id" title="lemma">subSubFormula</span></a> (<a id="f:25" class="idref" href="#f:25"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v1:26" class="idref" href="#v1:26"><span class="id" title="binder">v1</span></a> <a id="v2:27" class="idref" href="#v2:27"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:28" class="idref" href="#s1:28"><span class="id" title="binder">s1</span></a> <a id="s2:29" class="idref" href="#s2:29"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#v1:26"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:27"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:26"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:29"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="T:30" class="idref" href="#T:30"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.System"><span class="id" title="variable">System</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#T:30"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.folLogic2.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:26"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:28"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:27"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:29"><span class="id" title="variable">s2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.folLogic2.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:27"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:29"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.folLogic2.html#v1:26"><span class="id" title="variable">v1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s1:28"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#v2:27"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#s2:29"><span class="id" title="variable">s2</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folLogic2.html#More_Logic_Rules"><span class="id" title="section">More_Logic_Rules</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.folLogic3.html b/theories/html/hydras.Ackermann.folLogic3.html
new file mode 100644
index 00000000..c32b2a64
--- /dev/null
+++ b/theories/html/hydras.Ackermann.folLogic3.html
@@ -0,0 +1,163 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.folLogic3</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folLogic3</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+folLogic2.3
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <a class="idref" href="hydras.Ackermann.folLogic2.html#"><span class="id" title="library">folLogic2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a> <a class="idref" href="hydras.Ackermann.misc.html#"><span class="id" title="library">misc</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Equality_Logic_Rules" class="idref" href="#Equality_Logic_Rules"><span class="id" title="section">Equality_Logic_Rules</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Equality_Logic_Rules.L" class="idref" href="#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="abbreviation">Formula</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="abbreviation">Formulas</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="System" class="idref" href="#System"><span class="id" title="abbreviation">System</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Term" class="idref" href="#Term"><span class="id" title="abbreviation">Term</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="abbreviation">Terms</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Equality_Logic_Rules.Prf" class="idref" href="#Equality_Logic_Rules.Prf"><span class="id" title="variable">Prf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Equality_Logic_Rules.SysPrf" class="idref" href="#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqRefl" class="idref" href="#eqRefl"><span class="id" title="lemma">eqRefl</span></a> (<a id="T:2" class="idref" href="#T:2"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:2"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:3"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqSym" class="idref" href="#eqSym"><span class="id" title="lemma">eqSym</span></a> (<a id="T:4" class="idref" href="#T:4"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:4"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:6"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:4"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#b:6"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:5"><span class="id" title="variable">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eqTrans" class="idref" href="#eqTrans"><span class="id" title="lemma">eqTrans</span></a>  (<a id="T:13" class="idref" href="#T:13"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a> <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a> <a id="c:16" class="idref" href="#c:16"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:13"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:15"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:13"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#c:16"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:13"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#c:16"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="PairwiseEqual" class="idref" href="#PairwiseEqual"><span class="id" title="definition">PairwiseEqual</span></a> (<a id="T:23" class="idref" href="#T:23"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:24"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="ts:27" class="idref" href="#ts:27"><span class="id" title="binder">ts</span></a> <a id="ss:28" class="idref" href="#ss:28"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> 0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">x</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="ts:29" class="idref" href="#ts:29"><span class="id" title="binder">ts</span></a> <a id="ss:30" class="idref" href="#ss:30"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">x</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="a:31" class="idref" href="#a:31"><span class="id" title="binder">a</span></a>, <a id="b:32" class="idref" href="#b:32"><span class="id" title="binder">b</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#consTerms"><span class="id" title="lemma">consTerms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:29"><span class="id" title="variable">ts</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="c:33" class="idref" href="#c:33"><span class="id" title="binder">c</span></a>, <a id="d:34" class="idref" href="#d:34"><span class="id" title="binder">d</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#consTerms"><span class="id" title="lemma">consTerms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:30"><span class="id" title="variable">ss</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:23"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:31"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#c:33"><span class="id" title="variable">c</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#PairwiseEqual:25"><span class="id" title="definition">PairwiseEqual</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:23"><span class="id" title="variable">T</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:32"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#d:34"><span class="id" title="variable">d</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+TODO : specify and document 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Equality_Logic_Rules.termsMap" class="idref" href="#Equality_Logic_Rules.termsMap"><span class="id" title="variable">termsMap</span></a> (<a id="m:35" class="idref" href="#m:35"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:36" class="idref" href="#ts:36"><span class="id" title="binder">ts</span></a> <a id="ss:37" class="idref" href="#ss:37"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#m:35"><span class="id" title="variable">m</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="subAllnVars1" class="idref" href="#subAllnVars1"><span class="id" title="lemma">subAllnVars1</span></a> (<a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:43" class="idref" href="#ts:43"><span class="id" title="binder">ts</span></a> <a id="ss:44" class="idref" href="#ss:44"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:42"><span class="id" title="variable">a</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#ts:43"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#nVars"><span class="id" title="definition">nVars</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:42"><span class="id" title="variable">a</span></a>)) (<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.termsMap"><span class="id" title="variable">termsMap</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:43"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:44"><span class="id" title="variable">ss</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="subAllnVars2" class="idref" href="#subAllnVars2"><span class="id" title="lemma">subAllnVars2</span></a> (<a id="a:49" class="idref" href="#a:49"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:50" class="idref" href="#ts:50"><span class="id" title="binder">ts</span></a> <a id="ss:51" class="idref" href="#ss:51"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:49"><span class="id" title="variable">a</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#ss:51"><span class="id" title="variable">ss</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#nVars"><span class="id" title="definition">nVars</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:49"><span class="id" title="variable">a</span></a>)) (<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.termsMap"><span class="id" title="variable">termsMap</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:50"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:51"><span class="id" title="variable">ss</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Opaque</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Transparent</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">eq_nat_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="addPairwiseEquals" class="idref" href="#addPairwiseEquals"><span class="id" title="lemma">addPairwiseEquals</span></a> (<a id="T:56" class="idref" href="#T:56"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:58" class="idref" href="#ts:58"><span class="id" title="binder">ts</span></a> <a id="ss:59" class="idref" href="#ss:59"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#PairwiseEqual"><span class="id" title="definition">PairwiseEqual</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:56"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:58"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:59"><span class="id" title="variable">ss</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m0:60" class="idref" href="#m0:60"><span class="id" title="binder">m0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:61" class="idref" href="#q:61"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.folLogic3.html#q:61"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#m0:60"><span class="id" title="variable">m0</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#q:61"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.termsMap"><span class="id" title="variable">termsMap</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:58"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:59"><span class="id" title="variable">ss</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#q:61"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f0:62" class="idref" href="#f0:62"><span class="id" title="binder">f0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:56"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) <a class="idref" href="hydras.Ackermann.folLogic3.html#f0:62"><span class="id" title="variable">f0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="n:63" class="idref" href="#n:63"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="Hrecn:64" class="idref" href="#Hrecn:64"><span class="id" title="binder">Hrecn</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#(</span></a><a class="idref" href="hydras.Ackermann.folLogic3.html#n:63"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:63"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#n:63"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:63"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Hrecn:64"><span class="id" title="variable">Hrecn</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#n:57"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.folLogic3.html#m0:60"><span class="id" title="variable">m0</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:56"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#f0:62"><span class="id" title="variable">f0</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#m0:60"><span class="id" title="variable">m0</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="equalRelation" class="idref" href="#equalRelation"><span class="id" title="lemma">equalRelation</span></a> (<a id="T:65" class="idref" href="#T:65"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="r:66" class="idref" href="#r:66"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="ts:67" class="idref" href="#ts:67"><span class="id" title="binder">ts</span></a> <a id="ss:68" class="idref" href="#ss:68"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">_</span>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#PairwiseEqual"><span class="id" title="definition">PairwiseEqual</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:65"><span class="id" title="variable">T</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:67"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:68"><span class="id" title="variable">ss</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:65"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#r:66"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:67"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#r:66"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:68"><span class="id" title="variable">ss</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="equalFunction" class="idref" href="#equalFunction"><span class="id" title="lemma">equalFunction</span></a> (<a id="T:75" class="idref" href="#T:75"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="f:76" class="idref" href="#f:76"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="ts:77" class="idref" href="#ts:77"><span class="id" title="binder">ts</span></a> <a id="ss:78" class="idref" href="#ss:78"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">_</span>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#PairwiseEqual"><span class="id" title="definition">PairwiseEqual</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:75"><span class="id" title="variable">T</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:77"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:78"><span class="id" title="variable">ss</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:75"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#f:76"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:77"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#f:76"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ss:78"><span class="id" title="variable">ss</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subWithEqualsTerm" class="idref" href="#subWithEqualsTerm"><span class="id" title="lemma">subWithEqualsTerm</span></a> (<a id="a:85" class="idref" href="#a:85"><span class="id" title="binder">a</span></a> <a id="b:86" class="idref" href="#b:86"><span class="id" title="binder">b</span></a> <a id="t:87" class="idref" href="#t:87"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="v:88" class="idref" href="#v:88"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="T:89" class="idref" href="#T:89"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:89"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:85"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:86"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:89"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#t:87"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:88"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:85"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#t:87"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:88"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:86"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subWithEqualsTerms" class="idref" href="#subWithEqualsTerms"><span class="id" title="lemma">subWithEqualsTerms</span></a> (<a id="a:94" class="idref" href="#a:94"><span class="id" title="binder">a</span></a> <a id="b:95" class="idref" href="#b:95"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="n:96" class="idref" href="#n:96"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:97" class="idref" href="#ts:97"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#n:96"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:98" class="idref" href="#v:98"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:99" class="idref" href="#T:99"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:99"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:94"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:95"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#PairwiseEqual"><span class="id" title="definition">PairwiseEqual</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:99"><span class="id" title="variable">T</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:97"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:98"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:94"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#ts:97"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:98"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:95"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subWithEquals" class="idref" href="#subWithEquals"><span class="id" title="lemma">subWithEquals</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:100" class="idref" href="#f:100"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="v:101" class="idref" href="#v:101"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:102" class="idref" href="#a:102"><span class="id" title="binder">a</span></a> <a id="b:103" class="idref" href="#b:103"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>) (<a id="T:104" class="idref" href="#T:104"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:104"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folLogic3.html#a:102"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:103"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#T:104"><span class="id" title="variable">T</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#f:100"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:101"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#a:102"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#f:100"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#v:101"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#b:103"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules"><span class="id" title="section">Equality_Logic_Rules</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.folProof</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folProof</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+folProof.v
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ProofH" class="idref" href="#ProofH"><span class="id" title="section">ProofH</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="ProofH.L" class="idref" href="#ProofH.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="ProofH.Formula" class="idref" href="#ProofH.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="ProofH.Formulas" class="idref" href="#ProofH.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="ProofH.System" class="idref" href="#ProofH.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="ProofH.Term" class="idref" href="#ProofH.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="ProofH.Terms" class="idref" href="#ProofH.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nVars" class="idref" href="#nVars"><span class="id" title="definition">nVars</span></a> (<a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#n:2"><span class="id" title="variable">n</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProof.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n0</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">let</span> (<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a>,<a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a>) := <a class="idref" href="hydras.Ackermann.folProof.html#nVars:3"><span class="id" title="definition">nVars</span></a> <span class="id" title="var">n0</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<span class="id" title="var">n0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n0</span>)) <a class="idref" href="hydras.Ackermann.folProof.html#a:5"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n0</span>))) <a class="idref" href="hydras.Ackermann.folProof.html#b:6"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ProofH.Example" class="idref" href="#ProofH.Example"><span class="id" title="section">Example</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Example"><span class="id" title="section">Example</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="AxmEq4" class="idref" href="#AxmEq4"><span class="id" title="definition">AxmEq4</span></a> (<a id="R:7" class="idref" href="#R:7"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+TODO : An example in PA 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="AxmEq5" class="idref" href="#AxmEq5"><span class="id" title="definition">AxmEq5</span></a> (<a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Prf" class="idref" href="#Prf"><span class="id" title="inductive">Prf</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formulas"><span class="id" title="variable">Formulas</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+| <a id="AXM" class="idref" href="#AXM"><span class="id" title="constructor">AXM</span></a> : <span class="id" title="keyword">∀</span> <a id="A:19" class="idref" href="#A:19"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.folProof.html#A:19"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:19"><span class="id" title="variable">A</span></a><br/>
+| <a id="MP" class="idref" href="#MP"><span class="id" title="constructor">MP</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="Hyp1:20" class="idref" href="#Hyp1:20"><span class="id" title="binder">Hyp1</span></a> <a id="Hyp2:21" class="idref" href="#Hyp2:21"><span class="id" title="binder">Hyp2</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formulas"><span class="id" title="variable">Formulas</span></a>) (<a id="A:22" class="idref" href="#A:22"><span class="id" title="binder">A</span></a> <a id="B:23" class="idref" href="#B:23"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp1:20"><span class="id" title="variable">Hyp1</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#A:22"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:23"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp2:21"><span class="id" title="variable">Hyp2</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:22"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#Hyp1:20"><span class="id" title="variable">Hyp1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp2:21"><span class="id" title="variable">Hyp2</span></a>) <a class="idref" href="hydras.Ackermann.folProof.html#B:23"><span class="id" title="variable">B</span></a><br/>
+| <a id="GEN" class="idref" href="#GEN"><span class="id" title="constructor">GEN</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="Hyp:24" class="idref" href="#Hyp:24"><span class="id" title="binder">Hyp</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formulas"><span class="id" title="variable">Formulas</span></a>) (<a id="A:25" class="idref" href="#A:25"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:26" class="idref" href="#v:26"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:26"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp:24"><span class="id" title="variable">Hyp</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp:24"><span class="id" title="variable">Hyp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:25"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp:24"><span class="id" title="variable">Hyp</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:26"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:25"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="IMP1" class="idref" href="#IMP1"><span class="id" title="constructor">IMP1</span></a> : <span class="id" title="keyword">∀</span> <a id="A:27" class="idref" href="#A:27"><span class="id" title="binder">A</span></a> <a id="B:28" class="idref" href="#B:28"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:28"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:27"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="IMP2" class="idref" href="#IMP2"><span class="id" title="constructor">IMP2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:29" class="idref" href="#A:29"><span class="id" title="binder">A</span></a> <a id="B:30" class="idref" href="#B:30"><span class="id" title="binder">B</span></a> <a id="C:31" class="idref" href="#C:31"><span class="id" title="binder">C</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProof.html#A:29"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:30"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#C:31"><span class="id" title="variable">C</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProof.html#A:29"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:30"><span class="id" title="variable">B</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:29"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#C:31"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="CP" class="idref" href="#CP"><span class="id" title="constructor">CP</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a> <a id="B:33" class="idref" href="#B:33"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:33"><span class="id" title="variable">B</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:33"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:32"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="FA1" class="idref" href="#FA1"><span class="id" title="constructor">FA1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:34" class="idref" href="#A:34"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:35" class="idref" href="#v:35"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:36" class="idref" href="#t:36"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:35"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:34"><span class="id" title="variable">A</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:34"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:35"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#t:36"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="FA2" class="idref" href="#FA2"><span class="id" title="constructor">FA2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:37" class="idref" href="#A:37"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:38" class="idref" href="#v:38"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:38"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:37"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#A:37"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:38"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:37"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="FA3" class="idref" href="#FA3"><span class="id" title="constructor">FA3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a> <a id="B:40" class="idref" href="#B:40"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:41" class="idref" href="#v:41"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:41"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:40"><span class="id" title="variable">B</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:41"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#A:39"><span class="id" title="variable">A</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#v:41"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#B:40"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">fol</span><br/>
+| <a id="EQ1" class="idref" href="#EQ1"><span class="id" title="constructor">EQ1</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span><br/>
+| <a id="EQ2" class="idref" href="#EQ2"><span class="id" title="constructor">EQ2</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span><br/>
+| <a id="EQ3" class="idref" href="#EQ3"><span class="id" title="constructor">EQ3</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span><br/>
+| <a id="EQ4" class="idref" href="#EQ4"><span class="id" title="constructor">EQ4</span></a> : <span class="id" title="keyword">∀</span> <a id="R:42" class="idref" href="#R:42"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#AxmEq4"><span class="id" title="definition">AxmEq4</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#R:42"><span class="id" title="variable">R</span></a>)<br/>
+| <a id="EQ5" class="idref" href="#EQ5"><span class="id" title="constructor">EQ5</span></a> : <span class="id" title="keyword">∀</span> <a id="f:43" class="idref" href="#f:43"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#Prf:17"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.folProof.html#AxmEq5"><span class="id" title="definition">AxmEq5</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:43"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="SysPrf" class="idref" href="#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a id="T:44" class="idref" href="#T:44"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.System"><span class="id" title="variable">System</span></a>) (<a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="Hyp:46" class="idref" href="#Hyp:46"><span class="id" title="binder">Hyp</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formulas"><span class="id" title="variable">Formulas</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="prf:47" class="idref" href="#prf:47"><span class="id" title="binder">prf</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp:46"><span class="id" title="variable">Hyp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:45"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:48" class="idref" href="#g:48"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#g:48"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Hyp:46"><span class="id" title="variable">Hyp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folProof.html#T:44"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#g:48"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Inconsistent" class="idref" href="#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> (<a id="T:49" class="idref" href="#T:49"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.System"><span class="id" title="variable">System</span></a>) := <span class="id" title="keyword">∀</span> <a id="f:50" class="idref" href="#f:50"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#T:49"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:50"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Consistent" class="idref" href="#Consistent"><span class="id" title="definition">Consistent</span></a> (<a id="T:51" class="idref" href="#T:51"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.System"><span class="id" title="variable">System</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#ProofH.Formula"><span class="id" title="variable">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#T:51"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:52"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="independent" class="idref" href="#independent"><span class="id" title="definition">independent</span></a> <a id="T:53" class="idref" href="#T:53"><span class="id" title="binder">T</span></a> <a id="f:54" class="idref" href="#f:54"><span class="id" title="binder">f</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#T:53"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:54"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#T:53"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#f:54"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProof.html#ProofH"><span class="id" title="section">ProofH</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.folProof.html#independent"><span class="id" title="definition">independent</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="undecidable" class="idref" href="#undecidable"><span class="id" title="abbreviation">undecidable</span></a> := (<a class="idref" href="hydras.Ackermann.folProof.html#independent"><span class="id" title="definition">independent</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.folProp.html b/theories/html/hydras.Ackermann.folProp.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.folProp.html
@@ -0,0 +1,574 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.folProp</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folProp</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fol_Properties" class="idref" href="#Fol_Properties"><span class="id" title="section">Fol_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Fol_Properties.L" class="idref" href="#Fol_Properties.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="abbreviation">Formula</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="abbreviation">Formulas</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="System" class="idref" href="#System"><span class="id" title="abbreviation">System</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Term" class="idref" href="#Term"><span class="id" title="abbreviation">Term</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="abbreviation">Terms</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Fol_Properties.lt_depth" class="idref" href="#Fol_Properties.lt_depth"><span class="id" title="variable">lt_depth</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#lt_depth"><span class="id" title="definition">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fol_Properties.Free_Variables" class="idref" href="#Fol_Properties.Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="freeVarT" class="idref" href="#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#s:2"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span> ⇒ <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs:6"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">f</span>) <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">with</span> <a id="freeVarTs" class="idref" href="#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ss:4" class="idref" href="#ss:4"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:3"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ss</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#ss:4"><span class="id" title="variable">ss</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <span class="id" title="var">t</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT:5"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs:6"><span class="id" title="definition">freeVarTs</span></a> <span class="id" title="var">m</span> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarTApply" class="idref" href="#freeVarTApply"><span class="id" title="lemma">freeVarTApply</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="ts:10" class="idref" href="#ts:10"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">_</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#ts:10"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folProp.html#ts:10"><span class="id" title="variable">ts</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="freeVarF" class="idref" href="#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#A:11"><span class="id" title="variable">A</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">t</span> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">s</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF:12"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF:12"><span class="id" title="definition">freeVarF</span></a> <span class="id" title="var">B</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF:12"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#A:11"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <span class="id" title="var">v</span> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF:12"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#A:11"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ClosedSystem" class="idref" href="#ClosedSystem"><span class="id" title="definition">ClosedSystem</span></a> (<a id="T:14" class="idref" href="#T:14"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:15" class="idref" href="#v:15"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:16" class="idref" href="#f:16"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folProp.html#T:14"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:15"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:16"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="closed" class="idref" href="#closed"><span class="id" title="definition">closed</span></a> (<a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>):=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:18" class="idref" href="#v:18"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:18"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:17"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="closeList" class="idref" href="#closeList"><span class="id" title="definition">closeList</span></a> (<a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#l:19"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#a:20"><span class="id" title="variable">a</span></a><br/>
+|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">v</span> <span class="id" title="var">l</span> ⇒  <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">∀</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Ackermann.folProp.html#closeList:21"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:19"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:20"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7"><span class="id" title="notation">}</span></a> <a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a"><span class="id" title="notation">f</span></a><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="close" class="idref" href="#close"><span class="id" title="definition">close</span></a> (<a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nodup"><span class="id" title="definition">nodup</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:23"><span class="id" title="variable">x</span></a>)) <a class="idref" href="hydras.Ackermann.folProp.html#x:23"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarClosedList1" class="idref" href="#freeVarClosedList1"><span class="id" title="lemma">freeVarClosedList1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:24" class="idref" href="#l:24"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="v:25" class="idref" href="#v:25"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:25"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:24"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:25"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:24"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:26"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarClosedList2" class="idref" href="#freeVarClosedList2"><span class="id" title="lemma">freeVarClosedList2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:27" class="idref" href="#l:27"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="v:28" class="idref" href="#v:28"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:28"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:27"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:29"><span class="id" title="variable">x</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:28"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:29"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarClosed" class="idref" href="#freeVarClosed"><span class="id" title="lemma">freeVarClosed</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="x:30" class="idref" href="#x:30"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:31" class="idref" href="#v:31"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:31"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#close"><span class="id" title="definition">close</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:30"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="freeVarListFormula" class="idref" href="#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> (<a id="l:32" class="idref" href="#l:32"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#l:32"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;| <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula:33"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:32"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarListFormulaApp" class="idref" href="#freeVarListFormulaApp"><span class="id" title="lemma">freeVarListFormulaApp</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a> <a id="b:36" class="idref" href="#b:36"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:36"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:36"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_freeVarListFormula" class="idref" href="#In_freeVarListFormula"><span class="id" title="lemma">In_freeVarListFormula</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:37" class="idref" href="#v:37"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:38" class="idref" href="#f:38"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="F:39" class="idref" href="#F:39"><span class="id" title="binder">F</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:37"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:38"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#F:39"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:37"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#F:39"><span class="id" title="variable">F</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_freeVarListFormulaE" class="idref" href="#In_freeVarListFormulaE"><span class="id" title="lemma">In_freeVarListFormulaE</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:40" class="idref" href="#v:40"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="F:41" class="idref" href="#F:41"><span class="id" title="binder">F</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">fol.Formulas</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:40"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarListFormula"><span class="id" title="definition">freeVarListFormula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#F:41"><span class="id" title="variable">F</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:40"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:42"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#F:41"><span class="id" title="variable">F</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="In_freeVarSys" class="idref" href="#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> (<a id="v:43" class="idref" href="#v:43"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:44" class="idref" href="#T:44"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:43"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:45"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folProp.html#T:44"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:45"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="notInFreeVarSys" class="idref" href="#notInFreeVarSys"><span class="id" title="lemma">notInFreeVarSys</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:46" class="idref" href="#x:46"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:46"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Ensembles.Empty_set</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.Free_Variables"><span class="id" title="section">Free_Variables</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fol_Properties.Substitution" class="idref" href="#Fol_Properties.Substitution"><span class="id" title="section">Substitution</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="substT" class="idref" href="#substT"><span class="id" title="definition">substT</span></a> (<a id="s:47" class="idref" href="#s:47"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="x:48" class="idref" href="#x:48"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="t:49" class="idref" href="#t:49"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">s</span>} : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#s:47"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:48"><span class="id" title="variable">x</span></a> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#t:49"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs:55"><span class="id" title="definition">substTs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span> <a class="idref" href="hydras.Ackermann.folProp.html#x:48"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:49"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">with</span> <a id="substTs" class="idref" href="#substTs"><span class="id" title="definition">substTs</span></a> (<a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ss:51" class="idref" href="#ss:51"><span class="id" title="binder">ss</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:50"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:52" class="idref" href="#x:52"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:53" class="idref" href="#t:53"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ss</span>} : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:50"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#ss:51"><span class="id" title="variable">ss</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n0</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n0:57"><span class="id" title="variable">n0</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> <span class="id" title="var">ts</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substT:54"><span class="id" title="definition">substT</span></a> <span class="id" title="var">s</span> <a class="idref" href="hydras.Ackermann.folProp.html#x:52"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:53"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substTs:55"><span class="id" title="definition">substTs</span></a> <span class="id" title="var">m</span> <span class="id" title="var">ts</span> <a class="idref" href="hydras.Ackermann.folProp.html#x:52"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:53"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermVar1" class="idref" href="#subTermVar1"><span class="id" title="lemma">subTermVar1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:59" class="idref" href="#v:59"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>), <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:59"><span class="id" title="variable">v</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:60"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:60"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermVar2" class="idref" href="#subTermVar2"><span class="id" title="lemma">subTermVar2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:61" class="idref" href="#v:61"><span class="id" title="binder">v</span></a> <a id="x:62" class="idref" href="#x:62"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:63" class="idref" href="#s:63"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#v:61"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:62"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:62"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:61"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:63"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:62"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermApply" class="idref" href="#subTermApply"><span class="id" title="lemma">subTermApply</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:64" class="idref" href="#f:64"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="ts:65" class="idref" href="#ts:65"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:64"><span class="id" title="variable">f</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="v:66" class="idref" href="#v:66"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:67" class="idref" href="#s:67"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:64"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#ts:65"><span class="id" title="variable">ts</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:66"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:67"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:64"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folProp.html#ts:65"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:66"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:67"><span class="id" title="variable">s</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="newVar" class="idref" href="#newVar"><span class="id" title="definition">newVar</span></a> (<a id="l:68" class="idref" href="#l:68"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_right"><span class="id" title="definition">fold_right</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:68"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="newVar2" class="idref" href="#newVar2"><span class="id" title="lemma">newVar2</span></a> : <span class="id" title="keyword">∀</span> (<a id="l:69" class="idref" href="#l:69"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:69"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:69"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="newVar1" class="idref" href="#newVar1"><span class="id" title="lemma">newVar1</span></a> : <span class="id" title="keyword">∀</span> <a id="l:73" class="idref" href="#l:73"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#l:73"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#l:73"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="substituteFormulaImp" class="idref" href="#substituteFormulaImp"><span class="id" title="definition">substituteFormulaImp</span></a> (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="frec:76" class="idref" href="#frec:76"><span class="id" title="binder">frec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:75" class="idref" href="#y:75"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:75"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="g:77" class="idref" href="#g:77"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="grec:79" class="idref" href="#grec:79"><span class="id" title="binder">grec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:78" class="idref" href="#y:78"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:78"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:80" class="idref" href="#p:80"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:81" class="idref" href="#y:81"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:81"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#frec:76"><span class="id" title="variable">frec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#p:80"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">prf1</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#grec:79"><span class="id" title="variable">grec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#p:80"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">g'</span> <span class="id" title="var">prf2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="y:84" class="idref" href="#y:84"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:84"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a>))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">g'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_ind_r"><span class="id" title="definition">eq_ind_r</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:83" class="idref" href="#n:83"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:83"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">g'</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_ind_r"><span class="id" title="definition">eq_ind_r</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n:82" class="idref" href="#n:82"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#n:82"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:74"><span class="id" title="variable">f</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:77"><span class="id" title="variable">g</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">prf2</span>) <span class="id" title="var">prf1</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="substituteFormulaImpNice" class="idref" href="#substituteFormulaImpNice"><span class="id" title="lemma">substituteFormulaImpNice</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:85" class="idref" href="#f:85"><span class="id" title="binder">f</span></a> <a id="g:86" class="idref" href="#g:86"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="z1:88" class="idref" href="#z1:88"><span class="id" title="binder">z1</span></a> <a id="z2:89" class="idref" href="#z2:89"><span class="id" title="binder">z2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:87" class="idref" href="#y:87"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:87"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:85"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:90" class="idref" href="#q:90"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProp.html#z1:88"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:90"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:89"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:90"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="z3:92" class="idref" href="#z3:92"><span class="id" title="binder">z3</span></a> <a id="z4:93" class="idref" href="#z4:93"><span class="id" title="binder">z4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:91" class="idref" href="#y:91"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:91"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:86"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:94" class="idref" href="#q:94"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProp.html#z3:92"><span class="id" title="variable">z3</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:94"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z4:93"><span class="id" title="variable">z4</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:94"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:95" class="idref" href="#q:95"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaImp"><span class="id" title="definition">substituteFormulaImp</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:85"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z1:88"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:86"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z3:92"><span class="id" title="variable">z3</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:95"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaImp"><span class="id" title="definition">substituteFormulaImp</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:85"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:89"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#g:86"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z4:93"><span class="id" title="variable">z4</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:95"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="substituteFormulaNot" class="idref" href="#substituteFormulaNot"><span class="id" title="definition">substituteFormulaNot</span></a> (<a id="f:96" class="idref" href="#f:96"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="frec:98" class="idref" href="#frec:98"><span class="id" title="binder">frec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:97" class="idref" href="#y:97"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:97"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:96"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:99" class="idref" href="#p:99"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:100" class="idref" href="#y:100"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:100"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:96"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#frec:98"><span class="id" title="variable">frec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#p:99"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">prf1</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> (<span class="id" title="keyword">fun</span> <a id="y:102" class="idref" href="#y:102"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:102"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:96"><span class="id" title="variable">f</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">f'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_ind_r"><span class="id" title="definition">eq_ind_r</span></a> (<span class="id" title="keyword">fun</span> <a id="n:101" class="idref" href="#n:101"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:101"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:96"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:96"><span class="id" title="variable">f</span></a>))) <span class="id" title="var">prf1</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="substituteFormulaNotNice" class="idref" href="#substituteFormulaNotNice"><span class="id" title="lemma">substituteFormulaNotNice</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:103" class="idref" href="#f:103"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="z1:105" class="idref" href="#z1:105"><span class="id" title="binder">z1</span></a> <a id="z2:106" class="idref" href="#z2:106"><span class="id" title="binder">z2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:104" class="idref" href="#y:104"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:104"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:103"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="q:107" class="idref" href="#q:107"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProp.html#z1:105"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:107"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:106"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:107"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:108" class="idref" href="#q:108"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaNot"><span class="id" title="definition">substituteFormulaNot</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:103"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z1:105"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:108"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaNot"><span class="id" title="definition">substituteFormulaNot</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:103"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:106"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:108"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="substituteFormulaForall" class="idref" href="#substituteFormulaForall"><span class="id" title="definition">substituteFormulaForall</span></a> (<a id="n:109" class="idref" href="#n:109"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:110" class="idref" href="#f:110"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;(<a id="frec:113" class="idref" href="#frec:113"><span class="id" title="binder">frec</span></a> : <span class="id" title="keyword">∀</span> <a id="b:111" class="idref" href="#b:111"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.lt_depth"><span class="id" title="variable">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:111"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:112" class="idref" href="#y:112"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:112"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:111"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="p:114" class="idref" href="#p:114"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:115" class="idref" href="#y:115"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:115"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#p:114"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> (<span class="id" title="keyword">fun</span> <a id="y:117" class="idref" href="#y:117"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:117"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">s</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="nv:118" class="idref" href="#nv:118"><span class="id" title="binder">nv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (<span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#frec:113"><span class="id" title="variable">frec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depthForall"><span class="id" title="lemma">depthForall</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:118"><span class="id" title="variable">nv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">prf1</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#frec:113"><span class="id" title="variable">frec</span></a> <span class="id" title="var">f'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#eqDepth"><span class="id" title="lemma">eqDepth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <span class="id" title="var">f'</span> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a> <span class="id" title="var">prf1</span>) (<a class="idref" href="hydras.Ackermann.fol.html#depthForall"><span class="id" title="lemma">depthForall</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Ackermann.folProp.html#p:114"><span class="id" title="variable">p</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">f''</span> <span class="id" title="var">prf2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="y:120" class="idref" href="#y:120"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:120"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:118"><span class="id" title="variable">nv</span></a> <span class="id" title="var">f''</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_ind_r"><span class="id" title="definition">eq_ind_r</span></a> (<span class="id" title="keyword">fun</span> <a id="n:119" class="idref" href="#n:119"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:119"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#trans_eq"><span class="id" title="abbreviation">trans_eq</span></a> <span class="id" title="var">prf2</span> <span class="id" title="var">prf1</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.folProp.html#frec:113"><span class="id" title="variable">frec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depthForall"><span class="id" title="lemma">depthForall</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#p:114"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">prf1</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> (<span class="id" title="keyword">fun</span> <a id="y:122" class="idref" href="#y:122"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:122"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:109"><span class="id" title="variable">n</span></a> <span class="id" title="var">f'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_ind_r"><span class="id" title="definition">eq_ind_r</span></a> (<span class="id" title="keyword">fun</span> <a id="n:121" class="idref" href="#n:121"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:121"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:110"><span class="id" title="variable">f</span></a>))) <span class="id" title="var">prf1</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="substituteFormulaForallNice" class="idref" href="#substituteFormulaForallNice"><span class="id" title="lemma">substituteFormulaForallNice</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:123" class="idref" href="#v:123"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:124" class="idref" href="#a:124"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="z1:127" class="idref" href="#z1:127"><span class="id" title="binder">z1</span></a> <a id="z2:128" class="idref" href="#z2:128"><span class="id" title="binder">z2</span></a> : <span class="id" title="keyword">∀</span> <a id="b:125" class="idref" href="#b:125"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.lt_depth"><span class="id" title="variable">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:125"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:123"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:124"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:126" class="idref" href="#y:126"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:126"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:125"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="b:129" class="idref" href="#b:129"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="q:130" class="idref" href="#q:130"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.lt_depth"><span class="id" title="variable">lt_depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:129"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:123"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:124"><span class="id" title="variable">a</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="r:131" class="idref" href="#r:131"><span class="id" title="binder">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>), <a class="idref" href="hydras.Ackermann.folProp.html#z1:127"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:129"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:130"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:131"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:128"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#b:129"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:130"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:131"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="q:132" class="idref" href="#q:132"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaForall"><span class="id" title="definition">substituteFormulaForall</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:123"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:124"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z1:127"><span class="id" title="variable">z1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:132"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaForall"><span class="id" title="definition">substituteFormulaForall</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:123"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#a:124"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#z2:128"><span class="id" title="variable">z2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#q:132"><span class="id" title="variable">q</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="substituteFormulaHelp" class="idref" href="#substituteFormulaHelp"><span class="id" title="definition">substituteFormulaHelp</span></a> (<a id="f:133" class="idref" href="#f:133"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:134" class="idref" href="#v:134"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:135" class="idref" href="#s:135"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="y:136" class="idref" href="#y:136"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#y:136"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:133"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="substF" class="idref" href="#substF"><span class="id" title="definition">substF</span></a> (<a id="f:141" class="idref" href="#f:141"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:142" class="idref" href="#v:142"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:143" class="idref" href="#s:143"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substituteFormulaHelp"><span class="id" title="definition">substituteFormulaHelp</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:141"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:142"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:143"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaEqual" class="idref" href="#subFormulaEqual"><span class="id" title="lemma">subFormulaEqual</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="t1:144" class="idref" href="#t1:144"><span class="id" title="binder">t1</span></a> <a id="t2:145" class="idref" href="#t2:145"><span class="id" title="binder">t2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:146" class="idref" href="#v:146"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:147" class="idref" href="#s:147"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#t1:144"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t2:145"><span class="id" title="variable">t2</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:146"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:147"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t1:144"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:146"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:147"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t2:145"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:146"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:147"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaRelation" class="idref" href="#subFormulaRelation"><span class="id" title="lemma">subFormulaRelation</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="r:148" class="idref" href="#r:148"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="ts:149" class="idref" href="#ts:149"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:148"><span class="id" title="variable">r</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="v:150" class="idref" href="#v:150"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:151" class="idref" href="#s:151"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:148"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#ts:149"><span class="id" title="variable">ts</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:150"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:151"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:148"><span class="id" title="variable">r</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#r:148"><span class="id" title="variable">r</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#ts:149"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:150"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:151"><span class="id" title="variable">s</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaImp" class="idref" href="#subFormulaImp"><span class="id" title="lemma">subFormulaImp</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:152" class="idref" href="#f1:152"><span class="id" title="binder">f1</span></a> <a id="f2:153" class="idref" href="#f2:153"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:154" class="idref" href="#v:154"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:155" class="idref" href="#s:155"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#f1:152"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:153"><span class="id" title="variable">f2</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:154"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:152"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:154"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:153"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:154"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:155"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaNot" class="idref" href="#subFormulaNot"><span class="id" title="lemma">subFormulaNot</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:232" class="idref" href="#f:232"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:233" class="idref" href="#v:233"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:234" class="idref" href="#s:234"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:232"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:233"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:234"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:232"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:233"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:234"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaForall" class="idref" href="#subFormulaForall"><span class="id" title="lemma">subFormulaForall</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:279" class="idref" href="#f:279"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="x:280" class="idref" href="#x:280"><span class="id" title="binder">x</span></a> <a id="v:281" class="idref" href="#v:281"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:282" class="idref" href="#s:282"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="nv:283" class="idref" href="#nv:283"><span class="id" title="binder">nv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#v:281"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:282"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:279"><span class="id" title="variable">f</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:279"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:281"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:282"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:281"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:279"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:282"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:279"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:281"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:282"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:283"><span class="id" title="variable">nv</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:279"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:280"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:283"><span class="id" title="variable">nv</span></a>) ) <a class="idref" href="hydras.Ackermann.folProp.html#v:281"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:282"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fol_Properties.Substitution.Extensions" class="idref" href="#Fol_Properties.Substitution.Extensions"><span class="id" title="section">Extensions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaOr" class="idref" href="#subFormulaOr"><span class="id" title="lemma">subFormulaOr</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:392" class="idref" href="#f1:392"><span class="id" title="binder">f1</span></a> <a id="f2:393" class="idref" href="#f2:393"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:394" class="idref" href="#v:394"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:395" class="idref" href="#s:395"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#f1:392"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:393"><span class="id" title="variable">f2</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:394"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:395"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:392"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:394"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:395"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:393"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:394"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:395"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaAnd" class="idref" href="#subFormulaAnd"><span class="id" title="lemma">subFormulaAnd</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:396" class="idref" href="#f1:396"><span class="id" title="binder">f1</span></a> <a id="f2:397" class="idref" href="#f2:397"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:398" class="idref" href="#v:398"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:399" class="idref" href="#s:399"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#f1:396"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:397"><span class="id" title="variable">f2</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#v:398"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:399"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:396"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:398"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:399"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:397"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:398"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:399"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaExist" class="idref" href="#subFormulaExist"><span class="id" title="lemma">subFormulaExist</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:400" class="idref" href="#f:400"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="x:401" class="idref" href="#x:401"><span class="id" title="binder">x</span></a> <a id="v:402" class="idref" href="#v:402"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:403" class="idref" href="#s:403"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="nv:404" class="idref" href="#nv:404"><span class="id" title="binder">nv</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#v:402"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:403"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:400"><span class="id" title="variable">f</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:400"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:402"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:403"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:402"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:400"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:403"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:400"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:402"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:403"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:404"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:400"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:401"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:404"><span class="id" title="variable">nv</span></a>)) <a class="idref" href="hydras.Ackermann.folProp.html#v:402"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:403"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaIff" class="idref" href="#subFormulaIff"><span class="id" title="lemma">subFormulaIff</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:405" class="idref" href="#f1:405"><span class="id" title="binder">f1</span></a> <a id="f2:406" class="idref" href="#f2:406"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:407" class="idref" href="#v:407"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:408" class="idref" href="#s:408"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:405"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:406"><span class="id" title="variable">f2</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:407"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:408"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:405"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:407"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:408"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:406"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:407"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:408"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaIfThenElse" class="idref" href="#subFormulaIfThenElse"><span class="id" title="lemma">subFormulaIfThenElse</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:409" class="idref" href="#f1:409"><span class="id" title="binder">f1</span></a> <a id="f2:410" class="idref" href="#f2:410"><span class="id" title="binder">f2</span></a> <a id="f3:411" class="idref" href="#f3:411"><span class="id" title="binder">f3</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:412" class="idref" href="#v:412"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:413" class="idref" href="#s:413"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:409"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:410"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f3:411"><span class="id" title="variable">f3</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:412"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:413"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f1:409"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:412"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:413"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f2:410"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:412"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:413"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f3:411"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:412"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:413"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Extensions"><span class="id" title="section">Extensions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaDepth" class="idref" href="#subFormulaDepth"><span class="id" title="lemma">subFormulaDepth</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:414" class="idref" href="#f:414"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:415" class="idref" href="#v:415"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:416" class="idref" href="#s:416"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:414"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:415"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:416"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#depth"><span class="id" title="definition">depth</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:414"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fol_Properties.Substitution.Substitution_Properties" class="idref" href="#Fol_Properties.Substitution.Substitution_Properties"><span class="id" title="section">Substitution_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermId" class="idref" href="#subTermId"><span class="id" title="lemma">subTermId</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:417" class="idref" href="#t:417"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:418" class="idref" href="#v:418"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:417"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:418"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:418"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:417"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermsId" class="idref" href="#subTermsId"><span class="id" title="lemma">subTermsId</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:423" class="idref" href="#n:423"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:424" class="idref" href="#ts:424"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:423"><span class="id" title="variable">n</span></a>) (<a id="v:425" class="idref" href="#v:425"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#n:423"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#ts:424"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:425"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:425"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#ts:424"><span class="id" title="variable">ts</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaId" class="idref" href="#subFormulaId"><span class="id" title="lemma">subFormulaId</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:426" class="idref" href="#f:426"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="v:427" class="idref" href="#v:427"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:426"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:427"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:427"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:426"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaForall2" class="idref" href="#subFormulaForall2"><span class="id" title="lemma">subFormulaForall2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:428" class="idref" href="#f:428"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="x:429" class="idref" href="#x:429"><span class="id" title="binder">x</span></a> <a id="v:430" class="idref" href="#v:430"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:431" class="idref" href="#s:431"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="nv:432" class="idref" href="#nv:432"><span class="id" title="binder">nv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:432"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:431"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#nv:432"><span class="id" title="variable">nv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:430"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:432"><span class="id" title="variable">nv</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:429"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:428"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:429"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:428"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:430"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:431"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:429"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:430"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:429"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:428"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:432"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:428"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:429"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:432"><span class="id" title="variable">nv</span></a>)) <a class="idref" href="hydras.Ackermann.folProp.html#v:430"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:431"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaExist2" class="idref" href="#subFormulaExist2"><span class="id" title="lemma">subFormulaExist2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:433" class="idref" href="#f:433"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>) (<a id="x:434" class="idref" href="#x:434"><span class="id" title="binder">x</span></a> <a id="v:435" class="idref" href="#v:435"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:436" class="idref" href="#s:436"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="nv:437" class="idref" href="#nv:437"><span class="id" title="binder">nv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:437"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:436"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#nv:437"><span class="id" title="variable">nv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:435"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:437"><span class="id" title="variable">nv</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:434"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:433"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:433"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#v:435"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:436"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:435"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:433"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:437"><span class="id" title="variable">nv</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:433"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:434"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#nv:437"><span class="id" title="variable">nv</span></a>)) <a class="idref" href="hydras.Ackermann.folProp.html#v:435"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#s:436"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="substExHC" class="idref" href="#substExHC"><span class="id" title="lemma">substExHC</span></a>  (<a id="A:438" class="idref" href="#A:438"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.folProp.html#Formula"><span class="id" title="abbreviation">Formula</span></a>) (<a id="v:439" class="idref" href="#v:439"><span class="id" title="binder">v</span></a> <a id="x:440" class="idref" href="#x:440"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="t:441" class="idref" href="#t:441"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Ackermann.folProp.html#Term"><span class="id" title="abbreviation">Term</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#v:439"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:440"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:439"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:441"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:439"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#A:438"><span class="id" title="variable">A</span></a>) <a class="idref" href="hydras.Ackermann.folProp.html#x:440"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:441"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:439"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#A:438"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#x:440"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#t:441"><span class="id" title="variable">t</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Substitution_Properties"><span class="id" title="section">Substitution_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution"><span class="id" title="section">Substitution</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Sentence" class="idref" href="#Sentence"><span class="id" title="definition">Sentence</span></a> (<a id="f:442" class="idref" href="#f:442"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Ackermann.folProp.html#Formula"><span class="id" title="abbreviation">Formula</span></a>) := <br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="v:443" class="idref" href="#v:443"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#v:443"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#f:442"><span class="id" title="variable">f</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folProp.html#Fol_Properties"><span class="id" title="section">Fol_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#closed"><span class="id" title="definition">closed</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> {<span class="id" title="var">L</span> <span class="id" title="var">n</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> .<br/>
+</div>
+
+<div class="doc">
+compatibility with older names 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use substF")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="substituteFormula" class="idref" href="#substituteFormula"><span class="id" title="abbreviation">substituteFormula</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use substT")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="substituteTerm" class="idref" href="#substituteTerm"><span class="id" title="abbreviation">substituteTerm</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use substTs")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="substituteTerms" class="idref" href="#substituteTerms"><span class="id" title="abbreviation">substituteTerms</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use freeVarF")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="freeVarFormula" class="idref" href="#freeVarFormula"><span class="id" title="abbreviation">freeVarFormula</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use freeVarT")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="freeVarTerm" class="idref" href="#freeVarTerm"><span class="id" title="abbreviation">freeVarTerm</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use freeVarTs")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="freeVarTerms" class="idref" href="#freeVarTerms"><span class="id" title="abbreviation">freeVarTerms</span></a> := <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>.<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+to replace with a single recursive custom notation ? 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF2" class="idref" href="#substF2"><span class="id" title="abbreviation">substF2</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span>) <span class="id" title="var">v2</span> <span class="id" title="var">t2</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF3" class="idref" href="#substF3"><span class="id" title="abbreviation">substF3</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF2"><span class="id" title="abbreviation">substF2</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span>) <span class="id" title="var">v3</span> <span class="id" title="var">t3</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF4" class="idref" href="#substF4"><span class="id" title="abbreviation">substF4</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF3"><span class="id" title="abbreviation">substF3</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span>) <span class="id" title="var">v4</span> <span class="id" title="var">t4</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF5" class="idref" href="#substF5"><span class="id" title="abbreviation">substF5</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF4"><span class="id" title="abbreviation">substF4</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span>) <span class="id" title="var">v5</span> <span class="id" title="var">t5</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF6" class="idref" href="#substF6"><span class="id" title="abbreviation">substF6</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF5"><span class="id" title="abbreviation">substF5</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span>) <span class="id" title="var">v6</span> <span class="id" title="var">t6</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span> <a id="substF7" class="idref" href="#substF7"><span class="id" title="abbreviation">substF7</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span> <span class="id" title="var">v7</span> <span class="id" title="var">t7</span>:=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF6"><span class="id" title="abbreviation">substF6</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span>) <span class="id" title="var">v7</span> <span class="id" title="var">t7</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span><br/>
+&nbsp;&nbsp;<a id="substF8" class="idref" href="#substF8"><span class="id" title="abbreviation">substF8</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span> <span class="id" title="var">v7</span> <span class="id" title="var">t7</span> <span class="id" title="var">v8</span> <span class="id" title="var">t8</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF2"><span class="id" title="abbreviation">substF2</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF6"><span class="id" title="abbreviation">substF6</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">v7</span> <span class="id" title="var">t7</span> <span class="id" title="var">v8</span> <span class="id" title="var">t8</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Notation</span><br/>
+&nbsp;&nbsp;<a id="substF9" class="idref" href="#substF9"><span class="id" title="abbreviation">substF9</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span> <span class="id" title="var">v7</span> <span class="id" title="var">t7</span> <span class="id" title="var">v8</span> <span class="id" title="var">t8</span> <span class="id" title="var">v9</span> <span class="id" title="var">t9</span>:=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF3"><span class="id" title="abbreviation">substF3</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF6"><span class="id" title="abbreviation">substF6</span></a> <span class="id" title="var">e</span> <span class="id" title="var">v1</span> <span class="id" title="var">t1</span> <span class="id" title="var">v2</span> <span class="id" title="var">t2</span> <span class="id" title="var">v3</span> <span class="id" title="var">t3</span> <span class="id" title="var">v4</span> <span class="id" title="var">t4</span> <span class="id" title="var">v5</span> <span class="id" title="var">t5</span> <span class="id" title="var">v6</span> <span class="id" title="var">t6</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">v7</span> <span class="id" title="var">t7</span> <span class="id" title="var">v8</span> <span class="id" title="var">t8</span> <span class="id" title="var">v9</span> <span class="id" title="var">t9</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> {<span class="id" title="var">L</span>} <span class="id" title="var">_</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> {<span class="id" title="var">L</span> <span class="id" title="var">n</span>} <span class="id" title="var">_</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.folReplace.html b/theories/html/hydras.Ackermann.folReplace.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.folReplace.html
@@ -0,0 +1,117 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.folReplace</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.folReplace</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Replacement" class="idref" href="#Replacement"><span class="id" title="section">Replacement</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Replacement.L" class="idref" href="#Replacement.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.Formula" class="idref" href="#Replacement.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.Formulas" class="idref" href="#Replacement.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.System" class="idref" href="#Replacement.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.Term" class="idref" href="#Replacement.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.Terms" class="idref" href="#Replacement.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Replacement.SysPrf" class="idref" href="#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceImp" class="idref" href="#reduceImp"><span class="id" title="lemma">reduceImp</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f1:2" class="idref" href="#f1:2"><span class="id" title="binder">f1</span></a> <a id="f2:3" class="idref" href="#f2:3"><span class="id" title="binder">f2</span></a> <a id="f3:4" class="idref" href="#f3:4"><span class="id" title="binder">f3</span></a> <a id="f4:5" class="idref" href="#f4:5"><span class="id" title="binder">f4</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:6" class="idref" href="#T:6"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:6"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:2"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:4"><span class="id" title="variable">f3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:6"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:3"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:5"><span class="id" title="variable">f4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:6"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:2"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:3"><span class="id" title="variable">f2</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:4"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:5"><span class="id" title="variable">f4</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceNot" class="idref" href="#reduceNot"><span class="id" title="lemma">reduceNot</span></a> (<a id="f1:17" class="idref" href="#f1:17"><span class="id" title="binder">f1</span></a> <a id="f2:18" class="idref" href="#f2:18"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:19" class="idref" href="#T:19"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:19"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:17"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:18"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:19"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:17"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:18"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="impForall" class="idref" href="#impForall"><span class="id" title="lemma">impForall</span></a> (<a id="f1:26" class="idref" href="#f1:26"><span class="id" title="binder">f1</span></a> <a id="f2:27" class="idref" href="#f2:27"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:28" class="idref" href="#v:28"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:29" class="idref" href="#T:29"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folReplace.html#v:28"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:29"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:29"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:26"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:27"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:29"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:28"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:26"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:28"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:27"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceForall" class="idref" href="#reduceForall"><span class="id" title="lemma">reduceForall</span></a> (<a id="f1:30" class="idref" href="#f1:30"><span class="id" title="binder">f1</span></a> <a id="f2:31" class="idref" href="#f2:31"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:32" class="idref" href="#v:32"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:33" class="idref" href="#T:33"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folReplace.html#v:32"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:33"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:33"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:30"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:31"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:33"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:32"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:30"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:32"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:31"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceOr" class="idref" href="#reduceOr"><span class="id" title="lemma">reduceOr</span></a> (<a id="f1:34" class="idref" href="#f1:34"><span class="id" title="binder">f1</span></a> <a id="f2:35" class="idref" href="#f2:35"><span class="id" title="binder">f2</span></a> <a id="f3:36" class="idref" href="#f3:36"><span class="id" title="binder">f3</span></a> <a id="f4:37" class="idref" href="#f4:37"><span class="id" title="binder">f4</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:38" class="idref" href="#T:38"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:34"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:36"><span class="id" title="variable">f3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:35"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:37"><span class="id" title="variable">f4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:38"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:34"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:35"><span class="id" title="variable">f2</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:36"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:37"><span class="id" title="variable">f4</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceAnd" class="idref" href="#reduceAnd"><span class="id" title="lemma">reduceAnd</span></a> (<a id="f1:49" class="idref" href="#f1:49"><span class="id" title="binder">f1</span></a> <a id="f2:50" class="idref" href="#f2:50"><span class="id" title="binder">f2</span></a> <a id="f3:51" class="idref" href="#f3:51"><span class="id" title="binder">f3</span></a> <a id="f4:52" class="idref" href="#f4:52"><span class="id" title="binder">f4</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:53" class="idref" href="#T:53"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:53"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:49"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:51"><span class="id" title="variable">f3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:53"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:50"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:52"><span class="id" title="variable">f4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:53"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:49"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:50"><span class="id" title="variable">f2</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#andH"><span class="id" title="definition">andH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:51"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:52"><span class="id" title="variable">f4</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iffExist" class="idref" href="#iffExist"><span class="id" title="lemma">iffExist</span></a> (<a id="f1:64" class="idref" href="#f1:64"><span class="id" title="binder">f1</span></a> <a id="f2:65" class="idref" href="#f2:65"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:66" class="idref" href="#v:66"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:67" class="idref" href="#T:67"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folReplace.html#v:66"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:67"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:64"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:65"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:67"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:66"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:64"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:66"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:65"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceExist" class="idref" href="#reduceExist"><span class="id" title="lemma">reduceExist</span></a> (<a id="f1:68" class="idref" href="#f1:68"><span class="id" title="binder">f1</span></a> <a id="f2:69" class="idref" href="#f2:69"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:70" class="idref" href="#v:70"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="T:71" class="idref" href="#T:71"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folReplace.html#v:70"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:71"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:71"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:68"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:69"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:71"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:70"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:68"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:70"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:69"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceIff" class="idref" href="#reduceIff"><span class="id" title="lemma">reduceIff</span></a> (<a id="f1:72" class="idref" href="#f1:72"><span class="id" title="binder">f1</span></a> <a id="f2:73" class="idref" href="#f2:73"><span class="id" title="binder">f2</span></a> <a id="f3:74" class="idref" href="#f3:74"><span class="id" title="binder">f3</span></a> <a id="f4:75" class="idref" href="#f4:75"><span class="id" title="binder">f4</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:76" class="idref" href="#T:76"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:72"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:74"><span class="id" title="variable">f3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:73"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:75"><span class="id" title="variable">f4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:76"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:72"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:73"><span class="id" title="variable">f2</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:74"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:75"><span class="id" title="variable">f4</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceIfThenElse" class="idref" href="#reduceIfThenElse"><span class="id" title="lemma">reduceIfThenElse</span></a> (<a id="f1:87" class="idref" href="#f1:87"><span class="id" title="binder">f1</span></a> <a id="f2:88" class="idref" href="#f2:88"><span class="id" title="binder">f2</span></a> <a id="f3:89" class="idref" href="#f3:89"><span class="id" title="binder">f3</span></a> <a id="f4:90" class="idref" href="#f4:90"><span class="id" title="binder">f4</span></a> <a id="f5:91" class="idref" href="#f5:91"><span class="id" title="binder">f5</span></a> <a id="f6:92" class="idref" href="#f6:92"><span class="id" title="binder">f6</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:93" class="idref" href="#T:93"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:87"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:90"><span class="id" title="variable">f4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:88"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f5:91"><span class="id" title="variable">f5</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:89"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f6:92"><span class="id" title="variable">f6</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:93"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:87"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:88"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f3:89"><span class="id" title="variable">f3</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#ifThenElseH"><span class="id" title="definition">ifThenElseH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f4:90"><span class="id" title="variable">f4</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f5:91"><span class="id" title="variable">f5</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f6:92"><span class="id" title="variable">f6</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceSub" class="idref" href="#reduceSub"><span class="id" title="lemma">reduceSub</span></a>  (<a id="T:94" class="idref" href="#T:94"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>) (<a id="v:95" class="idref" href="#v:95"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:96" class="idref" href="#s:96"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Term"><span class="id" title="variable">Term</span></a>) (<a id="f:97" class="idref" href="#f:97"><span class="id" title="binder">f</span></a> <a id="g:98" class="idref" href="#g:98"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:95"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:94"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:94"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f:97"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#g:98"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:94"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f:97"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:95"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#s:96"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#g:98"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:95"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#s:96"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceCloseList" class="idref" href="#reduceCloseList"><span class="id" title="lemma">reduceCloseList</span></a> (<a id="l:109" class="idref" href="#l:109"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f1:110" class="idref" href="#f1:110"><span class="id" title="binder">f1</span></a> <a id="f2:111" class="idref" href="#f2:111"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:112" class="idref" href="#T:112"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.System"><span class="id" title="variable">System</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:113" class="idref" href="#v:113"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#v:113"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#l:109"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.folReplace.html#v:113"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:112"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:110"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:111"><span class="id" title="variable">f2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#T:112"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#l:109"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f1:110"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#closeList"><span class="id" title="definition">closeList</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#l:109"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#f2:111"><span class="id" title="variable">f2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.folReplace.html#Replacement"><span class="id" title="section">Replacement</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.misc.html b/theories/html/hydras.Ackermann.misc.html
new file mode 100644
index 00000000..7b3f2b23
--- /dev/null
+++ b/theories/html/hydras.Ackermann.misc.html
@@ -0,0 +1,38 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.misc</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.misc</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Eqdep_dec</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Set</span> <span class="id" title="var">Asymmetric</span> <span class="id" title="var">Patterns</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.model.html b/theories/html/hydras.Ackermann.model.html
new file mode 100644
index 00000000..b46a5c7a
--- /dev/null
+++ b/theories/html/hydras.Ackermann.model.html
@@ -0,0 +1,223 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.model</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.model</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+model.v :
+
+<div class="paragraph"> </div>
+
+   Original script by Russel O'Connor
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.misc.html#"><span class="id" title="library">misc</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Model_Theory" class="idref" href="#Model_Theory"><span class="id" title="section">Model_Theory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Model_Theory.L" class="idref" href="#Model_Theory.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="naryFunc" class="idref" href="#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a id="A:2" class="idref" href="#A:2"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#n:3"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.model.html#A:2"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#A:2"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#naryFunc:4"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.model.html#A:2"><span class="id" title="variable">A</span></a> <span class="id" title="var">m</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="naryRel" class="idref" href="#naryRel"><span class="id" title="definition">naryRel</span></a> (<a id="A:6" class="idref" href="#A:6"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#n:7"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="keyword">Prop</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#A:6"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#naryRel:8"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.model.html#A:6"><span class="id" title="variable">A</span></a> <span class="id" title="var">m</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="Model" class="idref" href="#Model"><span class="id" title="record">Model</span></a> : <span class="id" title="keyword">Type</span> := <a id="model" class="idref" href="#model"><span class="id" title="constructor">model</span></a><br/>
+&nbsp;&nbsp;{<a id="U" class="idref" href="#U"><span class="id" title="projection">U</span></a> : <span class="id" title="keyword">Set</span>;<br/>
+&nbsp;&nbsp;&nbsp;<a id="func" class="idref" href="#func"><span class="id" title="projection">func</span></a> : <span class="id" title="keyword">∀</span> <a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.model.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.model.html#U:11"><span class="id" title="method">U</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:12"><span class="id" title="variable">f</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;<a id="rel" class="idref" href="#rel"><span class="id" title="projection">rel</span></a> : <span class="id" title="keyword">∀</span> <a id="r:14" class="idref" href="#r:14"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.model.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.model.html#U:11"><span class="id" title="method">U</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#r:14"><span class="id" title="variable">r</span></a>)}.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Model_Theory.M" class="idref" href="#Model_Theory.M"><span class="id" title="variable">M</span></a> : <a class="idref" href="hydras.Ackermann.model.html#Model"><span class="id" title="record">Model</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="interpTerm" class="idref" href="#interpTerm"><span class="id" title="definition">interpTerm</span></a> (<a id="value:17" class="idref" href="#value:17"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="t:18" class="idref" href="#t:18"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t</span>} : <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#t:18"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">v</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#value:17"><span class="id" title="variable">value</span></a> <span class="id" title="var">v</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpTerms:24"><span class="id" title="definition">interpTerms</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.model.html#func"><span class="id" title="projection">func</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a> <span class="id" title="var">f</span>) <a class="idref" href="hydras.Ackermann.model.html#value:17"><span class="id" title="variable">value</span></a> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="interpTerms" class="idref" href="#interpTerms"><span class="id" title="definition">interpTerms</span></a> (<a id="m:19" class="idref" href="#m:19"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:20" class="idref" href="#f:20"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.model.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#m:19"><span class="id" title="variable">m</span></a>) <br/>
+&nbsp;(<a id="value:21" class="idref" href="#value:21"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="ts:22" class="idref" href="#ts:22"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#m:19"><span class="id" title="variable">m</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#ts:22"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.model.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="f:28" class="idref" href="#f:28"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.Ackermann.model.html#f:28"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">ts</span> ⇒ <span class="id" title="keyword">fun</span> <a id="f:29" class="idref" href="#f:29"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpTerms:24"><span class="id" title="definition">interpTerms</span></a> <a class="idref" href="hydras.Ackermann.model.html#m:19"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.model.html#f:29"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm:23"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:21"><span class="id" title="variable">value</span></a> <span class="id" title="var">t</span>)) <a class="idref" href="hydras.Ackermann.model.html#value:21"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:22"><span class="id" title="variable">ts</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span> <a class="idref" href="hydras.Ackermann.model.html#f:20"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="interpRels" class="idref" href="#interpRels"><span class="id" title="definition">interpRels</span></a> (<a id="m:30" class="idref" href="#m:30"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="r:31" class="idref" href="#r:31"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.model.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#m:30"><span class="id" title="variable">m</span></a>) <br/>
+&nbsp;(<a id="value:32" class="idref" href="#value:32"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="ts:33" class="idref" href="#ts:33"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#m:30"><span class="id" title="variable">m</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#ts:33"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.model.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="r:37" class="idref" href="#r:37"><span class="id" title="binder">r</span></a> ⇒ <a class="idref" href="hydras.Ackermann.model.html#r:37"><span class="id" title="variable">r</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">ts</span> ⇒ <span class="id" title="keyword">fun</span> <a id="r:38" class="idref" href="#r:38"><span class="id" title="binder">r</span></a> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpRels:34"><span class="id" title="definition">interpRels</span></a> <a class="idref" href="hydras.Ackermann.model.html#m:30"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.model.html#r:38"><span class="id" title="variable">r</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:32"><span class="id" title="variable">value</span></a> <span class="id" title="var">t</span>)) <a class="idref" href="hydras.Ackermann.model.html#value:32"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:33"><span class="id" title="variable">ts</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span> <a class="idref" href="hydras.Ackermann.model.html#r:31"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="updateValue" class="idref" href="#updateValue"><span class="id" title="definition">updateValue</span></a> (<a id="value:39" class="idref" href="#value:39"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="n:40" class="idref" href="#n:40"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:41" class="idref" href="#v:41"><span class="id" title="binder">v</span></a> : <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="x:42" class="idref" href="#x:42"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:42"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#v:41"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#value:39"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:42"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="interpFormula" class="idref" href="#interpFormula"><span class="id" title="definition">interpFormula</span></a> (<a id="value:43" class="idref" href="#value:43"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">f</span>} :<br/>
+&nbsp;<span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#f:44"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">s</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpRels"><span class="id" title="definition">interpRels</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.model.html#rel"><span class="id" title="projection">rel</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a> <span class="id" title="var">r</span>) <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpFormula:45"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpFormula:45"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">B</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="hydras.Ackermann.model.html#interpFormula:45"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> ⇒ <span class="id" title="keyword">∀</span> <a id="x:47" class="idref" href="#x:47"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>, <a class="idref" href="hydras.Ackermann.model.html#interpFormula:45"><span class="id" title="definition">interpFormula</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:43"><span class="id" title="variable">value</span></a> <span class="id" title="var">v</span> <a class="idref" href="hydras.Ackermann.model.html#x:47"><span class="id" title="variable">x</span></a>) <span class="id" title="var">A</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarInterpTerm" class="idref" href="#freeVarInterpTerm"><span class="id" title="lemma">freeVarInterpTerm</span></a> (<a id="v1:48" class="idref" href="#v1:48"><span class="id" title="binder">v1</span></a> <a id="v2:49" class="idref" href="#v2:49"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="t:50" class="idref" href="#t:50"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:51" class="idref" href="#x:51"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:51"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.model.html#t:50"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:48"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:51"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:49"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:51"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:48"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#t:50"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:49"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#t:50"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarInterpRel" class="idref" href="#freeVarInterpRel"><span class="id" title="lemma">freeVarInterpRel</span></a> (<a id="v1:60" class="idref" href="#v1:60"><span class="id" title="binder">v1</span></a> <a id="v2:61" class="idref" href="#v2:61"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="ts:63" class="idref" href="#ts:63"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:62"><span class="id" title="variable">n</span></a>) (<a id="r:64" class="idref" href="#r:64"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.model.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:62"><span class="id" title="variable">n</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:65" class="idref" href="#x:65"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:65"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:63"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:60"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:65"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:61"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:65"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpRels"><span class="id" title="definition">interpRels</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.model.html#r:64"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:60"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:63"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpRels"><span class="id" title="definition">interpRels</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.model.html#r:64"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:61"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:63"><span class="id" title="variable">ts</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarInterpFormula" class="idref" href="#freeVarInterpFormula"><span class="id" title="lemma">freeVarInterpFormula</span></a> (<a id="v1:66" class="idref" href="#v1:66"><span class="id" title="binder">v1</span></a> <a id="v2:67" class="idref" href="#v2:67"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="g:68" class="idref" href="#g:68"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:69" class="idref" href="#x:69"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:69"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.model.html#g:68"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:66"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:69"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:67"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#x:69"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#v1:66"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.model.html#g:68"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#v2:67"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.model.html#g:68"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subInterpTerm" class="idref" href="#subInterpTerm"><span class="id" title="lemma">subInterpTerm</span></a> (<a id="value:70" class="idref" href="#value:70"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="t:71" class="idref" href="#t:71"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="v:72" class="idref" href="#v:72"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:73" class="idref" href="#s:73"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:70"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:72"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:70"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:73"><span class="id" title="variable">s</span></a>)) <a class="idref" href="hydras.Ackermann.model.html#t:71"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:70"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.model.html#t:71"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:72"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:73"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subInterpRel" class="idref" href="#subInterpRel"><span class="id" title="lemma">subInterpRel</span></a> (<a id="value:80" class="idref" href="#value:80"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:82" class="idref" href="#ts:82"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:81"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:83" class="idref" href="#v:83"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:84" class="idref" href="#s:84"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="r:85" class="idref" href="#r:85"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.model.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:81"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpRels"><span class="id" title="definition">interpRels</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.model.html#r:85"><span class="id" title="variable">r</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:80"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:83"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:80"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:84"><span class="id" title="variable">s</span></a>)) <a class="idref" href="hydras.Ackermann.model.html#ts:82"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpRels"><span class="id" title="definition">interpRels</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.model.html#r:85"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:80"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.model.html#ts:82"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:83"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:84"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subInterpFormula" class="idref" href="#subInterpFormula"><span class="id" title="lemma">subInterpFormula</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="value:86" class="idref" href="#value:86"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="f:87" class="idref" href="#f:87"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="v:88" class="idref" href="#v:88"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:89" class="idref" href="#s:89"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:86"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:88"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:86"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:89"><span class="id" title="variable">s</span></a>)) <a class="idref" href="hydras.Ackermann.model.html#f:87"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:86"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:87"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:88"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:89"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subInterpFormula1" class="idref" href="#subInterpFormula1"><span class="id" title="lemma">subInterpFormula1</span></a> (<a id="value:142" class="idref" href="#value:142"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="f:143" class="idref" href="#f:143"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="v:144" class="idref" href="#v:144"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:145" class="idref" href="#s:145"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:142"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:144"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:142"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:145"><span class="id" title="variable">s</span></a>)) <a class="idref" href="hydras.Ackermann.model.html#f:143"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:142"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:143"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:144"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:145"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subInterpFormula2" class="idref" href="#subInterpFormula2"><span class="id" title="lemma">subInterpFormula2</span></a> (<a id="value:146" class="idref" href="#value:146"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="f:147" class="idref" href="#f:147"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="v:148" class="idref" href="#v:148"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:149" class="idref" href="#s:149"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:146"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:147"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:148"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:149"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> (<a class="idref" href="hydras.Ackermann.model.html#updateValue"><span class="id" title="definition">updateValue</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:146"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:148"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:146"><span class="id" title="variable">value</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:149"><span class="id" title="variable">s</span></a>)) <a class="idref" href="hydras.Ackermann.model.html#f:147"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nnHelp" class="idref" href="#nnHelp"><span class="id" title="definition">nnHelp</span></a> (<a id="f:150" class="idref" href="#f:150"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#f:150"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <span class="id" title="var">t</span> <span class="id" title="var">s</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnHelp:151"><span class="id" title="definition">nnHelp</span></a> <span class="id" title="var">A</span>) (<a class="idref" href="hydras.Ackermann.model.html#nnHelp:151"><span class="id" title="definition">nnHelp</span></a> <span class="id" title="var">B</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <span class="id" title="var">A</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnHelp:151"><span class="id" title="definition">nnHelp</span></a> <span class="id" title="var">A</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">v</span> <span class="id" title="var">A</span> ⇒ (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.model.html#nnHelp:151"><span class="id" title="definition">nnHelp</span></a> <span class="id" title="var">A</span>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="nnTranslate" class="idref" href="#nnTranslate"><span class="id" title="definition">nnTranslate</span></a> (<a id="f:153" class="idref" href="#f:153"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnHelp"><span class="id" title="definition">nnHelp</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:153"><span class="id" title="variable">f</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarNNHelp" class="idref" href="#freeVarNNHelp"><span class="id" title="lemma">freeVarNNHelp</span></a> (<a id="f:154" class="idref" href="#f:154"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>):  <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:154"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnHelp"><span class="id" title="definition">nnHelp</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:154"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subNNHelp" class="idref" href="#subNNHelp"><span class="id" title="lemma">subNNHelp</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:155" class="idref" href="#f:155"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>) (<a id="v:156" class="idref" href="#v:156"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:157" class="idref" href="#s:157"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnHelp"><span class="id" title="definition">nnHelp</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:155"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.model.html#v:156"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:157"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.model.html#nnHelp"><span class="id" title="definition">nnHelp</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:155"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.model.html#v:156"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.model.html#s:157"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Model_Theory.Consistent_Theory" class="idref" href="#Model_Theory.Consistent_Theory"><span class="id" title="section">Consistent_Theory</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Model_Theory.Consistent_Theory.T" class="idref" href="#Model_Theory.Consistent_Theory.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="interpTermsVector" class="idref" href="#interpTermsVector"><span class="id" title="definition">interpTermsVector</span></a> (<a id="value:159" class="idref" href="#value:159"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a id="n:160" class="idref" href="#n:160"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="ts:161" class="idref" href="#ts:161"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#n:160"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:160"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.model.html#ts:161"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="hydras.Ackermann.model.html#n:163"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <span class="id" title="var">ts</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="hydras.Ackermann.model.html#interpTerm"><span class="id" title="definition">interpTerm</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:159"><span class="id" title="variable">value</span></a> <span class="id" title="var">t</span>) <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.model.html#interpTermsVector:162"><span class="id" title="definition">interpTermsVector</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:159"><span class="id" title="variable">value</span></a> <span class="id" title="var">m</span> <a class="idref" href="hydras.Ackermann.model.html#ts:161"><span class="id" title="variable">ts</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="preserveValue" class="idref" href="#preserveValue"><span class="id" title="lemma">preserveValue</span></a> (<a id="value:165" class="idref" href="#value:165"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:166" class="idref" href="#f:166"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:166"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:165"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnTranslate"><span class="id" title="definition">nnTranslate</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:166"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:167" class="idref" href="#g:167"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.model.html#g:167"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:165"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnTranslate"><span class="id" title="definition">nnTranslate</span></a> <a class="idref" href="hydras.Ackermann.model.html#g:167"><span class="id" title="variable">g</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ModelConsistent" class="idref" href="#ModelConsistent"><span class="id" title="lemma">ModelConsistent</span></a> (<a id="value:222" class="idref" href="#value:222"><span class="id" title="binder">value</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#U"><span class="id" title="projection">U</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.M"><span class="id" title="variable">M</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:223" class="idref" href="#f:223"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:223"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.model.html#interpFormula"><span class="id" title="definition">interpFormula</span></a> <a class="idref" href="hydras.Ackermann.model.html#value:222"><span class="id" title="variable">value</span></a> (<a class="idref" href="hydras.Ackermann.model.html#nnTranslate"><span class="id" title="definition">nnTranslate</span></a> <a class="idref" href="hydras.Ackermann.model.html#f:223"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Consistent"><span class="id" title="definition">Consistent</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory.T"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory"><span class="id" title="section">Consistent_Theory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.model.html#Model_Theory"><span class="id" title="section">Model_Theory</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.prLogic.html b/theories/html/hydras.Ackermann.prLogic.html
new file mode 100644
index 00000000..363c0f05
--- /dev/null
+++ b/theories/html/hydras.Ackermann.prLogic.html
@@ -0,0 +1,60 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.prLogic</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.prLogic</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+prLogic.v
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor
+
+<div class="paragraph"> </div>
+
+     Although this module doesn't depend on FOL, the following lemmas 
+     are JUST  helpers to prove that the encoding of basic FOL connectives are PR 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="codeForallIsPR" class="idref" href="#codeForallIsPR"><span class="id" title="lemma">codeForallIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="b:2" class="idref" href="#b:2"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 3 (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.prLogic.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.prLogic.html#b:2"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeNotIsPR" class="idref" href="#codeNotIsPR"><span class="id" title="instance">codeNotIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.code.html#codeNot"><span class="id" title="definition">codeNot</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="codeImpIsPR" class="idref" href="#codeImpIsPR"><span class="id" title="instance">codeImpIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.code.html#codeImp"><span class="id" title="definition">codeImp</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.primRec.html b/theories/html/hydras.Ackermann.primRec.html
new file mode 100644
index 00000000..5a9d61df
--- /dev/null
+++ b/theories/html/hydras.Ackermann.primRec.html
@@ -0,0 +1,950 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.primRec</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.primRec</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Primitive Recursive functions  Russel O'Connor 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Unicode.Utf8.html#"><span class="id" title="library">Utf8</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.misc.html#"><span class="id" title="library">misc</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.EqNat.html#"><span class="id" title="library">EqNat</span></a>.<br/>
+<span class="id" title="keyword">From</span>  <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab305"></a><h1 class="section">Definitions</h1>
+
+<div class="paragraph"> </div>
+
+ <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> : data type of primitive recursive functions of arity <span class="inlinecode"><span class="id" title="var">n</span></span> 
+    <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> : m-tuples of <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="PrimRec" class="idref" href="#PrimRec"><span class="id" title="inductive">PrimRec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="succFunc" class="idref" href="#succFunc"><span class="id" title="constructor">succFunc</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> 1<br/>
+&nbsp;&nbsp;| <a id="zeroFunc" class="idref" href="#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> 0<br/>
+&nbsp;&nbsp;| <a id="projFunc" class="idref" href="#projFunc"><span class="id" title="constructor">projFunc</span></a> : <span class="id" title="keyword">∀</span> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> <a id="m:6" class="idref" href="#m:6"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#m:6"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:5"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;| <a id="composeFunc" class="idref" href="#composeFunc"><span class="id" title="constructor">composeFunc</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> <a id="m:8" class="idref" href="#m:8"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:9" class="idref" href="#g:9"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs:2"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:7"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:8"><span class="id" title="variable">m</span></a>) (<a id="h:10" class="idref" href="#h:10"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:8"><span class="id" title="variable">m</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:7"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;| <a id="primRecFunc" class="idref" href="#primRecFunc"><span class="id" title="constructor">primRecFunc</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:12" class="idref" href="#g:12"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:11"><span class="id" title="variable">n</span></a>) (<a id="h:13" class="idref" href="#h:13"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:11"><span class="id" title="variable">n</span></a>))),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:11"><span class="id" title="variable">n</span></a>)<br/>
+<span class="id" title="keyword">with</span> <a id="PrimRecs" class="idref" href="#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="PRnil" class="idref" href="#PRnil"><span class="id" title="constructor">PRnil</span></a> : <span class="id" title="keyword">∀</span> <a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs:2"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:14"><span class="id" title="variable">n</span></a> 0<br/>
+&nbsp;&nbsp;| <a id="PRcons" class="idref" href="#PRcons"><span class="id" title="constructor">PRcons</span></a> : <span class="id" title="keyword">∀</span> <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> <a id="m:16" class="idref" href="#m:16"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec:1"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:15"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs:2"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:15"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:16"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs:2"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:15"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:16"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="0a87cec6db386ad672945a4ad4643c14" class="idref" href="#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">&quot;</span></a>f '=x=' g" := (<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <span class="id" title="var">_</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="PRNotations" class="idref" href="#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">pr_scope</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">pr_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">pr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.::pr_scope:x_'::'_x" class="idref" href="#PRNotations.::pr_scope:x_'::'_x"><span class="id" title="notation">&quot;</span></a>h :: t" := (<a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">h</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;: <span class="id" title="var">pr_scope</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="7270cfdd80e3adbb7f438dd90c04a50e" class="idref" href="#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">&quot;</span></a>[ x ]" := (<a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">x</span>  (<a class="idref" href="hydras.Ackermann.primRec.html#PRnil"><span class="id" title="constructor">PRnil</span></a> <span class="id" title="var">_</span>)) : <span class="id" title="var">pr_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="0d92c10e9c6b3f86aa7104ae219ede74" class="idref" href="#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">&quot;</span></a>[ x ; y ; .. ; z ]" := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">x</span>  (<a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">y</span> .. (<a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">z</span>  (<a class="idref" href="hydras.Ackermann.primRec.html#PRnil"><span class="id" title="constructor">PRnil</span></a> <span class="id" title="var">_</span>)) ..)) : <span class="id" title="var">pr_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.PRcomp" class="idref" href="#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <span class="id" title="var">f</span> <span class="id" title="var">v</span> := (<a class="idref" href="hydras.Ackermann.primRec.html#composeFunc"><span class="id" title="constructor">composeFunc</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">v</span> <span class="id" title="var">f</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.PRrec" class="idref" href="#PRNotations.PRrec"><span class="id" title="abbreviation">PRrec</span></a> <span class="id" title="var">f0</span> <span class="id" title="var">fS</span> := (<a class="idref" href="hydras.Ackermann.primRec.html#primRecFunc"><span class="id" title="constructor">primRecFunc</span></a> <span class="id" title="var">_</span> <span class="id" title="var">f0</span> <span class="id" title="var">fS</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Popular projections 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi1_1" class="idref" href="#PRNotations.pi1_1"><span class="id" title="abbreviation">pi1_1</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 1 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 1)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi1_2" class="idref" href="#PRNotations.pi1_2"><span class="id" title="abbreviation">pi1_2</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 2 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 2)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi2_2" class="idref" href="#PRNotations.pi2_2"><span class="id" title="abbreviation">pi2_2</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 2 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_S"><span class="id" title="constructor">le_S</span></a> 1 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 1))).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi1_3" class="idref" href="#PRNotations.pi1_3"><span class="id" title="abbreviation">pi1_3</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 3 2 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 3)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi2_3" class="idref" href="#PRNotations.pi2_3"><span class="id" title="abbreviation">pi2_3</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 3 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_S"><span class="id" title="constructor">le_S</span></a> 2 2 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 2))).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="PRNotations.pi3_3" class="idref" href="#PRNotations.pi3_3"><span class="id" title="abbreviation">pi3_3</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 3 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_S"><span class="id" title="constructor">le_S</span></a> 1 2 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_S"><span class="id" title="constructor">le_S</span></a> 1 1 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 1)))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="PrimRec_PrimRecs_rec" class="idref" href="#PrimRec_PrimRecs_rec"><span class="id" title="definition">PrimRec_PrimRecs_rec</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a id="PrimRecs_PrimRec_rec" class="idref" href="#PrimRecs_PrimRec_rec"><span class="id" title="definition">PrimRecs_PrimRec_rec</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a>  <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Set</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec_PrimRecs_rec"><span class="id" title="definition">PrimRec_PrimRecs_rec</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs_PrimRec_rec"><span class="id" title="definition">PrimRecs_PrimRec_rec</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="PrimRec_PrimRecs_ind" class="idref" href="#PrimRec_PrimRecs_ind"><span class="id" title="definition">PrimRec_PrimRecs_ind</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a id="PrimRecs_PrimRec_ind" class="idref" href="#PrimRecs_PrimRec_ind"><span class="id" title="definition">PrimRecs_PrimRec_ind</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a>  <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec_PrimRecs_ind"><span class="id" title="definition">PrimRec_PrimRecs_ind</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs_PrimRec_ind"><span class="id" title="definition">PrimRecs_PrimRec_ind</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec_PrimRecs_ind"><span class="id" title="definition">PrimRec_PrimRecs_ind</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab306"></a><h2 class="section">Semantics</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab307"></a><h3 class="section">Constants</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalConstFunc" class="idref" href="#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> (<a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="m:18" class="idref" href="#m:18"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:17"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#n:17"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:20"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#m:18"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc:19"><span class="id" title="definition">evalConstFunc</span></a> <span class="id" title="var">n'</span> <a class="idref" href="hydras.Ackermann.primRec.html#m:18"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab308"></a><h3 class="section">Projections</h3>
+
+   The parameters are numbered in opposite order.
+   So proj(2,0)(a,b) = b. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalProjFunc" class="idref" href="#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> (<a id="n:21" class="idref" href="#n:21"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m:22" class="idref" href="#m:22"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#m:22"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:21"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:21"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#n:21"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<span class="id" title="keyword">∀</span> <a id="m:25" class="idref" href="#m:25"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#m:25"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:24"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="keyword">fun</span> (<a id="m:26" class="idref" href="#m:26"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:27" class="idref" href="#l:27"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:26"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> 0) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False_rec"><span class="id" title="definition">False_rec</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.nlt_0_r"><span class="id" title="lemma">Nat.nlt_0_r</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#l:27"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="m:28" class="idref" href="#m:28"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:28"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:28"><span class="id" title="variable">m</span></a> <span class="id" title="var">n'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <span class="id" title="keyword">fun</span> <a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:30"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">l1</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc:23"><span class="id" title="definition">evalProjFunc</span></a> <span class="id" title="var">n'</span> <a class="idref" href="hydras.Ackermann.primRec.html#m:28"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Compat815.html#Compat815.le_lt_or_eq"><span class="id" title="lemma">Compat815.le_lt_or_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Compat815.html#Compat815.lt_n_Sm_le"><span class="id" title="lemma">Compat815.lt_n_Sm_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#l:29"><span class="id" title="variable">l</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#or_introl"><span class="id" title="constructor">or_introl</span></a> <span class="id" title="var">l2</span> ⇒ <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#or_intror"><span class="id" title="constructor">or_intror</span></a> <span class="id" title="var">l2</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False_ind"><span class="id" title="definition">False_ind</span></a> <span class="id" title="var">_</span> (<span class="id" title="var">l1</span> <span class="id" title="var">l2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Irrelevance of the proof that <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">n</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="evalProjFuncInd" class="idref" href="#evalProjFuncInd"><span class="id" title="lemma">evalProjFuncInd</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> <a id="m:32" class="idref" href="#m:32"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p1:33" class="idref" href="#p1:33"><span class="id" title="binder">p1</span></a> <a id="p2:34" class="idref" href="#p2:34"><span class="id" title="binder">p2</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:32"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:31"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:32"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p1:33"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:32"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p2:34"><span class="id" title="variable">p2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Applies an m-ary function to the vector l 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalList" class="idref" href="#evalList"><span class="id" title="definition">evalList</span></a> (<a id="m:35" class="idref" href="#m:35"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:36" class="idref" href="#l:36"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:35"><span class="id" title="variable">m</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:35"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#l:36"><span class="id" title="variable">l</span></a>  <span class="id" title="tactic">in</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <span class="id" title="var">_</span> <span class="id" title="var">m</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:39"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 0 ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#x:40"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">l'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="x:41" class="idref" href="#x:41"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#evalList:37"><span class="id" title="definition">evalList</span></a> <span class="id" title="var">n</span> <span class="id" title="var">l'</span> (<a class="idref" href="hydras.Ackermann.primRec.html#x:41"><span class="id" title="variable">x</span></a> <span class="id" title="var">a</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalOneParamList" class="idref" href="#evalOneParamList"><span class="id" title="definition">evalOneParamList</span></a> (<a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a> <a id="m:43" class="idref" href="#m:43"><span class="id" title="binder">m</span></a> <a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:45" class="idref" href="#l:45"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:42"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Ackermann.primRec.html#m:43"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;{<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:42"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:43"><span class="id" title="variable">m</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#l:45"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <span class="id" title="var">_</span> <span class="id" title="var">m</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:42"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:48"><span class="id" title="variable">m</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a>  (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:42"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">f</span> <span class="id" title="var">m'</span> <span class="id" title="var">l'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> (<span class="id" title="var">f</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:44"><span class="id" title="variable">a</span></a>) <span class="id" title="var">m'</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalOneParamList:46"><span class="id" title="definition">evalOneParamList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:42"><span class="id" title="variable">n</span></a> <span class="id" title="var">m'</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:44"><span class="id" title="variable">a</span></a> <span class="id" title="var">l'</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab309"></a><h3 class="section">Function composition</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalComposeFunc" class="idref" href="#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> (<a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m:50" class="idref" href="#m:50"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:49"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:50"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:50"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:49"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#n:49"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">return</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="m:53" class="idref" href="#m:53"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:52"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:53"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:53"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:52"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#evalList"><span class="id" title="definition">evalList</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="m:54" class="idref" href="#m:54"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:55" class="idref" href="#l:55"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>)) <a class="idref" href="hydras.Ackermann.primRec.html#m:54"><span class="id" title="variable">m</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:56" class="idref" href="#f:56"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:54"><span class="id" title="variable">m</span></a>) (<a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc:51"><span class="id" title="definition">evalComposeFunc</span></a> <span class="id" title="var">n'</span> <a class="idref" href="hydras.Ackermann.primRec.html#m:54"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalOneParamList"><span class="id" title="definition">evalOneParamList</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l:55"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#f:56"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="compose2" class="idref" href="#compose2"><span class="id" title="definition">compose2</span></a> (<a id="n:58" class="idref" href="#n:58"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:58"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:58"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:58"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#n:58"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:60"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:60"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:60"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="keyword">fun</span> (<a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:62" class="idref" href="#g:62"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:62"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:61"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="f:63" class="idref" href="#f:63"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>)) (<a id="g:64" class="idref" href="#g:64"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>))) (<a id="a:65" class="idref" href="#a:65"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#compose2:59"><span class="id" title="definition">compose2</span></a> <span class="id" title="var">n'</span> (<a class="idref" href="hydras.Ackermann.primRec.html#f:63"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:65"><span class="id" title="variable">a</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:66" class="idref" href="#x:66"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:64"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:66"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:65"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab310"></a><h3 class="section">Primitive recursion</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalPrimRecFunc" class="idref" href="#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> (<a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:68" class="idref" href="#g:68"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:67"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;(<a id="h:69" class="idref" href="#h:69"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:67"><span class="id" title="variable">n</span></a>))) (<a id="a:70" class="idref" href="#a:70"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">a</span>} : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:67"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:70"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:68"><span class="id" title="variable">g</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">a'</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc:71"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:68"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h:69"><span class="id" title="variable">h</span></a> <span class="id" title="var">a'</span>) (<a class="idref" href="hydras.Ackermann.primRec.html#h:69"><span class="id" title="variable">h</span></a> <span class="id" title="var">a'</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ The interpretation function 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="evalPrimRec" class="idref" href="#evalPrimRec"><span class="id" title="definition">evalPrimRec</span></a> (<a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:73"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">f</span>} : <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:73"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#f:74"><span class="id" title="variable">f</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <span class="id" title="var">n</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:81"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">pf</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:73"><span class="id" title="variable">n</span></a> <span class="id" title="var">m</span> <span class="id" title="var">pf</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#composeFunc"><span class="id" title="constructor">composeFunc</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">f</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:73"><span class="id" title="variable">n</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecs:79"><span class="id" title="definition">evalPrimRecs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">l</span>) (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec:78"><span class="id" title="definition">evalPrimRec</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#f:74"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#primRecFunc"><span class="id" title="constructor">primRecFunc</span></a> <span class="id" title="var">n</span> <span class="id" title="var">g</span> <span class="id" title="var">h</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:73"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec:78"><span class="id" title="definition">evalPrimRec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">g</span>) (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec:78"><span class="id" title="definition">evalPrimRec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">h</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="evalPrimRecs" class="idref" href="#evalPrimRecs"><span class="id" title="definition">evalPrimRecs</span></a> (<a id="n:75" class="idref" href="#n:75"><span class="id" title="binder">n</span></a> <a id="m:76" class="idref" href="#m:76"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="fs:77" class="idref" href="#fs:77"><span class="id" title="binder">fs</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:75"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:76"><span class="id" title="variable">m</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">fs</span>} :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:75"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:76"><span class="id" title="variable">m</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#fs:77"><span class="id" title="variable">fs</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:84"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:83"><span class="id" title="variable">m</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#PRnil"><span class="id" title="constructor">PRnil</span></a> <span class="id" title="var">a</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a>  (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <span class="id" title="var">a</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.primRec.html#PRcons"><span class="id" title="constructor">PRcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">g</span> <span class="id" title="var">gs</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec:78"><span class="id" title="definition">evalPrimRec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">g</span>) <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecs:79"><span class="id" title="definition">evalPrimRecs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">gs</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="PReval" class="idref" href="#PReval"><span class="id" title="abbreviation">PReval</span></a> <span class="id" title="var">f</span> := (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec"><span class="id" title="definition">evalPrimRec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">f</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="PRevalN" class="idref" href="#PRevalN"><span class="id" title="abbreviation">PRevalN</span></a> <span class="id" title="var">fs</span> := (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecs"><span class="id" title="definition">evalPrimRecs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">fs</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">p</span></span> is a correct implementation of <span class="inlinecode"><span class="id" title="var">f</span></span> in <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PRcorrect" class="idref" href="#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> {<a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}(<a id="p:86" class="idref" href="#p:86"><span class="id" title="binder">p</span></a>:<a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:85"><span class="id" title="variable">n</span></a>)(<a id="f:87" class="idref" href="#f:87"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:85"><span class="id" title="variable">n</span></a>) := <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p:86"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:87"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="extEqualVectorGeneral" class="idref" href="#extEqualVectorGeneral"><span class="id" title="definition">extEqualVectorGeneral</span></a> (<a id="n:88" class="idref" href="#n:88"><span class="id" title="binder">n</span></a> <a id="m:89" class="idref" href="#m:89"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:90" class="idref" href="#l:90"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:88"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:89"><span class="id" title="variable">m</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m':91" class="idref" href="#m':91"><span class="id" title="binder">m'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l':92" class="idref" href="#l':92"><span class="id" title="binder">l'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:88"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m':91"><span class="id" title="variable">m'</span></a>), <span class="id" title="keyword">Prop</span>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Every element of l is extensionally equal to the element of l' at the same position 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="extEqualVector" class="idref" href="#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> <a id="n:93" class="idref" href="#n:93"><span class="id" title="binder">n</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m:94" class="idref" href="#m:94"><span class="id" title="binder">m</span></a> (<a id="l:95" class="idref" href="#l:95"><span class="id" title="binder">l</span></a> <a id="l':96" class="idref" href="#l':96"><span class="id" title="binder">l'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:93"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:94"><span class="id" title="variable">m</span></a>), <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualVectorRefl" class="idref" href="#extEqualVectorRefl"><span class="id" title="lemma">extEqualVectorRefl</span></a> (<a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a> <a id="m:98" class="idref" href="#m:98"><span class="id" title="binder">m</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:99" class="idref" href="#l:99"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:97"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:98"><span class="id" title="variable">m</span></a>), <a class="idref" href="hydras.Ackermann.primRec.html#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:98"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l:99"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l:99"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualOneParamList" class="idref" href="#extEqualOneParamList"><span class="id" title="lemma">extEqualOneParamList</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:100" class="idref" href="#n:100"><span class="id" title="binder">n</span></a> <a id="m:101" class="idref" href="#m:101"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l1:102" class="idref" href="#l1:102"><span class="id" title="binder">l1</span></a> <a id="l2:103" class="idref" href="#l2:103"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:100"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Ackermann.primRec.html#m:101"><span class="id" title="variable">m</span></a>) (<a id="c:104" class="idref" href="#c:104"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:100"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:101"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l1:102"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l2:103"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:101"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalOneParamList"><span class="id" title="definition">evalOneParamList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:101"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:104"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l1:102"><span class="id" title="variable">l1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalOneParamList"><span class="id" title="definition">evalOneParamList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:101"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:104"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l2:103"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualCompose" class="idref" href="#extEqualCompose"><span class="id" title="lemma">extEqualCompose</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:105" class="idref" href="#n:105"><span class="id" title="binder">n</span></a> <a id="m:106" class="idref" href="#m:106"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l1:107" class="idref" href="#l1:107"><span class="id" title="binder">l1</span></a> <a id="l2:108" class="idref" href="#l2:108"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:105"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:106"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f1:109" class="idref" href="#f1:109"><span class="id" title="binder">f1</span></a> <a id="f2:110" class="idref" href="#f2:110"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:106"><span class="id" title="variable">m</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:105"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:106"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l1:107"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l2:108"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#f1:109"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f2:110"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:105"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:106"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l1:107"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f1:109"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:105"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:106"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#l2:108"><span class="id" title="variable">l2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f2:110"><span class="id" title="variable">f2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualCompose2" class="idref" href="#extEqualCompose2"><span class="id" title="lemma">extEqualCompose2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:111" class="idref" href="#n:111"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f1:112" class="idref" href="#f1:112"><span class="id" title="binder">f1</span></a> <a id="f2:113" class="idref" href="#f2:113"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:111"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#f1:112"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f2:113"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g1:114" class="idref" href="#g1:114"><span class="id" title="binder">g1</span></a> <a id="g2:115" class="idref" href="#g2:115"><span class="id" title="binder">g2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:111"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#g1:114"><span class="id" title="variable">g1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g2:115"><span class="id" title="variable">g2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:111"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f1:112"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g1:114"><span class="id" title="variable">g1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:111"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f2:113"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g2:115"><span class="id" title="variable">g2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="extEqualPrimRec" class="idref" href="#extEqualPrimRec"><span class="id" title="lemma">extEqualPrimRec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:116" class="idref" href="#n:116"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g1:117" class="idref" href="#g1:117"><span class="id" title="binder">g1</span></a> <a id="g2:118" class="idref" href="#g2:118"><span class="id" title="binder">g2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:116"><span class="id" title="variable">n</span></a>) (<a id="h1:119" class="idref" href="#h1:119"><span class="id" title="binder">h1</span></a> <a id="h2:120" class="idref" href="#h2:120"><span class="id" title="binder">h2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:116"><span class="id" title="variable">n</span></a>))),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#g1:117"><span class="id" title="variable">g1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g2:118"><span class="id" title="variable">g2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h1:119"><span class="id" title="variable">h1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h2:120"><span class="id" title="variable">h2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g1:117"><span class="id" title="variable">g1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h1:119"><span class="id" title="variable">h1</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:116"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g2:118"><span class="id" title="variable">g2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h2:120"><span class="id" title="variable">h2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab311"></a><h2 class="section">Predicates "to be primitive recursive"</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="isPR" class="idref" href="#isPR"><span class="id" title="record">isPR</span></a> (<a id="n:121" class="idref" href="#n:121"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:122" class="idref" href="#f:122"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:121"><span class="id" title="variable">n</span></a>) : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="is_pr" class="idref" href="#is_pr"><span class="id" title="projection">is_pr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="p:124" class="idref" href="#p:124"><span class="id" title="binder">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:121"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:121"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p:124"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#f:122"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fun2PR" class="idref" href="#fun2PR"><span class="id" title="definition">fun2PR</span></a> {<a id="n:126" class="idref" href="#n:126"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}(<a id="f:127" class="idref" href="#f:127"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:126"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;{<a id="p:128" class="idref" href="#p:128"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#f:127"><span class="id" title="variable">f</span></a>}: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:126"><span class="id" title="variable">n</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p:128"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="isPRrel" class="idref" href="#isPRrel"><span class="id" title="record">isPRrel</span></a> (<a id="n:129" class="idref" href="#n:129"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:130" class="idref" href="#R:130"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:129"><span class="id" title="variable">n</span></a>) : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="is_pr_rel" class="idref" href="#is_pr_rel"><span class="id" title="projection">is_pr_rel</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:129"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:129"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:130"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="succIsPR" class="idref" href="#succIsPR"><span class="id" title="instance">succIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="const0_NIsPR" class="idref" href="#const0_NIsPR"><span class="id" title="instance">const0_NIsPR</span></a> (<a id="n:133" class="idref" href="#n:133"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 0 <a class="idref" href="hydras.Ackermann.primRec.html#n:133"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="const1_NIsPR" class="idref" href="#const1_NIsPR"><span class="id" title="instance">const1_NIsPR</span></a> <a id="n:134" class="idref" href="#n:134"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#n:134"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab312"></a><h2 class="section">Usual projections (in curried form) are primitive recursive</h2>
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="idIsPR" class="idref" href="#idIsPR"><span class="id" title="instance">idIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="x:135" class="idref" href="#x:135"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#x:135"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi1_2IsPR" class="idref" href="#pi1_2IsPR"><span class="id" title="instance">pi1_2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:136" class="idref" href="#a:136"><span class="id" title="binder">a</span></a> <a id="b:137" class="idref" href="#b:137"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#a:136"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi2_2IsPR" class="idref" href="#pi2_2IsPR"><span class="id" title="instance">pi2_2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:138" class="idref" href="#a:138"><span class="id" title="binder">a</span></a> <a id="b:139" class="idref" href="#b:139"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#b:139"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi1_3IsPR" class="idref" href="#pi1_3IsPR"><span class="id" title="instance">pi1_3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:140" class="idref" href="#a:140"><span class="id" title="binder">a</span></a> <a id="b:141" class="idref" href="#b:141"><span class="id" title="binder">b</span></a> <a id="c:142" class="idref" href="#c:142"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#a:140"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi2_3IsPR" class="idref" href="#pi2_3IsPR"><span class="id" title="instance">pi2_3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:143" class="idref" href="#a:143"><span class="id" title="binder">a</span></a> <a id="b:144" class="idref" href="#b:144"><span class="id" title="binder">b</span></a> <a id="c:145" class="idref" href="#c:145"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#b:144"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi3_3IsPR" class="idref" href="#pi3_3IsPR"><span class="id" title="instance">pi3_3IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:146" class="idref" href="#a:146"><span class="id" title="binder">a</span></a> <a id="b:147" class="idref" href="#b:147"><span class="id" title="binder">b</span></a> <a id="c:148" class="idref" href="#c:148"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#c:148"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi1_4IsPR" class="idref" href="#pi1_4IsPR"><span class="id" title="instance">pi1_4IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:149" class="idref" href="#a:149"><span class="id" title="binder">a</span></a> <a id="b:150" class="idref" href="#b:150"><span class="id" title="binder">b</span></a> <a id="c:151" class="idref" href="#c:151"><span class="id" title="binder">c</span></a> <a id="d:152" class="idref" href="#d:152"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#a:149"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi2_4IsPR" class="idref" href="#pi2_4IsPR"><span class="id" title="instance">pi2_4IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:153" class="idref" href="#a:153"><span class="id" title="binder">a</span></a> <a id="b:154" class="idref" href="#b:154"><span class="id" title="binder">b</span></a> <a id="c:155" class="idref" href="#c:155"><span class="id" title="binder">c</span></a> <a id="d:156" class="idref" href="#d:156"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#b:154"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi3_4IsPR" class="idref" href="#pi3_4IsPR"><span class="id" title="instance">pi3_4IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:157" class="idref" href="#a:157"><span class="id" title="binder">a</span></a> <a id="b:158" class="idref" href="#b:158"><span class="id" title="binder">b</span></a> <a id="c:159" class="idref" href="#c:159"><span class="id" title="binder">c</span></a> <a id="d:160" class="idref" href="#d:160"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#c:159"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="pi4_4IsPR" class="idref" href="#pi4_4IsPR"><span class="id" title="instance">pi4_4IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:161" class="idref" href="#a:161"><span class="id" title="binder">a</span></a> <a id="b:162" class="idref" href="#b:162"><span class="id" title="binder">b</span></a> <a id="c:163" class="idref" href="#c:163"><span class="id" title="binder">c</span></a> <a id="d:164" class="idref" href="#d:164"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#d:164"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab313"></a><h2 class="section">Composition lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter01IsPR" class="idref" href="#filter01IsPR"><span class="id" title="instance">filter01IsPR</span></a> (<a id="g:165" class="idref" href="#g:165"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="H:166" class="idref" href="#H:166"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:165"><span class="id" title="variable">g</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:167" class="idref" href="#a:167"><span class="id" title="binder">a</span></a> <a id="b:168" class="idref" href="#b:168"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:165"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:168"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter10IsPR" class="idref" href="#filter10IsPR"><span class="id" title="instance">filter10IsPR</span></a> (<a id="g:173" class="idref" href="#g:173"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="H:174" class="idref" href="#H:174"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:173"><span class="id" title="variable">g</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:175" class="idref" href="#a:175"><span class="id" title="binder">a</span></a> <a id="b:176" class="idref" href="#b:176"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:173"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:175"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter100IsPR" class="idref" href="#filter100IsPR"><span class="id" title="instance">filter100IsPR</span></a>(<a id="g:181" class="idref" href="#g:181"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="H:182" class="idref" href="#H:182"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:181"><span class="id" title="variable">g</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:183" class="idref" href="#a:183"><span class="id" title="binder">a</span></a> <a id="b:184" class="idref" href="#b:184"><span class="id" title="binder">b</span></a> <a id="c:185" class="idref" href="#c:185"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:181"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:183"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter010IsPR" class="idref" href="#filter010IsPR"><span class="id" title="instance">filter010IsPR</span></a> (<a id="g:192" class="idref" href="#g:192"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="H:193" class="idref" href="#H:193"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:192"><span class="id" title="variable">g</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:194" class="idref" href="#a:194"><span class="id" title="binder">a</span></a> <a id="b:195" class="idref" href="#b:195"><span class="id" title="binder">b</span></a> <a id="c:196" class="idref" href="#c:196"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:192"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:195"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter001IsPR" class="idref" href="#filter001IsPR"><span class="id" title="instance">filter001IsPR</span></a> (<a id="g:203" class="idref" href="#g:203"><span class="id" title="binder">g</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="H:204" class="idref" href="#H:204"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:203"><span class="id" title="variable">g</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:205" class="idref" href="#a:205"><span class="id" title="binder">a</span></a> <a id="b:206" class="idref" href="#b:206"><span class="id" title="binder">b</span></a> <a id="c:207" class="idref" href="#c:207"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:203"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:207"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter011IsPR" class="idref" href="#filter011IsPR"><span class="id" title="instance">filter011IsPR</span></a> (<a id="g:214" class="idref" href="#g:214"><span class="id" title="binder">g</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="H:215" class="idref" href="#H:215"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:214"><span class="id" title="variable">g</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:216" class="idref" href="#a:216"><span class="id" title="binder">a</span></a> <a id="b:217" class="idref" href="#b:217"><span class="id" title="binder">b</span></a> <a id="c:218" class="idref" href="#c:218"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:214"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:217"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:218"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter110IsPR" class="idref" href="#filter110IsPR"><span class="id" title="instance">filter110IsPR</span></a>(<a id="g:231" class="idref" href="#g:231"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:231"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:232" class="idref" href="#a:232"><span class="id" title="binder">a</span></a> <a id="b:233" class="idref" href="#b:233"><span class="id" title="binder">b</span></a> <a id="c:234" class="idref" href="#c:234"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:231"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:232"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:233"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter101IsPR" class="idref" href="#filter101IsPR"><span class="id" title="instance">filter101IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:247" class="idref" href="#g:247"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:247"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="a:248" class="idref" href="#a:248"><span class="id" title="binder">a</span></a> <a id="b:249" class="idref" href="#b:249"><span class="id" title="binder">b</span></a> <a id="c:250" class="idref" href="#c:250"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:247"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:248"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:250"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter0011IsPR" class="idref" href="#filter0011IsPR"><span class="id" title="instance">filter0011IsPR</span></a> (<a id="g:263" class="idref" href="#g:263"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:263"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:264" class="idref" href="#a:264"><span class="id" title="binder">a</span></a> <a id="b:265" class="idref" href="#b:265"><span class="id" title="binder">b</span></a> <a id="c:266" class="idref" href="#c:266"><span class="id" title="binder">c</span></a> <a id="d:267" class="idref" href="#d:267"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:263"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:266"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#d:267"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter1000IsPR" class="idref" href="#filter1000IsPR"><span class="id" title="instance">filter1000IsPR</span></a> (<a id="g:284" class="idref" href="#g:284"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:284"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:285" class="idref" href="#a:285"><span class="id" title="binder">a</span></a> <a id="b:286" class="idref" href="#b:286"><span class="id" title="binder">b</span></a> <a id="c:287" class="idref" href="#c:287"><span class="id" title="binder">c</span></a> <a id="d:288" class="idref" href="#d:288"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:284"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:285"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter1011IsPR" class="idref" href="#filter1011IsPR"><span class="id" title="instance">filter1011IsPR</span></a> (<a id="g:297" class="idref" href="#g:297"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#g:297"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:298" class="idref" href="#a:298"><span class="id" title="binder">a</span></a> <a id="b:299" class="idref" href="#b:299"><span class="id" title="binder">b</span></a> <a id="c:300" class="idref" href="#c:300"><span class="id" title="binder">c</span></a> <a id="d:301" class="idref" href="#d:301"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:297"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:298"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:300"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#d:301"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="filter1100IsPR" class="idref" href="#filter1100IsPR"><span class="id" title="instance">filter1100IsPR</span></a> (<a id="g:326" class="idref" href="#g:326"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:326"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:327" class="idref" href="#a:327"><span class="id" title="binder">a</span></a> <a id="b:328" class="idref" href="#b:328"><span class="id" title="binder">b</span></a> <a id="c:329" class="idref" href="#c:329"><span class="id" title="binder">c</span></a> <a id="d:330" class="idref" href="#d:330"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:326"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:327"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:328"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose1_1IsPR" class="idref" href="#compose1_1IsPR"><span class="id" title="instance">compose1_1IsPR</span></a> (<a id="f:347" class="idref" href="#f:347"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f:347"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:348" class="idref" href="#g:348"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:348"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="x:349" class="idref" href="#x:349"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:348"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:347"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:349"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose1_2IsPR" class="idref" href="#compose1_2IsPR"><span class="id" title="instance">compose1_2IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:350" class="idref" href="#f:350"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f:350"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f':351" class="idref" href="#f':351"><span class="id" title="binder">f'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f':351"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:352" class="idref" href="#g:352"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:352"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="x:353" class="idref" href="#x:353"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:352"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:350"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:353"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f':351"><span class="id" title="variable">f'</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:353"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose1_3IsPR" class="idref" href="#compose1_3IsPR"><span class="id" title="instance">compose1_3IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:354" class="idref" href="#f1:354"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f1:354"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:355" class="idref" href="#f2:355"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f2:355"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f3:356" class="idref" href="#f3:356"><span class="id" title="binder">f3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f3:356"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:357" class="idref" href="#g:357"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#g:357"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="x:358" class="idref" href="#x:358"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:357"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:354"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:358"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:355"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:358"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f3:356"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:358"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose2_1IsPR" class="idref" href="#compose2_1IsPR"><span class="id" title="instance">compose2_1IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:359" class="idref" href="#f:359"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f:359"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:360" class="idref" href="#g:360"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:360"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="x:361" class="idref" href="#x:361"><span class="id" title="binder">x</span></a> <a id="y:362" class="idref" href="#y:362"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:360"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:359"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:361"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:362"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose2_2IsPR" class="idref" href="#compose2_2IsPR"><span class="id" title="instance">compose2_2IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:363" class="idref" href="#f:363"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f:363"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:364" class="idref" href="#g:364"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:364"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="h:365" class="idref" href="#h:365"><span class="id" title="binder">h</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#h:365"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="x:366" class="idref" href="#x:366"><span class="id" title="binder">x</span></a> <a id="y:367" class="idref" href="#y:367"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#h:365"><span class="id" title="variable">h</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:363"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:366"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:367"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#g:364"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:366"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:367"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose2_3IsPR" class="idref" href="#compose2_3IsPR"><span class="id" title="instance">compose2_3IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:368" class="idref" href="#f1:368"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f1:368"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:369" class="idref" href="#f2:369"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f2:369"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f3:370" class="idref" href="#f3:370"><span class="id" title="binder">f3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f3:370"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:371" class="idref" href="#g:371"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#g:371"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="x:372" class="idref" href="#x:372"><span class="id" title="binder">x</span></a> <a id="y:373" class="idref" href="#y:373"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:371"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:368"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:372"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:373"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:369"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:372"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:373"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f3:370"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:372"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:373"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose2_4IsPR" class="idref" href="#compose2_4IsPR"><span class="id" title="instance">compose2_4IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:374" class="idref" href="#f1:374"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f1:374"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:375" class="idref" href="#f2:375"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f2:375"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f3:376" class="idref" href="#f3:376"><span class="id" title="binder">f3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f3:376"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f4:377" class="idref" href="#f4:377"><span class="id" title="binder">f4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f4:377"><span class="id" title="variable">f4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:378" class="idref" href="#g:378"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#g:378"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="x:379" class="idref" href="#x:379"><span class="id" title="binder">x</span></a> <a id="y:380" class="idref" href="#y:380"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:378"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:374"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:380"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:375"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:380"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f3:376"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:380"><span class="id" title="variable">y</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f4:377"><span class="id" title="variable">f4</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:379"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:380"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose3_1IsPR" class="idref" href="#compose3_1IsPR"><span class="id" title="instance">compose3_1IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:381" class="idref" href="#f:381"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f:381"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:382" class="idref" href="#g:382"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:382"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="x:383" class="idref" href="#x:383"><span class="id" title="binder">x</span></a> <a id="y:384" class="idref" href="#y:384"><span class="id" title="binder">y</span></a> <a id="z:385" class="idref" href="#z:385"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:382"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:381"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:383"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:384"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:385"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose3_2IsPR" class="idref" href="#compose3_2IsPR"><span class="id" title="instance">compose3_2IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:386" class="idref" href="#f1:386"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f1:386"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:387" class="idref" href="#f2:387"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f2:387"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:388" class="idref" href="#g:388"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:388"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="x:389" class="idref" href="#x:389"><span class="id" title="binder">x</span></a> <a id="y:390" class="idref" href="#y:390"><span class="id" title="binder">y</span></a> <a id="z:391" class="idref" href="#z:391"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:388"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:386"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:389"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:390"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:391"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:387"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:389"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:390"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:391"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose3_3IsPR" class="idref" href="#compose3_3IsPR"><span class="id" title="instance">compose3_3IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:392" class="idref" href="#f1:392"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f1:392"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:393" class="idref" href="#f2:393"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f2:393"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f3:394" class="idref" href="#f3:394"><span class="id" title="binder">f3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f3:394"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:395" class="idref" href="#g:395"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#g:395"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="x:396" class="idref" href="#x:396"><span class="id" title="binder">x</span></a> <a id="y:397" class="idref" href="#y:397"><span class="id" title="binder">y</span></a> <a id="z:398" class="idref" href="#z:398"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:395"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:392"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:396"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:397"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:398"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:393"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:396"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:397"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:398"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f3:394"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:396"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:397"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:398"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose4_2IsPR" class="idref" href="#compose4_2IsPR"><span class="id" title="instance">compose4_2IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:399" class="idref" href="#f1:399"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f1:399"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:400" class="idref" href="#f2:400"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f2:400"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:401" class="idref" href="#g:401"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:401"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="w:402" class="idref" href="#w:402"><span class="id" title="binder">w</span></a> <a id="x:403" class="idref" href="#x:403"><span class="id" title="binder">x</span></a> <a id="y:404" class="idref" href="#y:404"><span class="id" title="binder">y</span></a> <a id="z:405" class="idref" href="#z:405"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:401"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:399"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#w:402"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:403"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:404"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:405"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:400"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#w:402"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:403"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:404"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:405"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose4_3IsPR" class="idref" href="#compose4_3IsPR"><span class="id" title="instance">compose4_3IsPR</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f1:406" class="idref" href="#f1:406"><span class="id" title="binder">f1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f1:406"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f2:407" class="idref" href="#f2:407"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f2:407"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f3:408" class="idref" href="#f3:408"><span class="id" title="binder">f3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f3:408"><span class="id" title="variable">f3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="g:409" class="idref" href="#g:409"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#g:409"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="w:410" class="idref" href="#w:410"><span class="id" title="binder">w</span></a> <a id="x:411" class="idref" href="#x:411"><span class="id" title="binder">x</span></a> <a id="y:412" class="idref" href="#y:412"><span class="id" title="binder">y</span></a> <a id="z:413" class="idref" href="#z:413"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:409"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f1:406"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#w:410"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:411"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:412"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:413"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f2:407"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#w:410"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:411"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:412"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:413"><span class="id" title="variable">z</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#f3:408"><span class="id" title="variable">f3</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#w:410"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:411"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:412"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#z:413"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="swapIsPR" class="idref" href="#swapIsPR"><span class="id" title="instance">swapIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:414" class="idref" href="#f:414"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f:414"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="x:415" class="idref" href="#x:415"><span class="id" title="binder">x</span></a> <a id="y:416" class="idref" href="#y:416"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#f:414"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:416"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:415"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="indIsPR" class="idref" href="#indIsPR"><span class="id" title="instance">indIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:425" class="idref" href="#f:425"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#f:425"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:426" class="idref" href="#g:426"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:427" class="idref" href="#a:427"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#g:426"><span class="id" title="variable">g</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:428" class="idref" href="#x:428"><span class="id" title="binder">x</span></a> <a id="y:429" class="idref" href="#y:429"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#f:425"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:428"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:429"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#a:427"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ind1ParamIsPR" class="idref" href="#ind1ParamIsPR"><span class="id" title="instance">ind1ParamIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:430" class="idref" href="#f:430"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#f:430"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:431" class="idref" href="#g:431"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:431"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:432" class="idref" href="#a:432"><span class="id" title="binder">a</span></a> <a id="b:433" class="idref" href="#b:433"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <a id="n:436" class="idref" href="#n:436"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#g:431"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:433"><span class="id" title="variable">b</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:434" class="idref" href="#x:434"><span class="id" title="binder">x</span></a> <a id="y:435" class="idref" href="#y:435"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#f:430"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:434"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:435"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:433"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#a:432"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ind2ParamIsPR" class="idref" href="#ind2ParamIsPR"><span class="id" title="instance">ind2ParamIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:437" class="idref" href="#f:437"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 4 <a class="idref" href="hydras.Ackermann.primRec.html#f:437"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:438" class="idref" href="#g:438"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#g:438"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:439" class="idref" href="#a:439"><span class="id" title="binder">a</span></a> <a id="b:440" class="idref" href="#b:440"><span class="id" title="binder">b</span></a> <a id="c:441" class="idref" href="#c:441"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <a id="n:444" class="idref" href="#n:444"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#g:438"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:440"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:441"><span class="id" title="variable">c</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:442" class="idref" href="#x:442"><span class="id" title="binder">x</span></a> <a id="y:443" class="idref" href="#y:443"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#f:437"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:442"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:443"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:440"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:441"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#a:439"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="plusIndIsPR" class="idref" href="#plusIndIsPR"><span class="id" title="instance">plusIndIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="n:445" class="idref" href="#n:445"><span class="id" title="binder">n</span></a> <a id="fn:446" class="idref" href="#fn:446"><span class="id" title="binder">fn</span></a> <a id="b:447" class="idref" href="#b:447"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#fn:446"><span class="id" title="variable">fn</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="plusIsPR" class="idref" href="#plusIsPR"><span class="id" title="instance">plusIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#plus"><span class="id" title="abbreviation">plus</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="multIndIsPR" class="idref" href="#multIndIsPR"><span class="id" title="instance">multIndIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="n:458" class="idref" href="#n:458"><span class="id" title="binder">n</span></a> <a id="fn:459" class="idref" href="#fn:459"><span class="id" title="binder">fn</span></a> <a id="b:460" class="idref" href="#b:460"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#fn:459"><span class="id" title="variable">fn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:460"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="multIsPR" class="idref" href="#multIsPR"><span class="id" title="instance">multIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#mult"><span class="id" title="abbreviation">mult</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="predIsPR" class="idref" href="#predIsPR"><span class="id" title="instance">predIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="minusIndIsPR" class="idref" href="#minusIndIsPR"><span class="id" title="instance">minusIndIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 (<span class="id" title="keyword">fun</span> <a id="n:479" class="idref" href="#n:479"><span class="id" title="binder">n</span></a> <a id="fn:480" class="idref" href="#fn:480"><span class="id" title="binder">fn</span></a> <a id="b:481" class="idref" href="#b:481"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#fn:480"><span class="id" title="variable">fn</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="minusIsPR" class="idref" href="#minusIsPR"><span class="id" title="instance">minusIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#minus"><span class="id" title="abbreviation">minus</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="notZero" class="idref" href="#notZero"><span class="id" title="definition">notZero</span></a> (<a id="a:502" class="idref" href="#a:502"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <a id="n:505" class="idref" href="#n:505"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 0 (<span class="id" title="keyword">fun</span> <a id="x:503" class="idref" href="#x:503"><span class="id" title="binder">x</span></a> <a id="y:504" class="idref" href="#y:504"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ 1) <a class="idref" href="hydras.Ackermann.primRec.html#a:502"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="notZeroIsPR" class="idref" href="#notZeroIsPR"><span class="id" title="instance">notZeroIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#notZero"><span class="id" title="definition">notZero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ltBool" class="idref" href="#ltBool"><span class="id" title="definition">ltBool</span></a> (<a id="a:506" class="idref" href="#a:506"><span class="id" title="binder">a</span></a> <a id="b:507" class="idref" href="#b:507"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:507"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:506"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltBoolTrue" class="idref" href="#ltBoolTrue"><span class="id" title="lemma">ltBoolTrue</span></a> : <span class="id" title="keyword">∀</span> <a id="a:508" class="idref" href="#a:508"><span class="id" title="binder">a</span></a> <a id="b:509" class="idref" href="#b:509"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:508"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:509"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:508"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:509"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ltBoolFalse" class="idref" href="#ltBoolFalse"><span class="id" title="lemma">ltBoolFalse</span></a> : <span class="id" title="keyword">∀</span> <a id="a:510" class="idref" href="#a:510"><span class="id" title="binder">a</span></a> <a id="b:511" class="idref" href="#b:511"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:510"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:511"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:510"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:511"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ltIsPR" class="idref" href="#ltIsPR"><span class="id" title="instance">ltIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="maxIsPR" class="idref" href="#maxIsPR"><span class="id" title="instance">maxIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="gtIsPR" class="idref" href="#gtIsPR"><span class="id" title="instance">gtIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:532" class="idref" href="#a:532"><span class="id" title="binder">a</span></a> <a id="b:533" class="idref" href="#b:533"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:533"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:532"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="replaceCompose2" class="idref" href="#replaceCompose2"><span class="id" title="lemma">replaceCompose2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:538" class="idref" href="#n:538"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:539" class="idref" href="#a:539"><span class="id" title="binder">a</span></a> <a id="b:540" class="idref" href="#b:540"><span class="id" title="binder">b</span></a> <a id="a':541" class="idref" href="#a':541"><span class="id" title="binder">a'</span></a> <a id="b':542" class="idref" href="#b':542"><span class="id" title="binder">b'</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a>) (<a id="c:543" class="idref" href="#c:543"><span class="id" title="binder">c</span></a> <a id="c':544" class="idref" href="#c':544"><span class="id" title="binder">c'</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:539"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a':541"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:540"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b':542"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#c:543"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c':544"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#a:539"><span class="id" title="variable">a</span></a> <span class="id" title="var">_</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#b:540"><span class="id" title="variable">b</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a>  (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a>)))) <a class="idref" href="hydras.Ackermann.primRec.html#c:543"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#a':541"><span class="id" title="variable">a'</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#b':542"><span class="id" title="variable">b'</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">Vector.nil</span></a>  (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:538"><span class="id" title="variable">n</span></a>)))) <a class="idref" href="hydras.Ackermann.primRec.html#c':544"><span class="id" title="variable">c'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="orRel" class="idref" href="#orRel"><span class="id" title="definition">orRel</span></a> (<a id="n:545" class="idref" href="#n:545"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:546" class="idref" href="#R:546"><span class="id" title="binder">R</span></a> <a id="S:547" class="idref" href="#S:547"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:545"><span class="id" title="variable">n</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:545"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="orRelPR" class="idref" href="#orRelPR"><span class="id" title="lemma">orRelPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:548" class="idref" href="#n:548"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:549" class="idref" href="#R:549"><span class="id" title="binder">R</span></a> <a id="R':550" class="idref" href="#R':550"><span class="id" title="binder">R'</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:548"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:548"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:549"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:548"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R':550"><span class="id" title="variable">R'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:548"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#orRel"><span class="id" title="definition">orRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:548"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:549"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R':550"><span class="id" title="variable">R'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="andRel" class="idref" href="#andRel"><span class="id" title="definition">andRel</span></a> (<a id="n:559" class="idref" href="#n:559"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:560" class="idref" href="#R:560"><span class="id" title="binder">R</span></a> <a id="S:561" class="idref" href="#S:561"><span class="id" title="binder">S</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:559"><span class="id" title="variable">n</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:559"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="andRelPR" class="idref" href="#andRelPR"><span class="id" title="lemma">andRelPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:562" class="idref" href="#n:562"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:563" class="idref" href="#R:563"><span class="id" title="binder">R</span></a> <a id="R':564" class="idref" href="#R':564"><span class="id" title="binder">R'</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:562"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:562"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:563"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:562"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R':564"><span class="id" title="variable">R'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:562"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#andRel"><span class="id" title="definition">andRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:562"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:563"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R':564"><span class="id" title="variable">R'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="notRel" class="idref" href="#notRel"><span class="id" title="definition">notRel</span></a> (<a id="n:565" class="idref" href="#n:565"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:566" class="idref" href="#R:566"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:565"><span class="id" title="variable">n</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:565"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="notRelPR" class="idref" href="#notRelPR"><span class="id" title="lemma">notRelPR</span></a>  (<a id="n:567" class="idref" href="#n:567"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="R:568" class="idref" href="#R:568"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:567"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:567"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:568"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:567"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#notRel"><span class="id" title="definition">notRel</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:567"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#R:568"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="bodd" class="idref" href="#bodd"><span class="id" title="definition">bodd</span></a> (<a id="n:577" class="idref" href="#n:577"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#n:577"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#bodd:578"><span class="id" title="definition">bodd</span></a> <span class="id" title="var">n'</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="boddIsPR" class="idref" href="#boddIsPR"><span class="id" title="instance">boddIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#bodd"><span class="id" title="definition">bodd</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_eqb_false" class="idref" href="#nat_eqb_false"><span class="id" title="lemma">nat_eqb_false</span></a> (<a id="a:586" class="idref" href="#a:586"><span class="id" title="binder">a</span></a> <a id="b:587" class="idref" href="#b:587"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.primRec.html#a:586"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:587"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:586"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:587"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="neqIsPR" class="idref" href="#neqIsPR"><span class="id" title="instance">neqIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="a:588" class="idref" href="#a:588"><span class="id" title="binder">a</span></a> <a id="b:589" class="idref" href="#b:589"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:588"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:589"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="eqIsPR" class="idref" href="#eqIsPR"><span class="id" title="instance">eqIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="leBool" class="idref" href="#leBool"><span class="id" title="definition">leBool</span></a> (<a id="a:598" class="idref" href="#a:598"><span class="id" title="binder">a</span></a> <a id="b:599" class="idref" href="#b:599"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#a:598"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:599"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="leIsPR" class="idref" href="#leIsPR"><span class="id" title="instance">leIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#leBool"><span class="id" title="definition">leBool</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Ignore_Params" class="idref" href="#Ignore_Params"><span class="id" title="section">Ignore_Params</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="ignoreParams" class="idref" href="#ignoreParams"><span class="id" title="definition">ignoreParams</span></a> (<a id="n:600" class="idref" href="#n:600"><span class="id" title="binder">n</span></a> <a id="m:601" class="idref" href="#m:601"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:602" class="idref" href="#f:602"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:600"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#m:601"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:600"><span class="id" title="variable">n</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#m:601"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#m:604"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:600"><span class="id" title="variable">n</span></a>)) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#f:602"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">x</span> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#ignoreParams:603"><span class="id" title="definition">ignoreParams</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:600"><span class="id" title="variable">n</span></a> <span class="id" title="var">x</span> <a class="idref" href="hydras.Ackermann.primRec.html#f:602"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="projectionListPR" class="idref" href="#projectionListPR"><span class="id" title="definition">projectionListPR</span></a> (<a id="n:605" class="idref" href="#n:605"><span class="id" title="binder">n</span></a> <a id="m:606" class="idref" href="#m:606"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:607" class="idref" href="#p:607"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:606"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:605"><span class="id" title="variable">n</span></a>) : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:605"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:606"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="projectionList" class="idref" href="#projectionList"><span class="id" title="definition">projectionList</span></a> (<a id="n:608" class="idref" href="#n:608"><span class="id" title="binder">n</span></a> <a id="m:609" class="idref" href="#m:609"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:610" class="idref" href="#p:610"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:609"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:608"><span class="id" title="variable">n</span></a>) : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:608"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:609"><span class="id" title="variable">m</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecs"><span class="id" title="definition">evalPrimRecs</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:608"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:609"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#projectionListPR"><span class="id" title="definition">projectionListPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:608"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:609"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p:610"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="projectionListInd" class="idref" href="#projectionListInd"><span class="id" title="lemma">projectionListInd</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:611" class="idref" href="#n:611"><span class="id" title="binder">n</span></a> <a id="m:612" class="idref" href="#m:612"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p1:613" class="idref" href="#p1:613"><span class="id" title="binder">p1</span></a> <a id="p2:614" class="idref" href="#p2:614"><span class="id" title="binder">p2</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:612"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:611"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#projectionList"><span class="id" title="definition">projectionList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:611"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:612"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p1:613"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#projectionList"><span class="id" title="definition">projectionList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:611"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:612"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p2:614"><span class="id" title="variable">p2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="projectionListApplyParam" class="idref" href="#projectionListApplyParam"><span class="id" title="lemma">projectionListApplyParam</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m:615" class="idref" href="#m:615"><span class="id" title="binder">m</span></a> <a id="n:616" class="idref" href="#n:616"><span class="id" title="binder">n</span></a> <a id="c:617" class="idref" href="#c:617"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p1:618" class="idref" href="#p1:618"><span class="id" title="binder">p1</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:615"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:616"><span class="id" title="variable">n</span></a>) (<a id="p2:619" class="idref" href="#p2:619"><span class="id" title="binder">p2</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#m:615"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:616"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#extEqualVector"><span class="id" title="definition">extEqualVector</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#projectionList"><span class="id" title="definition">projectionList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:616"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:615"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p1:618"><span class="id" title="variable">p1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalOneParamList"><span class="id" title="definition">evalOneParamList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:616"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:615"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:617"><span class="id" title="variable">c</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#projectionList"><span class="id" title="definition">projectionList</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:616"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:615"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p2:619"><span class="id" title="variable">p2</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="projectionListId" class="idref" href="#projectionListId"><span class="id" title="lemma">projectionListId</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:620" class="idref" href="#n:620"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:621" class="idref" href="#f:621"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a>) (<a id="p:622" class="idref" href="#p:622"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:621"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#projectionList"><span class="id" title="definition">projectionList</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:620"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#p:622"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#f:621"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="ignoreParamsIsPR" class="idref" href="#ignoreParamsIsPR"><span class="id" title="instance">ignoreParamsIsPR</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:631" class="idref" href="#n:631"><span class="id" title="binder">n</span></a> <a id="m:632" class="idref" href="#m:632"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:633" class="idref" href="#f:633"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:631"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#f:633"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.primRec.html#ignoreParams"><span class="id" title="definition">ignoreParams</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:631"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:632"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:633"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.primRec.html#Ignore_Params"><span class="id" title="section">Ignore_Params</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduce1stCompose" class="idref" href="#reduce1stCompose"><span class="id" title="lemma">reduce1stCompose</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="c:644" class="idref" href="#c:644"><span class="id" title="binder">c</span></a> <a id="n:645" class="idref" href="#n:645"><span class="id" title="binder">n</span></a> <a id="m:646" class="idref" href="#m:646"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="v:647" class="idref" href="#v:647"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:646"><span class="id" title="variable">m</span></a>) (<a id="g:648" class="idref" href="#g:648"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:646"><span class="id" title="variable">m</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:644"><span class="id" title="variable">c</span></a>) <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#v:647"><span class="id" title="variable">v</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#g:648"><span class="id" title="variable">g</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:645"><span class="id" title="variable">n</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#v:647"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#g:648"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:644"><span class="id" title="variable">c</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduce2ndCompose" class="idref" href="#reduce2ndCompose"><span class="id" title="lemma">reduce2ndCompose</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="c:649" class="idref" href="#c:649"><span class="id" title="binder">c</span></a> <a id="n:650" class="idref" href="#n:650"><span class="id" title="binder">n</span></a> <a id="m:651" class="idref" href="#m:651"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="v:652" class="idref" href="#v:652"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#m:651"><span class="id" title="variable">m</span></a>) (<a id="n0:653" class="idref" href="#n0:653"><span class="id" title="binder">n0</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:654" class="idref" href="#g:654"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#m:651"><span class="id" title="variable">m</span></a>))),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#n0:653"><span class="id" title="variable">n0</span></a> <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:649"><span class="id" title="variable">c</span></a>) <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#v:652"><span class="id" title="variable">v</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#g:654"><span class="id" title="variable">g</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:650"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#n0:653"><span class="id" title="variable">n0</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Ackermann.primRec.html#v:652"><span class="id" title="variable">v</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:655" class="idref" href="#x:655"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:654"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:655"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:649"><span class="id" title="variable">c</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="evalPrimRecParam" class="idref" href="#evalPrimRecParam"><span class="id" title="lemma">evalPrimRecParam</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:656" class="idref" href="#n:656"><span class="id" title="binder">n</span></a> <a id="c:657" class="idref" href="#c:657"><span class="id" title="binder">c</span></a> <a id="d:658" class="idref" href="#d:658"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:659" class="idref" href="#g:659"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:656"><span class="id" title="variable">n</span></a>)) (<a id="h:660" class="idref" href="#h:660"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:656"><span class="id" title="variable">n</span></a>)))),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:656"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#g:659"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#d:658"><span class="id" title="variable">d</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:661" class="idref" href="#x:661"><span class="id" title="binder">x</span></a> <a id="y:662" class="idref" href="#y:662"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#h:660"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:661"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#y:662"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#d:658"><span class="id" title="variable">d</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#c:657"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:656"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#g:659"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#h:660"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#c:657"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#d:658"><span class="id" title="variable">d</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose2IsPR" class="idref" href="#compose2IsPR"><span class="id" title="instance">compose2IsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:663" class="idref" href="#n:663"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:664" class="idref" href="#f:664"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a>) (<a id="p:665" class="idref" href="#p:665"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:664"><span class="id" title="variable">f</span></a>) (<a id="g:666" class="idref" href="#g:666"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="q:667" class="idref" href="#q:667"><span class="id" title="binder">q</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#g:666"><span class="id" title="variable">g</span></a>), <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:663"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:664"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:666"><span class="id" title="variable">g</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="compose1_NIsPR" class="idref" href="#compose1_NIsPR"><span class="id" title="instance">compose1_NIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:670" class="idref" href="#n:670"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:671" class="idref" href="#g:671"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:670"><span class="id" title="variable">n</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:670"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#g:671"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:672" class="idref" href="#f:672"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 1, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#f:672"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:670"><span class="id" title="variable">n</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:673" class="idref" href="#x:673"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:671"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#f:672"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:673"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="switchPR" class="idref" href="#switchPR"><span class="id" title="definition">switchPR</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 3 :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="n:674" class="idref" href="#n:674"><span class="id" title="binder">n</span></a> <a id="x:675" class="idref" href="#x:675"><span class="id" title="binder">x</span></a> <a id="y:676" class="idref" href="#y:676"><span class="id" title="binder">y</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#n:674"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#y:676"><span class="id" title="variable">y</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#x:675"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="switchIsPR" class="idref" href="#switchIsPR"><span class="id" title="instance">switchIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 3 <a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="boundedSearchHelp" class="idref" href="#boundedSearchHelp"><span class="id" title="definition">boundedSearchHelp</span></a> (<a id="P:700" class="idref" href="#P:700"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 1) (<a id="b:701" class="idref" href="#b:701"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">b</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#b:701"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">b'</span> ⇒ <span class="id" title="keyword">let</span> <a id="q:704" class="idref" href="#q:704"><span class="id" title="binder">q</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearchHelp:702"><span class="id" title="definition">boundedSearchHelp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:700"><span class="id" title="variable">P</span></a> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#q:704"><span class="id" title="variable">q</span></a>  <span class="id" title="var">b'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.primRec.html#P:700"><span class="id" title="variable">P</span></a> <span class="id" title="var">b'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> ⇒ <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#q:704"><span class="id" title="variable">q</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If there exists some <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" title="var">b</span></span> such that <span class="inlinecode">(<span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="var">x</span>)</span>, returns <span class="inlinecode"><span class="id" title="var">x</span></span>
+    Otherwise, returns <span class="inlinecode"><span class="id" title="var">b</span></span>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="boundedSearch" class="idref" href="#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> (<a id="P:705" class="idref" href="#P:705"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2) (<a id="b:706" class="idref" href="#b:706"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearchHelp"><span class="id" title="definition">boundedSearchHelp</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#P:705"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:706"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#b:706"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Examples" class="idref" href="#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Examples.sqrtHelp" class="idref" href="#Examples.sqrtHelp"><span class="id" title="definition">sqrtHelp</span></a> (<a id="n:707" class="idref" href="#n:707"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="x:708" class="idref" href="#x:708"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#x:708"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:708"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#n:707"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#n:707"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#2e0d8202c5a48097752d760d3a1db6a7"><span class="id" title="notation">/</span></a> 2)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Examples.exact_sqrt" class="idref" href="#Examples.exact_sqrt"><span class="id" title="definition">exact_sqrt</span></a>  (<a id="n:709" class="idref" href="#n:709"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="q:710" class="idref" href="#q:710"><span class="id" title="binder">q</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#Examples.sqrtHelp"><span class="id" title="definition">sqrtHelp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:709"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">if</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#q:710"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#q:710"><span class="id" title="variable">q</span></a>) <a class="idref" href="hydras.Ackermann.primRec.html#n:709"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#q:710"><span class="id" title="variable">q</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.primRec.html#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="boundedSearch1" class="idref" href="#boundedSearch1"><span class="id" title="lemma">boundedSearch1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:711" class="idref" href="#P:711"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2) (<a id="b:712" class="idref" href="#b:712"><span class="id" title="binder">b</span></a> <a id="x:713" class="idref" href="#x:713"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Ackermann.primRec.html#x:713"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:711"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:712"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#P:711"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:712"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:713"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundedSearch2" class="idref" href="#boundedSearch2"><span class="id" title="lemma">boundedSearch2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:714" class="idref" href="#P:714"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2) (<a id="b:715" class="idref" href="#b:715"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:714"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:715"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:715"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:714"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:715"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:714"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#b:715"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="boundSearchIsPR" class="idref" href="#boundSearchIsPR"><span class="id" title="instance">boundSearchIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:718" class="idref" href="#P:718"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#P:718"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#P:718"><span class="id" title="variable">P</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="iterate" class="idref" href="#iterate"><span class="id" title="definition">iterate</span></a> (<a id="g:815" class="idref" href="#g:815"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="keyword">fun</span> <a id="x:818" class="idref" href="#x:818"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#x:818"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="rec:816" class="idref" href="#rec:816"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="x:817" class="idref" href="#x:817"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#g:815"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#rec:816"><span class="id" title="variable">rec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#x:817"><span class="id" title="variable">x</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="iterateIsPR" class="idref" href="#iterateIsPR"><span class="id" title="instance">iterateIsPR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:819" class="idref" href="#g:819"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.primRec.html#g:819"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:820" class="idref" href="#n:820"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Ackermann.primRec.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:819"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:820"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="isPRTrans" class="idref" href="#isPRTrans"><span class="id" title="instance">isPRTrans</span></a> (<a id="n:821" class="idref" href="#n:821"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:822" class="idref" href="#f:822"><span class="id" title="binder">f</span></a> <a id="g:823" class="idref" href="#g:823"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:821"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#f:822"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:823"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:821"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:822"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:821"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:823"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="isPRextEqual" class="idref" href="#isPRextEqual"><span class="id" title="instance">isPRextEqual</span></a> (<a id="n:824" class="idref" href="#n:824"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:825" class="idref" href="#f:825"><span class="id" title="binder">f</span></a> <a id="g:826" class="idref" href="#g:826"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:824"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:824"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:825"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#f:825"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:826"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#n:824"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#g:826"><span class="id" title="variable">g</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.subAll.html b/theories/html/hydras.Ackermann.subAll.html
new file mode 100644
index 00000000..00111639
--- /dev/null
+++ b/theories/html/hydras.Ackermann.subAll.html
@@ -0,0 +1,329 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.subAll</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.subAll</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ subAll.v
+
+<div class="paragraph"> </div>
+
+    Original script by Russel O'Connor
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#"><span class="id" title="library">folLogic2</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#"><span class="id" title="library">subProp</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="SubAllVars" class="idref" href="#SubAllVars"><span class="id" title="section">SubAllVars</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="SubAllVars.L" class="idref" href="#SubAllVars.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formula" class="idref" href="#Formula"><span class="id" title="abbreviation">Formula</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Formulas" class="idref" href="#Formulas"><span class="id" title="abbreviation">Formulas</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="System" class="idref" href="#System"><span class="id" title="abbreviation">System</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Term" class="idref" href="#Term"><span class="id" title="abbreviation">Term</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Terms" class="idref" href="#Terms"><span class="id" title="abbreviation">Terms</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="SysPrf" class="idref" href="#SysPrf"><span class="id" title="abbreviation">SysPrf</span></a> := (<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="subAllTerm" class="idref" href="#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> (<a id="t:2" class="idref" href="#t:2"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.subAll.html#t:2"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <span class="id" title="var">x</span> ⇒ <span class="id" title="keyword">fun</span> <a id="m:8" class="idref" href="#m:8"><span class="id" title="binder">m</span></a> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#m:8"><span class="id" title="variable">m</span></a> <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> <span class="id" title="var">ts</span> ⇒ <span class="id" title="keyword">fun</span> <a id="m:9" class="idref" href="#m:9"><span class="id" title="binder">m</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms:6"><span class="id" title="definition">subAllTerms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:9"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<br/>
+&nbsp;<span class="id" title="keyword">with</span> <a id="subAllTerms" class="idref" href="#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> (<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:4" class="idref" href="#ts:4"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:3"><span class="id" title="variable">n</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">ts</span>} :<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:3"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#ts:4"><span class="id" title="variable">ts</span></a> <span class="id" title="tactic">in</span> (<a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">return</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:10"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">t</span> <span class="id" title="var">ss</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="m:12" class="idref" href="#m:12"><span class="id" title="binder">m</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm:5"><span class="id" title="definition">subAllTerm</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:12"><span class="id" title="variable">m</span></a>) (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms:6"><span class="id" title="definition">subAllTerms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ss</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:12"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllTerm_ext" class="idref" href="#subAllTerm_ext"><span class="id" title="lemma">subAllTerm_ext</span></a> (<a id="t:13" class="idref" href="#t:13"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m1:14" class="idref" href="#m1:14"><span class="id" title="binder">m1</span></a> <a id="m2:15" class="idref" href="#m2:15"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:16" class="idref" href="#m:16"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:16"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:13"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:14"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:16"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:15"><span class="id" title="variable">m2</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:16"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:13"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:14"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:13"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:15"><span class="id" title="variable">m2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllTerms_ext" class="idref" href="#subAllTerms_ext"><span class="id" title="lemma">subAllTerms_ext</span></a> (<a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:28" class="idref" href="#ts:28"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:27"><span class="id" title="variable">n</span></a>) (<a id="m1:29" class="idref" href="#m1:29"><span class="id" title="binder">m1</span></a> <a id="m2:30" class="idref" href="#m2:30"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:31" class="idref" href="#m:31"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:31"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:28"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:29"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:31"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:30"><span class="id" title="variable">m2</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:31"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:28"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:29"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:28"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:30"><span class="id" title="variable">m2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="freeVarMap" class="idref" href="#freeVarMap"><span class="id" title="definition">freeVarMap</span></a> (<a id="l:32" class="idref" href="#l:32"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.subAll.html#l:32"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <span class="id" title="keyword">fun</span> <a id="m:35" class="idref" href="#m:35"><span class="id" title="binder">m</span></a> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:35"><span class="id" title="variable">m</span></a> <span class="id" title="var">a</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#freeVarMap:33"><span class="id" title="definition">freeVarMap</span></a> <span class="id" title="var">l'</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:35"><span class="id" title="variable">m</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarMap_ext" class="idref" href="#freeVarMap_ext"><span class="id" title="lemma">freeVarMap_ext</span></a> (<a id="l:36" class="idref" href="#l:36"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f1:37" class="idref" href="#f1:37"><span class="id" title="binder">f1</span></a> <a id="f2:38" class="idref" href="#f2:38"><span class="id" title="binder">f2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:39" class="idref" href="#m:39"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:39"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#l:36"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f1:37"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:39"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f2:38"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:39"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#freeVarMap"><span class="id" title="definition">freeVarMap</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#l:36"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f1:37"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#freeVarMap"><span class="id" title="definition">freeVarMap</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#l:36"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f2:38"><span class="id" title="variable">f2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarMap1" class="idref" href="#freeVarMap1"><span class="id" title="lemma">freeVarMap1</span></a> (<a id="l:40" class="idref" href="#l:40"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="m:41" class="idref" href="#m:41"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:42" class="idref" href="#v:42"><span class="id" title="binder">v</span></a> <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:42"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:41"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:43"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:43"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#l:40"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:42"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#freeVarMap"><span class="id" title="definition">freeVarMap</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#l:40"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:41"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="subAllFormula" class="idref" href="#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subAll.html#Formula"><span class="id" title="abbreviation">Formula</span></a>) (<a id="m:45" class="idref" href="#m:45"><span class="id" title="binder">m</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#Term"><span class="id" title="abbreviation">Term</span></a>)) {<span class="id" title="keyword">struct</span> <span class="id" title="var">f</span>} : <a class="idref" href="hydras.Ackermann.subAll.html#Formula"><span class="id" title="abbreviation">Formula</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| (<span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <span class="id" title="var">s</span>)%<span class="id" title="var">fol</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>) (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <span class="id" title="var">s</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> <span class="id" title="var">ts</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">r</span> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <span class="id" title="var">_</span> <span class="id" title="var">ts</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;| (<span class="id" title="var">f</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">g</span>)%<span class="id" title="var">fol</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula:46"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula:46"><span class="id" title="definition">subAllFormula</span></a> <span class="id" title="var">g</span> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">f</span>)%<span class="id" title="var">fol</span> ⇒  (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula:46"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <span class="id" title="var">f</span>)%<span class="id" title="var">fol</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="nv:48" class="idref" href="#nv:48"><span class="id" title="binder">nv</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#newVar"><span class="id" title="definition">newVar</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#freeVarMap"><span class="id" title="definition">freeVarMap</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a>)%<span class="id" title="var">fol</span>) <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#nv:48"><span class="id" title="variable">nv</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula:46"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:44"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="v:49" class="idref" href="#v:49"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:49"><span class="id" title="variable">v</span></a> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#nv:48"><span class="id" title="variable">nv</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#m:45"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:49"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllFormula_ext" class="idref" href="#subAllFormula_ext"><span class="id" title="lemma">subAllFormula_ext</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:50" class="idref" href="#f:50"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m1:51" class="idref" href="#m1:51"><span class="id" title="binder">m1</span></a> <a id="m2:52" class="idref" href="#m2:52"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:53" class="idref" href="#m:53"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:53"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:50"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:51"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:53"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:52"><span class="id" title="variable">m2</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:53"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:51"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m2:52"><span class="id" title="variable">m2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllTerm1" class="idref" href="#freeVarSubAllTerm1"><span class="id" title="lemma">freeVarSubAllTerm1</span></a> (<a id="t:58" class="idref" href="#t:58"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:59" class="idref" href="#m:59"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:60" class="idref" href="#v:60"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:60"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:58"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:59"><span class="id" title="variable">m</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:61" class="idref" href="#n:61"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:61"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:58"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:60"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:59"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:61"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllTerms1" class="idref" href="#freeVarSubAllTerms1"><span class="id" title="lemma">freeVarSubAllTerms1</span></a> (<a id="n:68" class="idref" href="#n:68"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:69" class="idref" href="#ts:69"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:68"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;(<a id="m:70" class="idref" href="#m:70"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:71" class="idref" href="#v:71"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:71"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:68"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:69"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:70"><span class="id" title="variable">m</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#a:72"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:69"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:71"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:70"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#a:72"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllTerm2" class="idref" href="#freeVarSubAllTerm2"><span class="id" title="lemma">freeVarSubAllTerm2</span></a> (<a id="t:73" class="idref" href="#t:73"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:74" class="idref" href="#m:74"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:75" class="idref" href="#v:75"><span class="id" title="binder">v</span></a> <a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:73"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:75"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a>  (<a class="idref" href="hydras.Ackermann.subAll.html#m:74"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:76"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:75"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:73"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:74"><span class="id" title="variable">m</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllTerms2" class="idref" href="#freeVarSubAllTerms2"><span class="id" title="lemma">freeVarSubAllTerms2</span></a> (<a id="a:81" class="idref" href="#a:81"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:82" class="idref" href="#ts:82"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#a:81"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;(<a id="m:83" class="idref" href="#m:83"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:84" class="idref" href="#v:84"><span class="id" title="binder">v</span></a> <a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:85"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:82"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:84"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:83"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:85"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:84"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#a:81"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:82"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:83"><span class="id" title="variable">m</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllFormula1" class="idref" href="#freeVarSubAllFormula1"><span class="id" title="lemma">freeVarSubAllFormula1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:86" class="idref" href="#f:86"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:87" class="idref" href="#m:87"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:88" class="idref" href="#v:88"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:88"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:86"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:87"><span class="id" title="variable">m</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:89" class="idref" href="#n:89"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:89"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:86"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:88"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:87"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:89"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubAllFormula2" class="idref" href="#freeVarSubAllFormula2"><span class="id" title="lemma">freeVarSubAllFormula2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:92" class="idref" href="#f:92"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:93" class="idref" href="#m:93"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:94" class="idref" href="#v:94"><span class="id" title="binder">v</span></a> <a id="n:95" class="idref" href="#n:95"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:95"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:92"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:94"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:93"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:95"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:94"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:92"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:93"><span class="id" title="variable">m</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubAllTerm" class="idref" href="#subSubAllTerm"><span class="id" title="lemma">subSubAllTerm</span></a> (<a id="t:96" class="idref" href="#t:96"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:97" class="idref" href="#m:97"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:98" class="idref" href="#v:98"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:99" class="idref" href="#s:99"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:96"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:97"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:98"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:99"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:96"><span class="id" title="variable">t</span></a> (<span class="id" title="keyword">fun</span> <a id="n:100" class="idref" href="#n:100"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:97"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:100"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:98"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:99"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubAllTerms" class="idref" href="#subSubAllTerms"><span class="id" title="lemma">subSubAllTerms</span></a> (<a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:108" class="idref" href="#ts:108"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:107"><span class="id" title="variable">n</span></a>) (<a id="m:109" class="idref" href="#m:109"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) <br/>
+&nbsp;&nbsp;(<a id="v:110" class="idref" href="#v:110"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:111" class="idref" href="#s:111"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:107"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:108"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:109"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:110"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:107"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:108"><span class="id" title="variable">ts</span></a> (<span class="id" title="keyword">fun</span> <a id="n:112" class="idref" href="#n:112"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:109"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:112"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:110"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:111"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subSubAllFormula" class="idref" href="#subSubAllFormula"><span class="id" title="lemma">subSubAllFormula</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="T:113" class="idref" href="#T:113"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="f:114" class="idref" href="#f:114"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:115" class="idref" href="#m:115"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="v:116" class="idref" href="#v:116"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:117" class="idref" href="#s:117"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:113"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:114"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:115"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:116"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:117"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:114"><span class="id" title="variable">f</span></a> (<span class="id" title="keyword">fun</span> <a id="n:118" class="idref" href="#n:118"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:115"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:118"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#v:116"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:117"><span class="id" title="variable">s</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllTermId" class="idref" href="#subAllTermId"><span class="id" title="lemma">subAllTermId</span></a> (<a id="t:177" class="idref" href="#t:177"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:177"><span class="id" title="variable">t</span></a> (<span class="id" title="keyword">fun</span> <a id="x:178" class="idref" href="#x:178"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:178"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:177"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllTermsId" class="idref" href="#subAllTermsId"><span class="id" title="lemma">subAllTermsId</span></a>  (<a id="n:185" class="idref" href="#n:185"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:186" class="idref" href="#ts:186"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:185"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:186"><span class="id" title="variable">ts</span></a> (<span class="id" title="keyword">fun</span> <a id="x:187" class="idref" href="#x:187"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:187"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:186"><span class="id" title="variable">ts</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllFormulaId" class="idref" href="#subAllFormulaId"><span class="id" title="lemma">subAllFormulaId</span></a> (<a id="T:188" class="idref" href="#T:188"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="f:189" class="idref" href="#f:189"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:188"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:189"><span class="id" title="variable">f</span></a> (<span class="id" title="keyword">fun</span> <a id="x:190" class="idref" href="#x:190"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:190"><span class="id" title="variable">x</span></a>)) <a class="idref" href="hydras.Ackermann.subAll.html#f:189"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllSubAllTerm" class="idref" href="#subAllSubAllTerm"><span class="id" title="lemma">subAllSubAllTerm</span></a> (<a id="t:207" class="idref" href="#t:207"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) :<br/>
+&nbsp;<span class="id" title="keyword">∀</span>  (<a id="m1:208" class="idref" href="#m1:208"><span class="id" title="binder">m1</span></a> <a id="m2:209" class="idref" href="#m2:209"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:207"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:208"><span class="id" title="variable">m1</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:209"><span class="id" title="variable">m2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#t:207"><span class="id" title="variable">t</span></a> (<span class="id" title="keyword">fun</span> <a id="n:210" class="idref" href="#n:210"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m1:208"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:210"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:209"><span class="id" title="variable">m2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllSubAllTerms" class="idref" href="#subAllSubAllTerms"><span class="id" title="lemma">subAllSubAllTerms</span></a> (<a id="n:221" class="idref" href="#n:221"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:222" class="idref" href="#ts:222"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:221"><span class="id" title="variable">n</span></a>) (<a id="m1:223" class="idref" href="#m1:223"><span class="id" title="binder">m1</span></a> <a id="m2:224" class="idref" href="#m2:224"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:221"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:221"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:222"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:223"><span class="id" title="variable">m1</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:224"><span class="id" title="variable">m2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#subAllTerms"><span class="id" title="definition">subAllTerms</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:221"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#ts:222"><span class="id" title="variable">ts</span></a> (<span class="id" title="keyword">fun</span> <a id="n:225" class="idref" href="#n:225"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m1:223"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:225"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:224"><span class="id" title="variable">m2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllSubAllFormula" class="idref" href="#subAllSubAllFormula"><span class="id" title="lemma">subAllSubAllFormula</span></a> (<a id="T:226" class="idref" href="#T:226"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="f:227" class="idref" href="#f:227"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<span class="id" title="keyword">∀</span>  (<a id="m1:228" class="idref" href="#m1:228"><span class="id" title="binder">m1</span></a> <a id="m2:229" class="idref" href="#m2:229"><span class="id" title="binder">m2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:226"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:227"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m1:228"><span class="id" title="variable">m1</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:229"><span class="id" title="variable">m2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:227"><span class="id" title="variable">f</span></a> (<span class="id" title="keyword">fun</span> <a id="n:230" class="idref" href="#n:230"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m1:228"><span class="id" title="variable">m1</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:230"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#m2:229"><span class="id" title="variable">m2</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="SubAllVars.subAllCloseFrom" class="idref" href="#SubAllVars.subAllCloseFrom"><span class="id" title="section">subAllCloseFrom</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="closeFrom" class="idref" href="#closeFrom"><span class="id" title="definition">closeFrom</span></a> (<a id="a:249" class="idref" href="#a:249"><span class="id" title="binder">a</span></a> <a id="n:250" class="idref" href="#n:250"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="f:251" class="idref" href="#f:251"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Ackermann.subAll.html#n:250"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#f:251"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#a:249"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="hydras.Ackermann.subAll.html#closeFrom:252"><span class="id" title="definition">closeFrom</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#a:249"><span class="id" title="variable">a</span></a> <span class="id" title="var">m</span> <a class="idref" href="hydras.Ackermann.subAll.html#f:251"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Opaque</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="liftCloseFrom" class="idref" href="#liftCloseFrom"><span class="id" title="lemma">liftCloseFrom</span></a>  (<a id="n:254" class="idref" href="#n:254"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;<span class="id" title="keyword">∀</span>(<a id="f:255" class="idref" href="#f:255"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="T:256" class="idref" href="#T:256"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="m:257" class="idref" href="#m:257"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:258" class="idref" href="#v:258"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:258"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:255"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:258"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:257"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#n:254"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:257"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:256"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#closeFrom"><span class="id" title="definition">closeFrom</span></a> 0 <a class="idref" href="hydras.Ackermann.subAll.html#n:254"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:255"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:256"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#closeFrom"><span class="id" title="definition">closeFrom</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:257"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:254"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:255"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:259" class="idref" href="#x:259"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:254"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:259"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:259"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:257"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:259"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllCloseFrom1" class="idref" href="#subAllCloseFrom1"><span class="id" title="lemma">subAllCloseFrom1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:294" class="idref" href="#n:294"><span class="id" title="binder">n</span></a> <a id="m:295" class="idref" href="#m:295"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="map:296" class="idref" href="#map:296"><span class="id" title="binder">map</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="f:297" class="idref" href="#f:297"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="T:298" class="idref" href="#T:298"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:299" class="idref" href="#v:299"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subAll.html#v:299"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:294"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="w:300" class="idref" href="#w:300"><span class="id" title="binder">w</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#w:300"><span class="id" title="variable">w</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#map:296"><span class="id" title="variable">map</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:295"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:299"><span class="id" title="variable">v</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#w:300"><span class="id" title="variable">w</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:295"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:298"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#closeFrom"><span class="id" title="definition">closeFrom</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:295"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:294"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:297"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:298"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:297"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:301" class="idref" href="#x:301"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#m:295"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:301"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#m:295"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:294"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#x:301"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:301"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#map:296"><span class="id" title="variable">map</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:301"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:301"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllCloseFrom" class="idref" href="#subAllCloseFrom"><span class="id" title="lemma">subAllCloseFrom</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:314" class="idref" href="#n:314"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="m:315" class="idref" href="#m:315"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="f:316" class="idref" href="#f:316"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="T:317" class="idref" href="#T:317"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:317"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#closeFrom"><span class="id" title="definition">closeFrom</span></a> 0 <a class="idref" href="hydras.Ackermann.subAll.html#n:314"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:316"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:317"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#f:316"><span class="id" title="variable">f</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:318" class="idref" href="#x:318"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#n:314"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:318"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:318"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#m:315"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:318"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduceSubAll" class="idref" href="#reduceSubAll"><span class="id" title="lemma">reduceSubAll</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="T:339" class="idref" href="#T:339"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="map:340" class="idref" href="#map:340"><span class="id" title="binder">map</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="A:341" class="idref" href="#A:341"><span class="id" title="binder">A</span></a> <a id="B:342" class="idref" href="#B:342"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:343" class="idref" href="#v:343"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#In_freeVarSys"><span class="id" title="definition">In_freeVarSys</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:343"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:339"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:339"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:341"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#B:342"><span class="id" title="variable">B</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:339"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:341"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#map:340"><span class="id" title="variable">map</span></a>) (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#B:342"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#map:340"><span class="id" title="variable">map</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.subAllCloseFrom"><span class="id" title="section">subAllCloseFrom</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subToSubAll" class="idref" href="#subToSubAll"><span class="id" title="lemma">subToSubAll</span></a> (<a id="T:356" class="idref" href="#T:356"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="A:357" class="idref" href="#A:357"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:358" class="idref" href="#v:358"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;(<a id="s:359" class="idref" href="#s:359"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:356"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:357"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:358"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:359"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:357"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:360" class="idref" href="#x:360"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:358"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:360"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#s:359"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:360"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subAllSubFormula" class="idref" href="#subAllSubFormula"><span class="id" title="lemma">subAllSubFormula</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="T:369" class="idref" href="#T:369"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">fol.System</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="A:370" class="idref" href="#A:370"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="v:371" class="idref" href="#v:371"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:372" class="idref" href="#s:372"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>) (<a id="map:373" class="idref" href="#map:373"><span class="id" title="binder">map</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">fol.Term</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">folProof.SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#T:369"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:370"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:371"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:372"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Ackermann.subAll.html#map:373"><span class="id" title="variable">map</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#A:370"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:374" class="idref" href="#x:374"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#v:371"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:374"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#subAllTerm"><span class="id" title="definition">subAllTerm</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#s:372"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#map:373"><span class="id" title="variable">map</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.subAll.html#map:373"><span class="id" title="variable">map</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#x:374"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>))).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.subAll.html#SubAllVars"><span class="id" title="section">SubAllVars</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.subProp.html b/theories/html/hydras.Ackermann.subProp.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Ackermann.subProp.html
@@ -0,0 +1,218 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.subProp</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.subProp</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.ListExt.html#"><span class="id" title="library">ListExt</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folReplace.html#"><span class="id" title="library">folReplace</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Substitution_Properties" class="idref" href="#Substitution_Properties"><span class="id" title="section">Substitution_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Substitution_Properties.L" class="idref" href="#Substitution_Properties.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.Formula" class="idref" href="#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.Formulas" class="idref" href="#Substitution_Properties.Formulas"><span class="id" title="variable">Formulas</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formulas"><span class="id" title="definition">Formulas</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.System" class="idref" href="#Substitution_Properties.System"><span class="id" title="variable">System</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.Term" class="idref" href="#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.Terms" class="idref" href="#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Substitution_Properties.SysPrf" class="idref" href="#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> := <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerm1" class="idref" href="#freeVarSubTerm1"><span class="id" title="lemma">freeVarSubTerm1</span></a> (<a id="t:2" class="idref" href="#t:2"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="v:3" class="idref" href="#v:3"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:4" class="idref" href="#s:4"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:2"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:2"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:4"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerms1" class="idref" href="#freeVarSubTerms1"><span class="id" title="lemma">freeVarSubTerms1</span></a> (<a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:17" class="idref" href="#ts:17"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:16"><span class="id" title="variable">n</span></a>) (<a id="v:18" class="idref" href="#v:18"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:19" class="idref" href="#s:19"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:20" class="idref" href="#x:20"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:20"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:17"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v:18"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:20"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:20"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:17"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:18"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:19"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubFormula1" class="idref" href="#freeVarSubFormula1"><span class="id" title="lemma">freeVarSubFormula1</span></a> (<a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<span class="id" title="keyword">∀</span>  (<a id="v:22" class="idref" href="#v:22"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:23" class="idref" href="#s:23"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:24" class="idref" href="#x:24"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v:22"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:24"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:24"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:21"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:24"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:22"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:23"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerm2" class="idref" href="#freeVarSubTerm2"><span class="id" title="lemma">freeVarSubTerm2</span></a> (<a id="t:25" class="idref" href="#t:25"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="v:26" class="idref" href="#v:26"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:27" class="idref" href="#s:27"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:28"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:27"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:26"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:25"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:28"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:25"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:26"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:27"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerms2" class="idref" href="#freeVarSubTerms2"><span class="id" title="lemma">freeVarSubTerms2</span></a> (<a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:40" class="idref" href="#ts:40"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:39"><span class="id" title="variable">n</span></a>) (<a id="v:41" class="idref" href="#v:41"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:42" class="idref" href="#s:42"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:43" class="idref" href="#x:43"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:43"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:42"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:41"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:40"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:43"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:40"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:41"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:42"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubFormula2" class="idref" href="#freeVarSubFormula2"><span class="id" title="lemma">freeVarSubFormula2</span></a> (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="v:45" class="idref" href="#v:45"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:46" class="idref" href="#s:46"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:47" class="idref" href="#x:47"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:47"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:46"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:45"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:44"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:47"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:45"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:46"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerm3" class="idref" href="#freeVarSubTerm3"><span class="id" title="lemma">freeVarSubTerm3</span></a>  (<a id="t:48" class="idref" href="#t:48"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:49" class="idref" href="#v:49"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:50" class="idref" href="#s:50"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:51" class="idref" href="#x:51"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:51"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:48"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:49"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:50"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:51"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:49"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:48"><span class="id" title="variable">t</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:51"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:50"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerms3" class="idref" href="#freeVarSubTerms3"><span class="id" title="lemma">freeVarSubTerms3</span></a> (<a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:63" class="idref" href="#ts:63"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">fol.Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:62"><span class="id" title="variable">n</span></a>) (<a id="v:64" class="idref" href="#v:64"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="s:65" class="idref" href="#s:65"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:66" class="idref" href="#x:66"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:66"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:63"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:64"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:65"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:66"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:64"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:63"><span class="id" title="variable">ts</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:66"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:65"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubFormula3" class="idref" href="#freeVarSubFormula3"><span class="id" title="lemma">freeVarSubFormula3</span></a>  (<a id="f:67" class="idref" href="#f:67"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:68" class="idref" href="#v:68"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:69" class="idref" href="#s:69"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:70" class="idref" href="#x:70"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:70"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:67"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:68"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:69"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:70"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:68"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:67"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:70"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:69"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerm4" class="idref" href="#freeVarSubTerm4"><span class="id" title="lemma">freeVarSubTerm4</span></a> (<a id="t:71" class="idref" href="#t:71"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) :<br/>
+&nbsp;<span class="id" title="keyword">∀</span>  (<a id="v:72" class="idref" href="#v:72"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:73" class="idref" href="#s:73"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:74" class="idref" href="#x:74"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:74"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:71"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:72"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:73"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:72"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:71"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:74"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:71"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubTerms4" class="idref" href="#freeVarSubTerms4"><span class="id" title="lemma">freeVarSubTerms4</span></a> (<a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:86" class="idref" href="#ts:86"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:85"><span class="id" title="variable">n</span></a>) (<a id="v:87" class="idref" href="#v:87"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;(<a id="s:88" class="idref" href="#s:88"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:89" class="idref" href="#x:89"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:89"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#ts:86"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:87"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:88"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:87"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:86"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:89"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:86"><span class="id" title="variable">ts</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="freeVarSubFormula4" class="idref" href="#freeVarSubFormula4"><span class="id" title="lemma">freeVarSubFormula4</span></a>  (<a id="f:90" class="idref" href="#f:90"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:91" class="idref" href="#v:91"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:92" class="idref" href="#s:92"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="x:93" class="idref" href="#x:93"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:93"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:90"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:91"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:92"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:91"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:90"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#x:93"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:90"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermNil" class="idref" href="#subTermNil"><span class="id" title="lemma">subTermNil</span></a> (<a id="t:94" class="idref" href="#t:94"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="v:95" class="idref" href="#v:95"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:96" class="idref" href="#s:96"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:95"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:94"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:94"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:95"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:96"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:94"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermTrans" class="idref" href="#subTermTrans"><span class="id" title="lemma">subTermTrans</span></a>  (<a id="t:101" class="idref" href="#t:101"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="v1:102" class="idref" href="#v1:102"><span class="id" title="binder">v1</span></a> <a id="v2:103" class="idref" href="#v2:103"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:104" class="idref" href="#s:104"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:103"><span class="id" title="variable">v2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:102"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:101"><span class="id" title="variable">t</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:101"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:102"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:103"><span class="id" title="variable">v2</span></a>)) <a class="idref" href="hydras.Ackermann.subProp.html#v2:103"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:104"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:101"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:102"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:104"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermExch" class="idref" href="#subTermExch"><span class="id" title="lemma">subTermExch</span></a>  (<a id="t:109" class="idref" href="#t:109"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>) (<a id="v1:110" class="idref" href="#v1:110"><span class="id" title="binder">v1</span></a> <a id="v2:111" class="idref" href="#v2:111"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:112" class="idref" href="#s1:112"><span class="id" title="binder">s1</span></a> <a id="s2:113" class="idref" href="#s2:113"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v1:110"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:111"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:111"><span class="id" title="variable">v2</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:112"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:110"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:113"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:109"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:110"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:112"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v2:111"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:113"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#t:109"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:111"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:113"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v1:110"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:112"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermsNil" class="idref" href="#subTermsNil"><span class="id" title="lemma">subTermsNil</span></a>  (<a id="n:118" class="idref" href="#n:118"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:119" class="idref" href="#ts:119"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:118"><span class="id" title="variable">n</span></a>) (<a id="v:120" class="idref" href="#v:120"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:121" class="idref" href="#s:121"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:120"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:119"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:119"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:120"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:121"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:119"><span class="id" title="variable">ts</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermsTrans" class="idref" href="#subTermsTrans"><span class="id" title="lemma">subTermsTrans</span></a> (<a id="n:122" class="idref" href="#n:122"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:123" class="idref" href="#ts:123"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:122"><span class="id" title="variable">n</span></a>) (<a id="v1:124" class="idref" href="#v1:124"><span class="id" title="binder">v1</span></a> <a id="v2:125" class="idref" href="#v2:125"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:126" class="idref" href="#s:126"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:125"><span class="id" title="variable">v2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:124"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarTs"><span class="id" title="definition">freeVarTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:123"><span class="id" title="variable">ts</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:123"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:124"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:125"><span class="id" title="variable">v2</span></a>)) <a class="idref" href="hydras.Ackermann.subProp.html#v2:125"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:126"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:123"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:124"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:126"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subTermsExch" class="idref" href="#subTermsExch"><span class="id" title="lemma">subTermsExch</span></a>  (<a id="n:127" class="idref" href="#n:127"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:128" class="idref" href="#ts:128"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#n:127"><span class="id" title="variable">n</span></a>) (<a id="v1:129" class="idref" href="#v1:129"><span class="id" title="binder">v1</span></a> <a id="v2:130" class="idref" href="#v2:130"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;(<a id="s1:131" class="idref" href="#s1:131"><span class="id" title="binder">s1</span></a> <a id="s2:132" class="idref" href="#s2:132"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v1:129"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:130"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:130"><span class="id" title="variable">v2</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:131"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:129"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:132"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:128"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:129"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:131"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v2:130"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:132"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#ts:128"><span class="id" title="variable">ts</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:130"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:132"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v1:129"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:131"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="subFormulaNTEHelp" class="idref" href="#subFormulaNTEHelp"><span class="id" title="lemma">subFormulaNTEHelp</span></a> (<a id="f:133" class="idref" href="#f:133"><span class="id" title="binder">f</span></a> <a id="g:134" class="idref" href="#g:134"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="v:135" class="idref" href="#v:135"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:136" class="idref" href="#s:136"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>):<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#f:133"><span class="id" title="variable">f</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#g:134"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:133"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:135"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:136"><span class="id" title="variable">s</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#g:134"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:135"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:136"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="subFormulaNTE" class="idref" href="#subFormulaNTE"><span class="id" title="lemma">subFormulaNTE</span></a>  (<a id="f:137" class="idref" href="#f:137"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="T:138" class="idref" href="#T:138"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.System"><span class="id" title="variable">System</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="v:139" class="idref" href="#v:139"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:140" class="idref" href="#s:140"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:139"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:138"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:139"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:140"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="v1:141" class="idref" href="#v1:141"><span class="id" title="binder">v1</span></a> <a id="v2:142" class="idref" href="#v2:142"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:143" class="idref" href="#s:143"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:142"><span class="id" title="variable">v2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:141"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:138"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:141"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:142"><span class="id" title="variable">v2</span></a>)) <a class="idref" href="hydras.Ackermann.subProp.html#v2:142"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:143"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:141"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:143"><span class="id" title="variable">s</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="v1:144" class="idref" href="#v1:144"><span class="id" title="binder">v1</span></a> <a id="v2:145" class="idref" href="#v2:145"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:146" class="idref" href="#s1:146"><span class="id" title="binder">s1</span></a> <a id="s2:147" class="idref" href="#s2:147"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v1:144"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:145"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:145"><span class="id" title="variable">v2</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:146"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:144"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:147"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:138"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:144"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:146"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v2:145"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:147"><span class="id" title="variable">s2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:137"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:145"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:147"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v1:144"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:146"><span class="id" title="variable">s1</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaNil" class="idref" href="#subFormulaNil"><span class="id" title="lemma">subFormulaNil</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:156" class="idref" href="#f:156"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:157" class="idref" href="#T:157"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.System"><span class="id" title="variable">System</span></a>) (<a id="v:158" class="idref" href="#v:158"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:159" class="idref" href="#s:159"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:158"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:156"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:157"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#f:156"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v:158"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:159"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#f:156"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaTrans" class="idref" href="#subFormulaTrans"><span class="id" title="lemma">subFormulaTrans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:160" class="idref" href="#f:160"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:161" class="idref" href="#T:161"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.System"><span class="id" title="variable">System</span></a>) (<a id="v1:162" class="idref" href="#v1:162"><span class="id" title="binder">v1</span></a> <a id="v2:163" class="idref" href="#v2:163"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:164" class="idref" href="#s:164"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:163"><span class="id" title="variable">v2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#remove"><span class="id" title="definition">List.remove</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:162"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#f:160"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:161"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:162"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:163"><span class="id" title="variable">v2</span></a>)) <a class="idref" href="hydras.Ackermann.subProp.html#v2:163"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:164"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:162"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s:164"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="subFormulaExch" class="idref" href="#subFormulaExch"><span class="id" title="lemma">subFormulaExch</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:165" class="idref" href="#f:165"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula"><span class="id" title="variable">Formula</span></a>) (<a id="T:166" class="idref" href="#T:166"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.System"><span class="id" title="variable">System</span></a>) (<a id="v1:167" class="idref" href="#v1:167"><span class="id" title="binder">v1</span></a> <a id="v2:168" class="idref" href="#v2:168"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s1:169" class="idref" href="#s1:169"><span class="id" title="binder">s1</span></a> <a id="s2:170" class="idref" href="#s2:170"><span class="id" title="binder">s2</span></a> : <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.Term"><span class="id" title="variable">Term</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#v1:167"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:168"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:168"><span class="id" title="variable">v2</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:169"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:167"><span class="id" title="variable">v1</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:170"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf"><span class="id" title="variable">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#T:166"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#iffH"><span class="id" title="definition">iffH</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#f:165"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v1:167"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:169"><span class="id" title="variable">s1</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v2:168"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:170"><span class="id" title="variable">s2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>  <a class="idref" href="hydras.Ackermann.subProp.html#f:165"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#v2:168"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s2:170"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Ackermann.subProp.html#v1:167"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Ackermann.subProp.html#s1:169"><span class="id" title="variable">s1</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.subProp.html#Substitution_Properties"><span class="id" title="section">Substitution_Properties</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Ackermann.wConsistent.html b/theories/html/hydras.Ackermann.wConsistent.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Ackermann.wConsistent</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Ackermann.wConsistent</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wConsistent" class="idref" href="#wConsistent"><span class="id" title="definition">wConsistent</span></a> (<a id="T:1" class="idref" href="#T:1"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="f:2" class="idref" href="#f:2"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a>) (<a id="v:3" class="idref" href="#v:3"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#x:4"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:2"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#x:4"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:1"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:3"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:2"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:1"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:2"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:3"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#n:5"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wCon2Con" class="idref" href="#wCon2Con"><span class="id" title="lemma">wCon2Con</span></a> : <span class="id" title="keyword">∀</span> <a id="T:6" class="idref" href="#T:6"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>, <a class="idref" href="hydras.Ackermann.wConsistent.html#wConsistent"><span class="id" title="definition">wConsistent</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:6"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Consistent"><span class="id" title="definition">Consistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:6"><span class="id" title="variable">T</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wInconsistent" class="idref" href="#wInconsistent"><span class="id" title="definition">wInconsistent</span></a> (<a id="T:9" class="idref" href="#T:9"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#Formula"><span class="id" title="definition">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="v:11" class="idref" href="#v:11"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#x:12"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#freeVarF"><span class="id" title="definition">freeVarF</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:10"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:11"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#x:12"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:9"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:11"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:10"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:9"><span class="id" title="variable">T</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#v:11"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#n:13"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="notCon2wNotCon" class="idref" href="#notCon2wNotCon"><span class="id" title="lemma">notCon2wNotCon</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="T:14" class="idref" href="#T:14"><span class="id" title="binder">T</span></a> : <a class="idref" href="hydras.Ackermann.LNN.html#System"><span class="id" title="definition">System</span></a>, <a class="idref" href="hydras.Ackermann.folProof.html#Inconsistent"><span class="id" title="definition">Inconsistent</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:14"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#wInconsistent"><span class="id" title="definition">wInconsistent</span></a> <a class="idref" href="hydras.Ackermann.wConsistent.html#T:14"><span class="id" title="variable">T</span></a>.<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<title>hydras.Ackermann.wellFormed</title>
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+
+<h1 class="libtitle">Library hydras.Ackermann.wellFormed</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.code.html#"><span class="id" title="library">code</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.codeList.html#"><span class="id" title="library">codeList</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Well_Formed_Term" class="idref" href="#Well_Formed_Term"><span class="id" title="section">Well_Formed_Term</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Well_Formed_Term.L" class="idref" href="#Well_Formed_Term.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Context</span> `(<a id="cL:4" class="idref" href="#cL:4"><span class="id" title="binder">cL</span></a> : <a class="idref" href="hydras.Ackermann.code.html#Lcode"><span class="id" title="class">Lcode</span></a>  <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a> <a id="cf:2" class="idref" href="#cf:2"><span class="id" title="binder">cf</span></a> <a id="cr:3" class="idref" href="#cr:3"><span class="id" title="binder">cr</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Well_Formed_Term.codeArityF" class="idref" href="#Well_Formed_Term.codeArityF"><span class="id" title="variable">codeArityF</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityFIsPR:6" class="idref" href="#codeArityFIsPR:6"><span class="id" title="binder">codeArityFIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF"><span class="id" title="variable">codeArityF</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Well_Formed_Term.codeArityFIsCorrect1" class="idref" href="#Well_Formed_Term.codeArityFIsCorrect1"><span class="id" title="variable">codeArityFIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF"><span class="id" title="variable">codeArityF</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:7"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityF"><span class="id" title="projection">arityF</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:7"><span class="id" title="variable">f</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Well_Formed_Term.codeArityFIsCorrect2" class="idref" href="#Well_Formed_Term.codeArityFIsCorrect2"><span class="id" title="variable">codeArityFIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF"><span class="id" title="variable">codeArityF</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.cf"><span class="id" title="variable">cf</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:9"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Well_Formed_Term.Term" class="idref" href="#Well_Formed_Term.Term"><span class="id" title="variable">Term</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Well_Formed_Term.Terms" class="idref" href="#Well_Formed_Term.Terms"><span class="id" title="variable">Terms</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Terms"><span class="id" title="inductive">Terms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wellFormedTermTerms" class="idref" href="#wellFormedTermTerms"><span class="id" title="definition">wellFormedTermTerms</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="t:12" class="idref" href="#t:12"><span class="id" title="binder">t</span></a> <a id="recs:13" class="idref" href="#recs:13"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF"><span class="id" title="variable">codeArityF</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#codeLength"><span class="id" title="definition">codeLength</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>)) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:13"><span class="id" title="variable">recs</span></a>)) 1)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:13"><span class="id" title="variable">recs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:12"><span class="id" title="variable">t</span></a>))) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:13"><span class="id" title="variable">recs</span></a>)) 1)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wellFormedTerm" class="idref" href="#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> (<a id="t:14" class="idref" href="#t:14"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTermTerms"><span class="id" title="definition">wellFormedTermTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:14"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wellFormedTerms" class="idref" href="#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a> (<a id="ts:15" class="idref" href="#ts:15"><span class="id" title="binder">ts</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTermTerms"><span class="id" title="definition">wellFormedTermTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#ts:15"><span class="id" title="variable">ts</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lengthTerms" class="idref" href="#lengthTerms"><span class="id" title="lemma">lengthTerms</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:17" class="idref" href="#ts:17"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:16"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.Ackermann.codeList.html#codeLength"><span class="id" title="definition">codeLength</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#ts:17"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:16"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedTermCorrect1" class="idref" href="#wellFormedTermCorrect1"><span class="id" title="lemma">wellFormedTermCorrect1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:22" class="idref" href="#t:22"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Term"><span class="id" title="variable">Term</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:22"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedTermsCorrect1" class="idref" href="#wellFormedTermsCorrect1"><span class="id" title="lemma">wellFormedTermsCorrect1</span></a> (<a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="ts:32" class="idref" href="#ts:32"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:31"><span class="id" title="variable">n</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a> (<a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#ts:32"><span class="id" title="variable">ts</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="multLemma1" class="idref" href="#multLemma1"><span class="id" title="lemma">multLemma1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:37" class="idref" href="#a:37"><span class="id" title="binder">a</span></a> <a id="b:38" class="idref" href="#b:38"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#a:37"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#b:38"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#a:37"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="multLemma2" class="idref" href="#multLemma2"><span class="id" title="lemma">multLemma2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:39" class="idref" href="#a:39"><span class="id" title="binder">a</span></a> <a id="b:40" class="idref" href="#b:40"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#a:39"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#b:40"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#b:40"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="wellFormedTermTermsCorrect2" class="idref" href="#wellFormedTermTermsCorrect2"><span class="id" title="lemma">wellFormedTermTermsCorrect2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:41"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="t:42" class="idref" href="#t:42"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Term"><span class="id" title="variable">Term</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:42"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:41"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:41"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="m:43" class="idref" href="#m:43"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="ts:44" class="idref" href="#ts:44"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#m:43"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#ts:44"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:41"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedTermCorrect2" class="idref" href="#wellFormedTermCorrect2"><span class="id" title="lemma">wellFormedTermCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:59" class="idref" href="#n:59"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:59"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="t:60" class="idref" href="#t:60"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Term"><span class="id" title="variable">Term</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerm"><span class="id" title="definition">codeTerm</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#t:60"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:59"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedTermsCorrect2" class="idref" href="#wellFormedTermsCorrect2"><span class="id" title="lemma">wellFormedTermsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:61" class="idref" href="#n:61"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:61"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="m:62" class="idref" href="#m:62"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="ts:63" class="idref" href="#ts:63"><span class="id" title="binder">ts</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms"><span class="id" title="variable">Terms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#m:62"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeTerms"><span class="id" title="definition">codeTerms</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#ts:63"><span class="id" title="variable">ts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:61"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="wellFormedTermTermsIsPR" class="idref" href="#wellFormedTermTermsIsPR"><span class="id" title="instance">wellFormedTermTermsIsPR</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTermTerms"><span class="id" title="definition">wellFormedTermTerms</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="wellFormedTermIsPR" class="idref" href="#wellFormedTermIsPR"><span class="id" title="instance">wellFormedTermIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="wellFormedTermsIsPR" class="idref" href="#wellFormedTermsIsPR"><span class="id" title="instance">wellFormedTermsIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Well_Formed_Term.Well_Formed_Formula" class="idref" href="#Well_Formed_Term.Well_Formed_Formula"><span class="id" title="section">Well_Formed_Formula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a id="Well_Formed_Term.Well_Formed_Formula.codeArityR" class="idref" href="#Well_Formed_Term.Well_Formed_Formula.codeArityR"><span class="id" title="variable">codeArityR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Context</span> (<a id="codeArityRIsPR:181" class="idref" href="#codeArityRIsPR:181"><span class="id" title="binder">codeArityRIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR"><span class="id" title="variable">codeArityR</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1" class="idref" href="#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1"><span class="id" title="variable">codeArityRIsCorrect1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="r:182" class="idref" href="#r:182"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR"><span class="id" title="variable">codeArityR</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#r:182"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#arityR"><span class="id" title="projection">arityR</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#r:182"><span class="id" title="variable">r</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span><br/>
+&nbsp;&nbsp;<a id="Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2" class="idref" href="#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2"><span class="id" title="variable">codeArityRIsCorrect2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:184" class="idref" href="#n:184"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR"><span class="id" title="variable">codeArityR</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:184"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="r:185" class="idref" href="#r:185"><span class="id" title="binder">r</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.cr"><span class="id" title="variable">cr</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#r:185"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:184"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Well_Formed_Term.Well_Formed_Formula.Formula" class="idref" href="#Well_Formed_Term.Well_Formed_Formula.Formula"><span class="id" title="variable">Formula</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L"><span class="id" title="variable">L</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="wellFormedFormula" class="idref" href="#wellFormedFormula"><span class="id" title="definition">wellFormedFormula</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#evalStrongRec"><span class="id" title="definition">evalStrongRec</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="f:187" class="idref" href="#f:187"><span class="id" title="binder">f</span></a> <a id="recs:188" class="idref" href="#recs:188"><span class="id" title="binder">recs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#switchPR"><span class="id" title="definition">switchPR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eqb"><span class="id" title="definition">Nat.eqb</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR"><span class="id" title="variable">codeArityR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.codeList.html#codeLength"><span class="id" title="definition">codeLength</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerms"><span class="id" title="definition">wellFormedTerms</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:188"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>)) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:188"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:188"><span class="id" title="variable">recs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> (<a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>))) <a class="idref" href="hydras.Ackermann.wellFormed.html#recs:188"><span class="id" title="variable">recs</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedTerm"><span class="id" title="definition">wellFormedTerm</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:187"><span class="id" title="variable">f</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedFormulaCorrect1" class="idref" href="#wellFormedFormulaCorrect1"><span class="id" title="lemma">wellFormedFormulaCorrect1</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="f:189" class="idref" href="#f:189"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.Formula"><span class="id" title="variable">Formula</span></a>, <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedFormula"><span class="id" title="definition">wellFormedFormula</span></a>  (<a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:189"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wellFormedFormulaCorrect2" class="idref" href="#wellFormedFormulaCorrect2"><span class="id" title="lemma">wellFormedFormulaCorrect2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:194" class="idref" href="#n:194"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedFormula"><span class="id" title="definition">wellFormedFormula</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:194"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:195" class="idref" href="#f:195"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.Formula"><span class="id" title="variable">Formula</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.code.html#codeFormula"><span class="id" title="definition">codeFormula</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#f:195"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.wellFormed.html#n:194"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="wellFormedFormulaIsPR" class="idref" href="#wellFormedFormulaIsPR"><span class="id" title="instance">wellFormedFormulaIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Ackermann.wellFormed.html#wellFormedFormula"><span class="id" title="definition">wellFormedFormula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula"><span class="id" title="section">Well_Formed_Formula</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Ackermann.wellFormed.html#Well_Formed_Term"><span class="id" title="section">Well_Formed_Term</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,430 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Canon</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Canon</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ Canonical Sequences of ordinals below epsilon0
+
+<div class="paragraph"> </div>
+
+Pierre Casteran, 
+       LaBRI and University of Bordeaux.
+
+<div class="paragraph"> </div>
+
+       After J. Ketonen and R. Solovay's paper
+  " Rapidly Growing Ramsey Functions" <i>in</i>
+    Annals of mathematics, Mar. 1981 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="module">Relation_Operators</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Prelude</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Epsilon0</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab87"></a><h1 class="section">Definitions</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="canon" class="idref" href="#canon"><span class="id" title="definition">canon</span></a> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:1"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">k</span>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">k</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma</span> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ (<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">T1.pred</span></a> <span class="id" title="var">gamma</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">gamma'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Canon.html#i:2"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma'</span> <span class="id" title="var">j</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon:3"><span class="id" title="definition">canon</span></a> <span class="id" title="var">gamma</span> <a class="idref" href="hydras.Epsilon0.Canon.html#i:2"><span class="id" title="variable">i</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">T1.pred</span></a> <span class="id" title="var">gamma</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">gamma'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Canon.html#i:2"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma</span> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma'</span> <span class="id" title="var">j</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">gamma</span> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon:3"><span class="id" title="definition">canon</span></a> <span class="id" title="var">gamma</span>  <a class="idref" href="hydras.Epsilon0.Canon.html#i:2"><span class="id" title="variable">i</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">n</span> <span class="id" title="keyword">beta</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:1"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon:3"><span class="id" title="definition">canon</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="hydras.Epsilon0.Canon.html#i:2"><span class="id" title="variable">i</span></a>)  <br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Canon_examples" class="idref" href="#Canon_examples"><span class="id" title="section">Canon_examples</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ppT1_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Close</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ppT1_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">canon</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">^</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">10</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">phi0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">10</span></a>.   
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon_examples"><span class="id" title="section">Canon_examples</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab88"></a><h1 class="section">Properties of canonical sequences</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_zero" class="idref" href="#canon_zero"><span class="id" title="lemma">canon_zero</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> :  <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_tail" class="idref" href="#canon_tail"><span class="id" title="lemma">canon_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> <a id="i:11" class="idref" href="#i:11"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:10"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#beta:10"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:10"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:11"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:9"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:10"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:11"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_lim1" class="idref" href="#canonS_lim1"><span class="id" title="lemma">canonS_lim1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a> <a id="lambda:13" class="idref" href="#lambda:13"><span class="id" title="binder">lambda</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:13"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:13"><span class="id" title="variable">lambda</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:13"><span class="id" title="variable">lambda</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:12"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:13"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:12"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim1" class="idref" href="#canon_lim1"><span class="id" title="lemma">canon_lim1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a> <a id="lambda:15" class="idref" href="#lambda:15"><span class="id" title="binder">lambda</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:15"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:15"><span class="id" title="variable">lambda</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:15"><span class="id" title="variable">lambda</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)  <a class="idref" href="hydras.Epsilon0.Canon.html#i:14"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:15"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:14"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Here 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_lim2" class="idref" href="#canonS_lim2"><span class="id" title="lemma">canonS_lim2</span></a>  <a id="i:16" class="idref" href="#i:16"><span class="id" title="binder">i</span></a> <a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="lambda:18" class="idref" href="#lambda:18"><span class="id" title="binder">lambda</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:18"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:18"><span class="id" title="variable">lambda</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:18"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:17"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:16"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:18"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:17"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:18"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:16"><span class="id" title="variable">i</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_lim2" class="idref" href="#canon0_lim2"><span class="id" title="lemma">canon0_lim2</span></a>   <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> <a id="lambda:20" class="idref" href="#lambda:20"><span class="id" title="binder">lambda</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:20"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:20"><span class="id" title="variable">lambda</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:20"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:19"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:20"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:19"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:20"><span class="id" title="variable">lambda</span></a> 0)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim2" class="idref" href="#canon_lim2"><span class="id" title="lemma">canon_lim2</span></a>  <a id="i:21" class="idref" href="#i:21"><span class="id" title="binder">i</span></a>  <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> <a id="lambda:23" class="idref" href="#lambda:23"><span class="id" title="binder">lambda</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:23"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:22"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:21"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:22"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:23"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:21"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lim3" class="idref" href="#canon_lim3"><span class="id" title="lemma">canon_lim3</span></a> <a id="i:24" class="idref" href="#i:24"><span class="id" title="binder">i</span></a> <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> <a id="alpha:26" class="idref" href="#alpha:26"><span class="id" title="binder">alpha</span></a> <a id="lambda:27" class="idref" href="#lambda:27"><span class="id" title="binder">lambda</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:27"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:27"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:25"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:27"><span class="id" title="variable">lambda</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:24"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:25"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:27"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:24"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_succ" class="idref" href="#canon_succ"><span class="id" title="lemma">canon_succ</span></a> <a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a> <a id="alpha:29" class="idref" href="#alpha:29"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:29"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:28"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:29"><span class="id" title="variable">alpha</span></a>.  <span class="id" title="keyword">Lemma</span> <a id="canonS_succ" class="idref" href="#canonS_succ"><span class="id" title="lemma">canonS_succ</span></a> <a id="i:30" class="idref" href="#i:30"><span class="id" title="binder">i</span></a> <a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:31"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:30"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:31"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_succ" class="idref" href="#canon0_succ"><span class="id" title="lemma">canon0_succ</span></a>  <a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:32"><span class="id" title="variable">alpha</span></a>) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:32"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_phi0_succ_eqn" class="idref" href="#canonS_phi0_succ_eqn"><span class="id" title="lemma">canonS_phi0_succ_eqn</span></a> <a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> <a id="gamma:34" class="idref" href="#gamma:34"><span class="id" title="binder">gamma</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:34"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:34"><span class="id" title="variable">gamma</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:33"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:34"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_phi0_succ_eqn" class="idref" href="#canon0_phi0_succ_eqn"><span class="id" title="lemma">canon0_phi0_succ_eqn</span></a>  <a id="gamma:35" class="idref" href="#gamma:35"><span class="id" title="binder">gamma</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:35"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:35"><span class="id" title="variable">gamma</span></a>)) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_cons_succ_eqn2" class="idref" href="#canonS_cons_succ_eqn2"><span class="id" title="lemma">canonS_cons_succ_eqn2</span></a> <a id="i:36" class="idref" href="#i:36"><span class="id" title="binder">i</span></a> <a id="n:37" class="idref" href="#n:37"><span class="id" title="binder">n</span></a> <a id="gamma:38" class="idref" href="#gamma:38"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:38"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:38"><span class="id" title="variable">gamma</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:37"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:36"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:38"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#n:37"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:38"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:36"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_cons_succ_eqn2" class="idref" href="#canon0_cons_succ_eqn2"><span class="id" title="lemma">canon0_cons_succ_eqn2</span></a> <a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> <a id="gamma:40" class="idref" href="#gamma:40"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:40"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:40"><span class="id" title="variable">gamma</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:39"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)  0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:40"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_SSn_zero" class="idref" href="#canon_SSn_zero"><span class="id" title="lemma">canon_SSn_zero</span></a> (<a id="i:41" class="idref" href="#i:41"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a> <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a>  ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:42"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:43"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:41"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:43"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:42"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:41"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+&nbsp;#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use canon_SSn_zero")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="canonSSn" class="idref" href="#canonSSn"><span class="id" title="abbreviation">canonSSn</span></a> := <a class="idref" href="hydras.Epsilon0.Canon.html#canon_SSn_zero"><span class="id" title="lemma">canon_SSn_zero</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_zero_inv" class="idref" href="#canonS_zero_inv"><span class="id" title="lemma">canonS_zero_inv</span></a> (<a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:44"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:45"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:44"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:44"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab89"></a><h2 class="section">Canonical sequences and the order LT</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_LT" class="idref" href="#canonS_LT"><span class="id" title="lemma">canonS_LT</span></a> <a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a> <a id="alpha:47" class="idref" href="#alpha:47"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:47"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:46"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:47"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon0_LT" class="idref" href="#canon0_LT"><span class="id" title="lemma">canon0_LT</span></a>  <a id="alpha:48" class="idref" href="#alpha:48"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:48"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:48"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:48"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:48"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_canon" class="idref" href="#nf_canon"><span class="id" title="lemma">nf_canon</span></a>  <a id="i:49" class="idref" href="#i:49"><span class="id" title="binder">i</span></a> <a id="alpha:50" class="idref" href="#alpha:50"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:50"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:50"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:49"><span class="id" title="variable">i</span></a>).<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_LT" class="idref" href="#canon_LT"><span class="id" title="lemma">canon_LT</span></a> <a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a> <a id="alpha:52" class="idref" href="#alpha:52"><span class="id" title="binder">alpha</span></a> :  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:52"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_lt" class="idref" href="#canon_lt"><span class="id" title="lemma">canon_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="i:53" class="idref" href="#i:53"><span class="id" title="binder">i</span></a> <a id="alpha:54" class="idref" href="#alpha:54"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:54"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:54"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:54"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:53"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:54"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_cons_not_zero" class="idref" href="#canonS_cons_not_zero"><span class="id" title="lemma">canonS_cons_not_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a> <a id="alpha:56" class="idref" href="#alpha:56"><span class="id" title="binder">alpha</span></a> <a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> <a id="beta:58" class="idref" href="#beta:58"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:58"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:55"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab90"></a><h2 class="section">Canonical sequences of limit ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_canonS_not_zero" class="idref" href="#T1limit_canonS_not_zero"><span class="id" title="lemma">T1limit_canonS_not_zero</span></a> <a id="i:59" class="idref" href="#i:59"><span class="id" title="binder">i</span></a> <a id="lambda:60" class="idref" href="#lambda:60"><span class="id" title="binder">lambda</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:60"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:60"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:60"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:59"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+Let lambda  be a limit ordinal. For any beta &lt; lambda, we can 
+compute some item of the canonical sequence  of lambda which is 
+greater than beta.
+
+<div class="paragraph"> </div>
+
+   It is a constructive way of expressing that lambda is the 
+limit of its  own canonical sequence 
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_limit_strong" class="idref" href="#canonS_limit_strong"><span class="id" title="lemma">canonS_limit_strong</span></a> <a id="lambda:61" class="idref" href="#lambda:61"><span class="id" title="binder">lambda</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:61"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:61"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:62" class="idref" href="#beta:62"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Canon.html#beta:62"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:61"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="i:63" class="idref" href="#i:63"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:62"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:61"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:63"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_limit_strong" class="idref" href="#canon_limit_strong"><span class="id" title="lemma">canon_limit_strong</span></a> <a id="lambda:72" class="idref" href="#lambda:72"><span class="id" title="binder">lambda</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:72"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:72"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:73" class="idref" href="#beta:73"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Canon.html#beta:73"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:72"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="i:74" class="idref" href="#i:74"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:73"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:72"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:74"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_limit_lub" class="idref" href="#canonS_limit_lub"><span class="id" title="lemma">canonS_limit_lub</span></a> (<a id="lambda:75" class="idref" href="#lambda:75"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:75"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:75"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> (<span class="id" title="keyword">fun</span> <a id="i:76" class="idref" href="#i:76"><span class="id" title="binder">i</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:75"><span class="id" title="variable">lambda</span></a>( <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:76"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Canon.html#lambda:75"><span class="id" title="variable">lambda</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canon_limit_mono" class="idref" href="#canon_limit_mono"><span class="id" title="lemma">canon_limit_mono</span></a> <a id="alpha:77" class="idref" href="#alpha:77"><span class="id" title="binder">alpha</span></a> <a id="i:78" class="idref" href="#i:78"><span class="id" title="binder">i</span></a> <a id="j:79" class="idref" href="#j:79"><span class="id" title="binder">j</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:77"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:77"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#i:78"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#j:79"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:77"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:78"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:77"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#j:79"><span class="id" title="variable">j</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_limit_mono" class="idref" href="#canonS_limit_mono"><span class="id" title="lemma">canonS_limit_mono</span></a> <a id="alpha:80" class="idref" href="#alpha:80"><span class="id" title="binder">alpha</span></a> <a id="i:81" class="idref" href="#i:81"><span class="id" title="binder">i</span></a> <a id="j:82" class="idref" href="#j:82"><span class="id" title="binder">j</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:80"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#j:82"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:80"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:81"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:80"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#j:82"><span class="id" title="variable">j</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_LE" class="idref" href="#canonS_LE"><span class="id" title="lemma">canonS_LE</span></a> <a id="alpha:83" class="idref" href="#alpha:83"><span class="id" title="binder">alpha</span></a> <a id="n:84" class="idref" href="#n:84"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:83"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:83"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:84"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:83"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:84"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+exercise  after Guillaume Melquiond   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Fuel.html#"><span class="id" title="library">Fuel</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span>  <a id="approx" class="idref" href="#approx"><span class="id" title="definition">approx</span></a> <a id="alpha:85" class="idref" href="#alpha:85"><span class="id" title="binder">alpha</span></a> <a id="beta:86" class="idref" href="#beta:86"><span class="id" title="binder">beta</span></a> <a id="fuel:87" class="idref" href="#fuel:87"><span class="id" title="binder">fuel</span></a> <a id="i:88" class="idref" href="#i:88"><span class="id" title="binder">i</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Canon.html#fuel:87"><span class="id" title="variable">fuel</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Fuel.html#FO"><span class="id" title="constructor">FO</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Prelude.Fuel.html#FS"><span class="id" title="constructor">Fuel.FS</span></a> <span class="id" title="var">f</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="gamma:91" class="idref" href="#gamma:91"><span class="id" title="binder">gamma</span></a> := <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:85"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:88"><span class="id" title="variable">i</span></a>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide"><span class="id" title="definition">decide</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:86"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:91"><span class="id" title="variable">gamma</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#i:88"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#gamma:91"><span class="id" title="variable">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.Epsilon0.Canon.html#approx:89"><span class="id" title="definition">approx</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:85"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:86"><span class="id" title="variable">beta</span></a>  (<span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:88"><span class="id" title="variable">i</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="approx_ok" class="idref" href="#approx_ok"><span class="id" title="lemma">approx_ok</span></a> <a id="alpha:92" class="idref" href="#alpha:92"><span class="id" title="binder">alpha</span></a> <a id="beta:93" class="idref" href="#beta:93"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="fuel:94" class="idref" href="#fuel:94"><span class="id" title="binder">fuel</span></a> <a id="i:95" class="idref" href="#i:95"><span class="id" title="binder">i</span></a> <a id="j:96" class="idref" href="#j:96"><span class="id" title="binder">j</span></a> <a id="gamma:97" class="idref" href="#gamma:97"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Epsilon0.Canon.html#approx"><span class="id" title="definition">approx</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:92"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:93"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#fuel:94"><span class="id" title="variable">fuel</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:95"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#j:96"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#gamma:97"><span class="id" title="variable">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#gamma:97"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:92"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#j:96"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:93"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#gamma:97"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab91"></a><h2 class="section">Adaptation to E0 (well formed terms of type <span class="inlinecode"><span class="id" title="var">T1</span></span> )</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="Canon" class="idref" href="#Canon"><span class="id" title="definition">Canon</span></a> (<a id="alpha:98" class="idref" href="#alpha:98"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>)(<a id="i:99" class="idref" href="#i:99"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (@<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:98"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Canon.html#i:99"><span class="id" title="variable">i</span></a>) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_Succ" class="idref" href="#Canon_Succ"><span class="id" title="lemma">Canon_Succ</span></a> <a id="beta:100" class="idref" href="#beta:100"><span class="id" title="binder">beta</span></a> <a id="n:101" class="idref" href="#n:101"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:100"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:101"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#beta:100"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_Omega" class="idref" href="#Canon_Omega"><span class="id" title="lemma">Canon_Omega</span></a> <a id="k:102" class="idref" href="#k:102"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#k:102"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#k:102"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon_Omega"><span class="id" title="lemma">Canon_Omega</span></a> : <span class="id" title="var">E0_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonSSn" class="idref" href="#CanonSSn"><span class="id" title="lemma">CanonSSn</span></a> (<a id="i:103" class="idref" href="#i:103"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:104" class="idref" href="#alpha:104"><span class="id" title="binder">alpha</span></a> <a id="n:105" class="idref" href="#n:105"><span class="id" title="binder">n</span></a>  , <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Cons"><span class="id" title="instance">Cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:104"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:105"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:103"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#Cons"><span class="id" title="instance">Cons</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#n:105"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:104"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:103"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonS_phi0_lim" class="idref" href="#CanonS_phi0_lim"><span class="id" title="lemma">CanonS_phi0_lim</span></a> <a id="alpha:106" class="idref" href="#alpha:106"><span class="id" title="binder">alpha</span></a> <a id="k:107" class="idref" href="#k:107"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:106"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:106"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#k:107"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:106"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#k:107"><span class="id" title="variable">k</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonS_lt" class="idref" href="#CanonS_lt"><span class="id" title="lemma">CanonS_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="i:108" class="idref" href="#i:108"><span class="id" title="binder">i</span></a> <a id="alpha:109" class="idref" href="#alpha:109"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:109"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:108"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:109"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_lt" class="idref" href="#Canon_lt"><span class="id" title="lemma">Canon_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="i:110" class="idref" href="#i:110"><span class="id" title="binder">i</span></a> <a id="alpha:111" class="idref" href="#alpha:111"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:110"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:111"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_of_limit_not_null" class="idref" href="#Canon_of_limit_not_null"><span class="id" title="lemma">Canon_of_limit_not_null</span></a> : <span class="id" title="keyword">∀</span> <a id="i:112" class="idref" href="#i:112"><span class="id" title="binder">i</span></a> <a id="alpha:113" class="idref" href="#alpha:113"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:113"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:112"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">CanonS_lt</span> <span class="id" title="var">Canon_lt</span> <span class="id" title="var">Canon_of_limit_not_null</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonS_phi0_Succ" class="idref" href="#CanonS_phi0_Succ"><span class="id" title="lemma">CanonS_phi0_Succ</span></a> <a id="alpha:114" class="idref" href="#alpha:114"><span class="id" title="binder">alpha</span></a> <a id="i:115" class="idref" href="#i:115"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:114"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:115"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#alpha:114"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#i:115"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.E0</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.E0</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+   Class of ordinals less than epsilon0 
+
+<div class="paragraph"> </div>
+
+   <span class="inlinecode"><span class="id" title="var">E0</span></span> is a class type for ordinal terms proved to be in normal form.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Max.html#"><span class="id" title="library">Max</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Operators_Properties.html#"><span class="id" title="library">Operators_Properties</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">Prelude.More_Arith</span></a>  <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Prelude.Restriction</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">Prelude.DecPreOrder</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a> <a class="idref" href="hydras.Prelude.OrdNotations.html#"><span class="id" title="library">OrdNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Epsilon0</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Create HintDb</span> <span class="id" title="var">E0</span>.<br/>
+<span class="id" title="keyword">Create HintDb</span> <span class="id" title="var">E0_rw</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">E0_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">e0</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="E0" class="idref" href="#E0"><span class="id" title="record">E0</span></a> : <span class="id" title="keyword">Type</span> := <a id="mkord" class="idref" href="#mkord"><span class="id" title="constructor">mkord</span></a> {<a id="cnf" class="idref" href="#cnf"><span class="id" title="projection">cnf</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>; <a id="cnf_ok" class="idref" href="#cnf_ok"><span class="id" title="projection">cnf_ok</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf:2"><span class="id" title="method">cnf</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">cnf_ok</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab34"></a><h2 class="section">Lifting functions from <span class="inlinecode"><span class="id" title="var">T1</span></span> to <span class="inlinecode"><span class="id" title="var">E0</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0lt" class="idref" href="#E0lt"><span class="id" title="definition">E0lt</span></a> (<a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> <a id="beta:5" class="idref" href="#beta:5"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) := <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">T1.LT</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:4"><span class="id" title="variable">alpha</span></a>) (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:5"><span class="id" title="variable">beta</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0le" class="idref" href="#E0le"><span class="id" title="definition">E0le</span></a> := <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="::E0_scope:x_'o&lt;'_x" class="idref" href="#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&quot;</span></a>o&lt;" := <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> : <span class="id" title="var">E0_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="b8cab4266e2151a748b18e18aa1598f9" class="idref" href="#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">&quot;</span></a>o&lt;=" := <a class="idref" href="hydras.Epsilon0.E0.html#E0le"><span class="id" title="definition">E0le</span></a> : <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="E0zero" class="idref" href="#E0zero"><span class="id" title="instance">E0zero</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> := @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="E0_omega" class="idref" href="#E0_omega"><span class="id" title="instance">E0_omega</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> := @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0zero")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Zero" class="idref" href="#Zero"><span class="id" title="abbreviation">Zero</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0_omega")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="_Omega" class="idref" href="#_Omega"><span class="id" title="abbreviation">_Omega</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0_succ" class="idref" href="#E0_succ"><span class="id" title="instance">E0_succ</span></a> (<a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0_succ")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Succ" class="idref" href="#Succ"><span class="id" title="abbreviation">Succ</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0limit" class="idref" href="#E0limit"><span class="id" title="definition">E0limit</span></a> (<a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=  <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:7"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0limit")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Limitb" class="idref" href="#Limitb"><span class="id" title="abbreviation">Limitb</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0is_succ" class="idref" href="#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> (<a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:8"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0is_succ")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="E0succb" class="idref" href="#E0succb"><span class="id" title="abbreviation">E0succb</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0one" class="idref" href="#E0one"><span class="id" title="instance">E0one</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>:= @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0add" class="idref" href="#E0add"><span class="id" title="instance">E0add</span></a> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1add</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span>) (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="keyword">beta</span>))<span class="id" title="var">_</span> .<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="829ecb68b8b34ee91d22b1ffdd27a15c" class="idref" href="#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.Epsilon0.E0.html#E0add"><span class="id" title="instance">E0add</span></a> : <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0add")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Plus" class="idref" href="#Plus"><span class="id" title="abbreviation">Plus</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0add"><span class="id" title="instance">E0add</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0_phi0" class="idref" href="#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="c17b9ed2f114699671bc84e75652badb" class="idref" href="#c17b9ed2f114699671bc84e75652badb"><span class="id" title="notation">&quot;</span></a>'E0_omega^'" := <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="Omega_term" class="idref" href="#Omega_term"><span class="id" title="instance">Omega_term</span></a> (<a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span>) <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Cons" class="idref" href="#Cons"><span class="id" title="instance">Cons</span></a> (<a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="beta:16" class="idref" href="#beta:16"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a><br/>
+&nbsp;&nbsp;:= (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">e0</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0finS" class="idref" href="#E0finS"><span class="id" title="instance">E0finS</span></a> (<a id="i:17" class="idref" href="#i:17"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>:= @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">i</span>)%<span class="id" title="var">t1</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0fin" class="idref" href="#E0fin"><span class="id" title="instance">E0fin</span></a> (<a id="i:18" class="idref" href="#i:18"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> ⇒ <a class="idref" href="hydras.Epsilon0.E0.html#E0finS"><span class="id" title="instance">E0finS</span></a> <span class="id" title="var">i</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0fin")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Fin" class="idref" href="#Fin"><span class="id" title="abbreviation">Fin</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">:</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">nat</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">&gt;-&gt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="E0mul" class="idref" href="#E0mul"><span class="id" title="instance">E0mul</span></a> (<a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a> <a id="beta:21" class="idref" href="#beta:21"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">t1</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use E0mul")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="Mult" class="idref" href="#Mult"><span class="id" title="abbreviation">Mult</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0mul"><span class="id" title="instance">E0mul</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="c28e4e4c810917276f0357982f044110" class="idref" href="#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.Epsilon0.E0.html#E0mul"><span class="id" title="instance">E0mul</span></a> : <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="Mult_i" class="idref" href="#Mult_i"><span class="id" title="instance">Mult_i</span></a>  (<a id="alpha:22" class="idref" href="#alpha:22"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <span class="id" title="var">n</span>)%<span class="id" title="var">t1</span>  <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab35"></a><h2 class="section">Lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab36"></a><h2 class="section">On equality on type <span class="inlinecode"><span class="id" title="var">E0</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_proof_unicity" class="idref" href="#nf_proof_unicity"><span class="id" title="lemma">nf_proof_unicity</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="alpha:26" class="idref" href="#alpha:26"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="H:27" class="idref" href="#H:27"><span class="id" title="binder">H</span></a> <a id="H':28" class="idref" href="#H':28"><span class="id" title="binder">H'</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:26"><span class="id" title="variable">alpha</span></a>), <a class="idref" href="hydras.Epsilon0.E0.html#H:27"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#H':28"><span class="id" title="variable">H'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_eq_intro" class="idref" href="#E0_eq_intro"><span class="id" title="lemma">E0_eq_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:29" class="idref" href="#alpha:29"><span class="id" title="binder">alpha</span></a> <a id="beta:30" class="idref" href="#beta:30"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:30"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:30"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="E0_eq_iff" class="idref" href="#E0_eq_iff"><span class="id" title="lemma">E0_eq_iff</span></a> (<a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a> <a id="beta:32" class="idref" href="#beta:32"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:32"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:32"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Le_iff" class="idref" href="#Le_iff"><span class="id" title="lemma">Le_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:33" class="idref" href="#alpha:33"><span class="id" title="binder">alpha</span></a> <a id="beta:34" class="idref" href="#beta:34"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0le"><span class="id" title="definition">E0le</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:33"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:34"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LE"><span class="id" title="definition">T1.LE</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:33"><span class="id" title="variable">alpha</span></a>) (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:34"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succb_Succ" class="idref" href="#Succb_Succ"><span class="id" title="lemma">Succb_Succ</span></a> <a id="alpha:35" class="idref" href="#alpha:35"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:36" class="idref" href="#beta:36"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:36"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">E0_eq_intro</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">orefl</span> := (<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_intro"><span class="id" title="lemma">E0_eq_intro</span></a>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">reflexivity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">ochange</span> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span> := (<span class="id" title="tactic">replace</span> <span class="id" title="var">alpha</span> <span class="id" title="keyword">with</span> <span class="id" title="keyword">beta</span>; [| <span class="id" title="var">orefl</span>]).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_eq_dec" class="idref" href="#E0_eq_dec"><span class="id" title="lemma">E0_eq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:37" class="idref" href="#alpha:37"><span class="id" title="binder">alpha</span></a> <a id="beta:38" class="idref" href="#beta:38"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#alpha:37"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:38"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#alpha:37"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:38"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab37"></a><h2 class="section">Adaptation to <span class="inlinecode"><span class="id" title="var">E0</span></span> of lemmas about <span class="inlinecode"><span class="id" title="var">T1</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="alpha_plus_zero" class="idref" href="#alpha_plus_zero"><span class="id" title="lemma">alpha_plus_zero</span></a> <a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:39"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.E0.html#alpha_plus_zero"><span class="id" title="lemma">alpha_plus_zero</span></a> : <span class="id" title="var">E0_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_phi0" class="idref" href="#cnf_phi0"><span class="id" title="lemma">cnf_phi0</span></a> (<a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:40"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:40"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_Succ" class="idref" href="#cnf_Succ"><span class="id" title="lemma">cnf_Succ</span></a> (<a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:41"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:41"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_Omega_term" class="idref" href="#cnf_Omega_term"><span class="id" title="lemma">cnf_Omega_term</span></a> (<a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#n:43"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:42"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.E0.html#n:43"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_Omega_term" class="idref" href="#T1limit_Omega_term"><span class="id" title="lemma">T1limit_Omega_term</span></a> <a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a> <a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:44"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:44"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:45"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_phi0" class="idref" href="#T1limit_phi0"><span class="id" title="lemma">T1limit_phi0</span></a> <a id="alpha:46" class="idref" href="#alpha:46"><span class="id" title="binder">alpha</span></a>  : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:46"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:46"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T1limit_phi0</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_Limit_Succ_dec" class="idref" href="#Zero_Limit_Succ_dec"><span class="id" title="definition">Zero_Limit_Succ_dec</span></a> (<a id="alpha:47" class="idref" href="#alpha:47"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:48" class="idref" href="#beta:48"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:48"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="E0_pred" class="idref" href="#E0_pred"><span class="id" title="definition">E0_pred</span></a> (<a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.E0.html#Zero_Limit_Succ_dec"><span class="id" title="definition">Zero_Limit_Succ_dec</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#alpha:49"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.E0.html#alpha:49"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> <span class="id" title="keyword">beta</span> <span class="id" title="var">_</span>) ⇒ <span class="id" title="keyword">beta</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> "mko" <span class="id" title="keyword">constr</span>(<span class="id" title="var">alpha</span>) := <span class="id" title="tactic">refine</span> (@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq_refl"><span class="id" title="constructor">eq_refl</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Two" class="idref" href="#Two"><span class="id" title="instance">Two</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=  <span class="id" title="keyword">ltac</span>:(<span class="id" title="var">mko</span> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> 2)).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Omega_2" class="idref" href="#Omega_2"><span class="id" title="instance">Omega_2</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<span class="id" title="keyword">ltac</span>:(<span class="id" title="var">mko</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>)%<span class="id" title="var">t1</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="E0_sto" class="idref" href="#E0_sto"><span class="id" title="instance">E0_sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="compare_E0" class="idref" href="#compare_E0"><span class="id" title="instance">compare_E0</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:50" class="idref" href="#alpha:50"><span class="id" title="binder">alpha</span></a> <a id="beta:51" class="idref" href="#beta:51"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:50"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:51"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> (<a id="alpha:52" class="idref" href="#alpha:52"><span class="id" title="binder">alpha</span></a> <a id="beta:53" class="idref" href="#beta:53"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:53"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:53"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0lt_wf" class="idref" href="#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">E0lt_wf</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lt_Succ_Le" class="idref" href="#Lt_Succ_Le"><span class="id" title="lemma">Lt_Succ_Le</span></a> (<a id="alpha:54" class="idref" href="#alpha:54"><span class="id" title="binder">alpha</span></a> <a id="beta:55" class="idref" href="#beta:55"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>):  <a class="idref" href="hydras.Epsilon0.E0.html#beta:55"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:54"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:55"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:54"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_pred_of_Succ" class="idref" href="#E0_pred_of_Succ"><span class="id" title="lemma">E0_pred_of_Succ</span></a> (<a id="alpha:56" class="idref" href="#alpha:56"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:56"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:56"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_pred_of_Succ"><span class="id" title="lemma">E0_pred_of_Succ</span></a>: <span class="id" title="var">E0_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_inj" class="idref" href="#Succ_inj"><span class="id" title="lemma">Succ_inj</span></a> <a id="alpha:57" class="idref" href="#alpha:57"><span class="id" title="binder">alpha</span></a> <a id="beta:58" class="idref" href="#beta:58"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:57"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:58"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:57"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:58"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="E0_pred_Lt" class="idref" href="#E0_pred_Lt"><span class="id" title="lemma">E0_pred_Lt</span></a> (<a id="alpha:59" class="idref" href="#alpha:59"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:59"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">E0_pred_Lt</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_Succb" class="idref" href="#Succ_Succb"><span class="id" title="lemma">Succ_Succb</span></a> (<a id="alpha:60" class="idref" href="#alpha:60"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:60"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Succ_Succb</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">ord_eq</span> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span> := <span class="id" title="tactic">assert</span> (<span class="id" title="var">alpha</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">beta</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_intro"><span class="id" title="lemma">E0_eq_intro</span></a> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">reflexivity</span>|].<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FinS_eq" class="idref" href="#FinS_eq"><span class="id" title="lemma">FinS_eq</span></a> <a id="i:61" class="idref" href="#i:61"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0finS"><span class="id" title="instance">E0finS</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:61"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_fin_rw" class="idref" href="#mult_fin_rw"><span class="id" title="lemma">mult_fin_rw</span></a> <a id="alpha:62" class="idref" href="#alpha:62"><span class="id" title="binder">alpha</span></a> (<a id="i:63" class="idref" href="#i:63"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:62"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:63"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#Mult_i"><span class="id" title="instance">Mult_i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:62"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:63"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="FinS_Succ_eq" class="idref" href="#FinS_Succ_eq"><span class="id" title="lemma">FinS_Succ_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="i:64" class="idref" href="#i:64"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0finS"><span class="id" title="instance">E0finS</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:64"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:64"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.E0.html#FinS_Succ_eq"><span class="id" title="lemma">FinS_Succ_eq</span></a> : <span class="id" title="var">E0_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Limit_not_Zero" class="idref" href="#Limit_not_Zero"><span class="id" title="lemma">Limit_not_Zero</span></a> <a id="alpha:65" class="idref" href="#alpha:65"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_not_Zero" class="idref" href="#Succ_not_Zero"><span class="id" title="lemma">Succ_not_Zero</span></a> <a id="alpha:66" class="idref" href="#alpha:66"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:66"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:66"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lt_Succ" class="idref" href="#Lt_Succ"><span class="id" title="lemma">Lt_Succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:67" class="idref" href="#alpha:67"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:67"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:67"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_not_T1limit" class="idref" href="#Succ_not_T1limit"><span class="id" title="lemma">Succ_not_T1limit</span></a> <a id="alpha:68" class="idref" href="#alpha:68"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:68"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:68"><span class="id" title="variable">alpha</span></a> .<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Limit_not_Zero</span> <span class="id" title="var">Succ_not_Zero</span> <span class="id" title="var">Lt_Succ</span> <span class="id" title="var">Succ_not_T1limit</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_Succ_inv" class="idref" href="#lt_Succ_inv"><span class="id" title="lemma">lt_Succ_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:69" class="idref" href="#alpha:69"><span class="id" title="binder">alpha</span></a> <a id="beta:70" class="idref" href="#beta:70"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#beta:70"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:69"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:70"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:69"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_Succ_le_2" class="idref" href="#lt_Succ_le_2"><span class="id" title="lemma">lt_Succ_le_2</span></a> (<a id="alpha:71" class="idref" href="#alpha:71"><span class="id" title="binder">alpha</span></a> <a id="beta:72" class="idref" href="#beta:72"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:71"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:72"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:71"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:72"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_lt_T1limit" class="idref" href="#Succ_lt_T1limit"><span class="id" title="lemma">Succ_lt_T1limit</span></a> <a id="alpha:73" class="idref" href="#alpha:73"><span class="id" title="binder">alpha</span></a> <a id="beta:74" class="idref" href="#beta:74"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#beta:74"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:74"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:73"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_eq_dec" class="idref" href="#le_lt_eq_dec"><span class="id" title="lemma">le_lt_eq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:75" class="idref" href="#alpha:75"><span class="id" title="binder">alpha</span></a> <a id="beta:76" class="idref" href="#beta:76"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#alpha:75"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:76"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#alpha:75"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:76"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.E0.html#alpha:75"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:76"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="E0_comp" class="idref" href="#E0_comp"><span class="id" title="instance">E0_comp</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Epsilon0" class="idref" href="#Epsilon0"><span class="id" title="instance">Epsilon0</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_limit_succ_dec" class="idref" href="#Zero_limit_succ_dec"><span class="id" title="definition">Zero_limit_succ_dec</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec"><span class="id" title="definition">ZeroLimitSucc_dec</span></a> .<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab38"></a><h2 class="section">Rewriting lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_rw" class="idref" href="#Succ_rw"><span class="id" title="lemma">Succ_rw</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:77" class="idref" href="#alpha:77"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:77"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:77"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Plus_rw" class="idref" href="#Plus_rw"><span class="id" title="lemma">Plus_rw</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:78" class="idref" href="#alpha:78"><span class="id" title="binder">alpha</span></a> <a id="beta:79" class="idref" href="#beta:79"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#alpha:78"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:79"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:78"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:79"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lt_trans" class="idref" href="#Lt_trans"><span class="id" title="lemma">Lt_trans</span></a> <a id="alpha:80" class="idref" href="#alpha:80"><span class="id" title="binder">alpha</span></a> <a id="beta:81" class="idref" href="#beta:81"><span class="id" title="binder">beta</span></a> <a id="gamma:82" class="idref" href="#gamma:82"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:82"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:82"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Le_trans" class="idref" href="#Le_trans"><span class="id" title="lemma">Le_trans</span></a> <a id="alpha:83" class="idref" href="#alpha:83"><span class="id" title="binder">alpha</span></a> <a id="beta:84" class="idref" href="#beta:84"><span class="id" title="binder">beta</span></a> <a id="gamma:85" class="idref" href="#gamma:85"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:83"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:84"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:84"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:85"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:83"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:85"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Omega_term_plus" class="idref" href="#Omega_term_plus"><span class="id" title="lemma">Omega_term_plus</span></a> <a id="alpha:86" class="idref" href="#alpha:86"><span class="id" title="binder">alpha</span></a> <a id="beta:87" class="idref" href="#beta:87"><span class="id" title="binder">beta</span></a> <a id="i:88" class="idref" href="#i:88"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:86"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#beta:87"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:86"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:86"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:88"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:87"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:86"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.E0.html#i:88"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:87"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_Cons" class="idref" href="#cnf_Cons"><span class="id" title="lemma">cnf_Cons</span></a> (<a id="alpha:89" class="idref" href="#alpha:89"><span class="id" title="binder">alpha</span></a> <a id="beta:90" class="idref" href="#beta:90"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) <a id="n:91" class="idref" href="#n:91"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:89"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:90"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:89"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Cons"><span class="id" title="instance">Cons</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:89"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#n:91"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:90"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:89"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.E0.html#n:91"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:90"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_plus" class="idref" href="#T1limit_plus"><span class="id" title="lemma">T1limit_plus</span></a> <a id="alpha:92" class="idref" href="#alpha:92"><span class="id" title="binder">alpha</span></a> <a id="beta:93" class="idref" href="#beta:93"><span class="id" title="binder">beta</span></a> <a id="i:94" class="idref" href="#i:94"><span class="id" title="binder">i</span></a>:<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.E0.html#beta:93"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:92"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:93"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a>  (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:92"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:94"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:93"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">e0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Succ_of_cons" class="idref" href="#Succ_of_cons"><span class="id" title="lemma">Succ_of_cons</span></a> <a id="alpha:95" class="idref" href="#alpha:95"><span class="id" title="binder">alpha</span></a> <a id="gamma:96" class="idref" href="#gamma:96"><span class="id" title="binder">gamma</span></a> <a id="i:97" class="idref" href="#i:97"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:96"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:97"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:96"><span class="id" title="variable">gamma</span></a>)%<span class="id" title="var">e0</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:97"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:96"><span class="id" title="variable">gamma</span></a>)%<span class="id" title="var">e0</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="Oplus" class="idref" href="#Oplus"><span class="id" title="instance">Oplus</span></a> (<a id="alpha:98" class="idref" href="#alpha:98"><span class="id" title="binder">alpha</span></a> <a id="beta:99" class="idref" href="#beta:99"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> :=<br/>
+@<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="var">alpha</span>) (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <span class="id" title="keyword">beta</span>)) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="bae485d6f56c6bf81e7b6584053562d1" class="idref" href="#bae485d6f56c6bf81e7b6584053562d1"><span class="id" title="notation">&quot;</span></a>O+" := <a class="idref" href="hydras.Epsilon0.E0.html#Oplus"><span class="id" title="instance">Oplus</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>): <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Oplus_assoc" class="idref" href="#Oplus_assoc"><span class="id" title="instance">Oplus_assoc</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#Oplus"><span class="id" title="instance">Oplus</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oPlus_rw" class="idref" href="#oPlus_rw"><span class="id" title="lemma">oPlus_rw</span></a> (<a id="alpha:100" class="idref" href="#alpha:100"><span class="id" title="binder">alpha</span></a> <a id="beta:101" class="idref" href="#beta:101"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#alpha:100"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#bae485d6f56c6bf81e7b6584053562d1"><span class="id" title="notation">O</span></a><a class="idref" href="hydras.Epsilon0.E0.html#bae485d6f56c6bf81e7b6584053562d1"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:101"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:100"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:101"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="L_3_plus_omega" class="idref" href="#L_3_plus_omega"><span class="id" title="definition">L_3_plus_omega</span></a> :  3 <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_correct" class="idref" href="#succ_correct"><span class="id" title="lemma">succ_correct</span></a> <a id="alpha:102" class="idref" href="#alpha:102"><span class="id" title="binder">alpha</span></a> <a id="beta:103" class="idref" href="#beta:103"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:103"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:102"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:103"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:102"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Le_refl" class="idref" href="#Le_refl"><span class="id" title="lemma">Le_refl</span></a> <a id="alpha:104" class="idref" href="#alpha:104"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:104"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lt_Le_incl" class="idref" href="#Lt_Le_incl"><span class="id" title="lemma">Lt_Le_incl</span></a> <a id="alpha:105" class="idref" href="#alpha:105"><span class="id" title="binder">alpha</span></a> <a id="beta:106" class="idref" href="#beta:106"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:105"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:106"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:105"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:106"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_Lt_irrefl" class="idref" href="#E0_Lt_irrefl"><span class="id" title="lemma">E0_Lt_irrefl</span></a> (<a id="alpha:107" class="idref" href="#alpha:107"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:107"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:107"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_Lt_Succ_inv" class="idref" href="#E0_Lt_Succ_inv"><span class="id" title="lemma">E0_Lt_Succ_inv</span></a> (<a id="alpha:108" class="idref" href="#alpha:108"><span class="id" title="binder">alpha</span></a> <a id="beta:109" class="idref" href="#beta:109"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:109"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:109"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:109"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_not_Lt_zero" class="idref" href="#E0_not_Lt_zero"><span class="id" title="lemma">E0_not_Lt_zero</span></a> <a id="alpha:110" class="idref" href="#alpha:110"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_inv" class="idref" href="#lt_omega_inv"><span class="id" title="lemma">lt_omega_inv</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:111" class="idref" href="#alpha:111"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>,  <a class="idref" href="hydras.Epsilon0.E0.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="i:112" class="idref" href="#i:112"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#i:112"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_lt_eq_lt" class="idref" href="#E0_lt_eq_lt"><span class="id" title="lemma">E0_lt_eq_lt</span></a> <a id="alpha:113" class="idref" href="#alpha:113"><span class="id" title="binder">alpha</span></a> <a id="beta:114" class="idref" href="#beta:114"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:113"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="E0_lt_ge" class="idref" href="#E0_lt_ge"><span class="id" title="lemma">E0_lt_ge</span></a> <a id="alpha:115" class="idref" href="#alpha:115"><span class="id" title="binder">alpha</span></a> <a id="beta:116" class="idref" href="#beta:116"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:115"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:116"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:116"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:115"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Limit_gt_Zero" class="idref" href="#Limit_gt_Zero"><span class="id" title="lemma">Limit_gt_Zero</span></a> (<a id="alpha:117" class="idref" href="#alpha:117"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:117"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:117"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono" class="idref" href="#phi0_mono"><span class="id" title="lemma">phi0_mono</span></a> <a id="alpha:118" class="idref" href="#alpha:118"><span class="id" title="binder">alpha</span></a> <a id="beta:119" class="idref" href="#beta:119"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#alpha:118"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:119"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:118"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:119"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="plus_assoc" class="idref" href="#plus_assoc"><span class="id" title="instance">plus_assoc</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0add"><span class="id" title="instance">E0add</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="mult_plus_distr_l" class="idref" href="#mult_plus_distr_l"><span class="id" title="lemma">mult_plus_distr_l</span></a> (<a id="alpha:120" class="idref" href="#alpha:120"><span class="id" title="binder">alpha</span></a> <a id="beta:121" class="idref" href="#beta:121"><span class="id" title="binder">beta</span></a> <a id="gamma:122" class="idref" href="#gamma:122"><span class="id" title="binder">gamma</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#alpha:120"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.E0.html#beta:121"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:122"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:120"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#beta:121"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#alpha:120"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#gamma:122"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex42" class="idref" href="#Ex42"><span class="id" title="definition">Ex42</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> 42 <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c17b9ed2f114699671bc84e75652badb"><span class="id" title="notation">E0_omega</span></a><a class="idref" href="hydras.Epsilon0.E0.html#c17b9ed2f114699671bc84e75652badb"><span class="id" title="notation">^2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c17b9ed2f114699671bc84e75652badb"><span class="id" title="notation">E0_omega</span></a><a class="idref" href="hydras.Epsilon0.E0.html#c17b9ed2f114699671bc84e75652badb"><span class="id" title="notation">^2</span></a>.  </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.Epsilon0.html b/theories/html/hydras.Epsilon0.Epsilon0.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Epsilon0.Epsilon0.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<h1 class="libtitle">Library hydras.Epsilon0.Epsilon0</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="library">Large_Sets</span></a>.<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Epsilon0rpo</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Epsilon0rpo</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Pierre Casteran 
+    LaBRI, University of Bordeaux
+
+<div class="paragraph"> </div>
+
+    Evelyne Contejean
+    LRI.
+
+<div class="paragraph"> </div>
+
+ Cantor "pre" Normal form
+ After Manolios and Vroon work on ACL2 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>  <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#"><span class="id" title="library">Logic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Restriction</span></a>   <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>  <a class="idref" href="hydras.rpo.term.html#"><span class="id" title="library">rpo.term</span></a>  <a class="idref" href="hydras.rpo.rpo.html#"><span class="id" title="library">rpo.rpo</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">Epsilon0.T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Create HintDb</span> <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt" class="idref" href="#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span>  <a id="Alt.Eps0_sig" class="idref" href="#Alt.Eps0_sig"><span class="id" title="module">Eps0_sig</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Signature"><span class="id" title="module">Signature</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Alt.Eps0_sig.symb0" class="idref" href="#Alt.Eps0_sig.symb0"><span class="id" title="inductive">symb0</span></a> : <span class="id" title="keyword">Set</span> := <a id="Alt.Eps0_sig.nat_0" class="idref" href="#Alt.Eps0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a> | <a id="Alt.Eps0_sig.nat_S" class="idref" href="#Alt.Eps0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> | <a id="Alt.Eps0_sig.ord_zero" class="idref" href="#Alt.Eps0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> | <a id="Alt.Eps0_sig.ord_cons" class="idref" href="#Alt.Eps0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Eps0_sig.symb" class="idref" href="#Alt.Eps0_sig.symb"><span class="id" title="definition">symb</span></a> := <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb0"><span class="id" title="inductive">symb0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.Eps0_sig.eq_symbol_dec" class="idref" href="#Alt.Eps0_sig.eq_symbol_dec"><span class="id" title="lemma">eq_symbol_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="f1:3" class="idref" href="#f1:3"><span class="id" title="binder">f1</span></a> <a id="f2:4" class="idref" href="#f2:4"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb"><span class="id" title="definition">symb</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f1:3"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f2:4"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f1:3"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f2:4"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+The arity of a symbol contains also the information about built-in theories as in CiME 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Alt.Eps0_sig.arity_type" class="idref" href="#Alt.Eps0_sig.arity_type"><span class="id" title="inductive">arity_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Alt.Eps0_sig.AC" class="idref" href="#Alt.Eps0_sig.AC"><span class="id" title="constructor">AC</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Alt.Eps0_sig.C" class="idref" href="#Alt.Eps0_sig.C"><span class="id" title="constructor">C</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Alt.Eps0_sig.Free" class="idref" href="#Alt.Eps0_sig.Free"><span class="id" title="constructor">Free</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Eps0_sig.arity" class="idref" href="#Alt.Eps0_sig.arity"><span class="id" title="definition">arity</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb"><span class="id" title="definition">symb</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.arity_type"><span class="id" title="inductive">arity_type</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f:7"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free"><span class="id" title="constructor">Free</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free"><span class="id" title="constructor">Free</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free"><span class="id" title="constructor">Free</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free"><span class="id" title="constructor">Free</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig"><span class="id" title="module">Eps0_sig</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab29"></a><h1 class="section">Module Type Variables.</h1>
+
+ There are almost no assumptions, except a decidable equality. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt.Vars" class="idref" href="#Alt.Vars"><span class="id" title="module">Vars</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Variables"><span class="id" title="module">Variables</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Alt.Vars.empty_set" class="idref" href="#Alt.Vars.empty_set"><span class="id" title="inductive">empty_set</span></a> : <span class="id" title="keyword">Set</span> := .<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Vars.var" class="idref" href="#Alt.Vars.var"><span class="id" title="definition">var</span></a> := <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.empty_set"><span class="id" title="inductive">empty_set</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.Vars.eq_variable_dec" class="idref" href="#Alt.Vars.eq_variable_dec"><span class="id" title="lemma">eq_variable_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="v1:11" class="idref" href="#v1:11"><span class="id" title="binder">v1</span></a> <a id="v2:12" class="idref" href="#v2:12"><span class="id" title="binder">v2</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.var"><span class="id" title="definition">var</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#v1:11"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#v2:12"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#v1:11"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#v2:12"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars"><span class="id" title="module">Vars</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span>  <a id="Alt.Eps0_prec" class="idref" href="#Alt.Eps0_prec"><span class="id" title="module">Eps0_prec</span></a> &lt;: <a class="idref" href="hydras.rpo.rpo.html#Precedence"><span class="id" title="module">Precedence</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Eps0_prec.A" class="idref" href="#Alt.Eps0_prec.A"><span class="id" title="definition">A</span></a> : <span class="id" title="keyword">Set</span> := <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb"><span class="id" title="definition">Eps0_sig.symb</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig"><span class="id" title="module">Eps0_sig</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Eps0_prec.prec" class="idref" href="#Alt.Eps0_prec.prec"><span class="id" title="definition">prec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A"><span class="id" title="definition">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> <a id="g:14" class="idref" href="#g:14"><span class="id" title="binder">g</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#f:13"><span class="id" title="variable">f</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#g:14"><span class="id" title="variable">g</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Alt.Eps0_prec.status_type" class="idref" href="#Alt.Eps0_prec.status_type"><span class="id" title="inductive">status_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Alt.Eps0_prec.Lex" class="idref" href="#Alt.Eps0_prec.Lex"><span class="id" title="constructor">Lex</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#status_type:17"><span class="id" title="inductive">status_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Alt.Eps0_prec.Mul" class="idref" href="#Alt.Eps0_prec.Mul"><span class="id" title="constructor">Mul</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#status_type:17"><span class="id" title="inductive">status_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Alt.Eps0_prec.status" class="idref" href="#Alt.Eps0_prec.status"><span class="id" title="definition">status</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A"><span class="id" title="definition">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.status_type"><span class="id" title="inductive">status_type</span></a> := <span class="id" title="keyword">fun</span> <a id="f:19" class="idref" href="#f:19"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.Lex"><span class="id" title="constructor">Lex</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.Eps0_prec.prec_dec" class="idref" href="#Alt.Eps0_prec.prec_dec"><span class="id" title="lemma">prec_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:20" class="idref" href="#a1:20"><span class="id" title="binder">a1</span></a> <a id="a2:21" class="idref" href="#a2:21"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A"><span class="id" title="definition">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a1:20"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a2:21"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a1:20"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a2:21"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.Eps0_prec.prec_antisym" class="idref" href="#Alt.Eps0_prec.prec_antisym"><span class="id" title="lemma">prec_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="s:22" class="idref" href="#s:22"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#s:22"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#s:22"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.Eps0_prec.prec_transitive" class="idref" href="#Alt.Eps0_prec.prec_transitive"><span class="id" title="lemma">prec_transitive</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A"><span class="id" title="definition">A</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec"><span class="id" title="definition">prec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec"><span class="id" title="module">Eps0_prec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt.Eps0_alg" class="idref" href="#Alt.Eps0_alg"><span class="id" title="module">Eps0_alg</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">Term</span></a> := <a class="idref" href="hydras.rpo.term.html#Make"><span class="id" title="module">term.Make</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig"><span class="id" title="module">Eps0_sig</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars"><span class="id" title="module">Vars</span></a>).<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt.Eps0_rpo" class="idref" href="#Alt.Eps0_rpo"><span class="id" title="module">Eps0_rpo</span></a> &lt;: <a class="idref" href="hydras.rpo.rpo.html#RPO"><span class="id" title="module">RPO</span></a> := <a class="idref" href="hydras.rpo.rpo.html#Make"><span class="id" title="module">rpo.Make</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_alg"><span class="id" title="module">Eps0_alg</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec"><span class="id" title="module">Eps0_prec</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_alg"><span class="id" title="module">Eps0_alg</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_rpo"><span class="id" title="module">Eps0_rpo</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig"><span class="id" title="module">Eps0_sig</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Alt.nat_2_term" class="idref" href="#Alt.nat_2_term"><span class="id" title="definition">nat_2_term</span></a> (<a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:23"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_0"><span class="id" title="constructor">nat_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_S"><span class="id" title="constructor">nat_S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#nat_2_term:24"><span class="id" title="definition">nat_2_term</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab30"></a><h1 class="section"> </h1>
+
+Every (representation of a) natural number is less than
+ any non zero ordinal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.nat_lt_cons" class="idref" href="#Alt.nat_lt_cons"><span class="id" title="lemma">nat_lt_cons</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="p:28" class="idref" href="#p:28"><span class="id" title="binder">p</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:26"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:27"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#p:28"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:29"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Alt.rpo_trans" class="idref" href="#Alt.rpo_trans"><span class="id" title="lemma">rpo_trans</span></a> (<a id="t:30" class="idref" href="#t:30"><span class="id" title="binder">t</span></a> <a id="t1:31" class="idref" href="#t1:31"><span class="id" title="binder">t1</span></a> <a id="t2:32" class="idref" href="#t2:32"><span class="id" title="binder">t2</span></a>: <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.term"><span class="id" title="inductive">term</span></a>): <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t:30"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t1:31"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t1:31"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t2:32"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t:30"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#t2:32"><span class="id" title="variable">t2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Alt.T1_2_term" class="idref" href="#Alt.T1_2_term"><span class="id" title="definition">T1_2_term</span></a> (<a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.term"><span class="id" title="inductive">term</span></a> := <br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:33"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.ord_zero"><span class="id" title="constructor">ord_zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#T1_2_term:34"><span class="id" title="definition">T1_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#T1_2_term:34"><span class="id" title="definition">T1_2_term</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Alt.T1_size" class="idref" href="#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> (<a id="o:36" class="idref" href="#o:36"><span class="id" title="binder">o</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:36"><span class="id" title="variable">o</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#T1_size:37"><span class="id" title="definition">T1_size</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#T1_size:37"><span class="id" title="definition">T1_size</span></a> <span class="id" title="var">b</span>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.T1_size1" class="idref" href="#Alt.T1_size1"><span class="id" title="lemma">T1_size1</span></a> <a id="a:39" class="idref" href="#a:39"><span class="id" title="binder">a</span></a> <a id="n:40" class="idref" href="#n:40"><span class="id" title="binder">n</span></a> <a id="b:41" class="idref" href="#b:41"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:39"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:41"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.T1_size2" class="idref" href="#Alt.T1_size2"><span class="id" title="lemma">T1_size2</span></a> <a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a> <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:43"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:44"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.T1_size3" class="idref" href="#Alt.T1_size3"><span class="id" title="lemma">T1_size3</span></a>  <a id="a:45" class="idref" href="#a:45"><span class="id" title="binder">a</span></a> <a id="n:46" class="idref" href="#n:46"><span class="id" title="binder">n</span></a> <a id="b:47" class="idref" href="#b:47"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:45"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:46"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:47"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T1_size2</span> <span class="id" title="var">T1_size3</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+let us recall subterm properties on T1 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.lt_subterm1" class="idref" href="#Alt.lt_subterm1"><span class="id" title="lemma">lt_subterm1</span></a> <a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="a':49" class="idref" href="#a':49"><span class="id" title="binder">a'</span></a>  <a id="n':50" class="idref" href="#n':50"><span class="id" title="binder">n'</span></a>  <a id="b':51" class="idref" href="#b':51"><span class="id" title="binder">b'</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:48"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a':49"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:48"><span class="id" title="variable">a</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a':49"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n':50"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b':51"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.lt_subterm2" class="idref" href="#Alt.lt_subterm2"><span class="id" title="lemma">lt_subterm2</span></a>  <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a> <a id="a':53" class="idref" href="#a':53"><span class="id" title="binder">a'</span></a> <a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a> <a id="n':55" class="idref" href="#n':55"><span class="id" title="binder">n'</span></a> <a id="b:56" class="idref" href="#b:56"><span class="id" title="binder">b</span></a> <a id="b':57" class="idref" href="#b':57"><span class="id" title="binder">b'</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:52"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a':53"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:54"><span class="id" title="variable">n</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:56"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a':53"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n':55"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b':57"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b:56"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a':53"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n':55"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#b':57"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nat_lt_cons</span> : <span class="id" title="var">rpo</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">lt_subterm2</span> <span class="id" title="var">lt_subterm1</span> : <span class="id" title="var">rpo</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T1_size3</span> <span class="id" title="var">T1_size2</span> <span class="id" title="var">T1_size1</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.nat_2_term_mono" class="idref" href="#Alt.nat_2_term_mono"><span class="id" title="lemma">nat_2_term_mono</span></a> (<a id="n:58" class="idref" href="#n:58"><span class="id" title="binder">n</span></a> <a id="n':59" class="idref" href="#n':59"><span class="id" title="binder">n'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:58"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n':59"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:58"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n':59"><span class="id" title="variable">n'</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Alt.lt_inc_rpo_0" class="idref" href="#Alt.lt_inc_rpo_0"><span class="id" title="lemma">lt_inc_rpo_0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:60" class="idref" href="#n:60"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="o':61" class="idref" href="#o':61"><span class="id" title="binder">o'</span></a> <a id="o:62" class="idref" href="#o:62"><span class="id" title="binder">o</span></a>, (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:62"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size"><span class="id" title="definition">T1_size</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':61"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#n:60"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:62"><span class="id" title="variable">o</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':61"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:62"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':61"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term"><span class="id" title="definition">T1_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:62"><span class="id" title="variable">o</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term"><span class="id" title="definition">T1_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':61"><span class="id" title="variable">o'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Alt.R1" class="idref" href="#Alt.R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_0"><span class="id" title="constructor">nat_0</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R1</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Alt.R2" class="idref" href="#Alt.R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.ord_zero"><span class="id" title="constructor">ord_zero</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R2</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Alt.R3" class="idref" href="#Alt.R3"><span class="id" title="lemma">R3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_S"><span class="id" title="constructor">nat_S</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R3</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Alt.R4" class="idref" href="#Alt.R4"><span class="id" title="lemma">R4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.ord_cons"><span class="id" title="constructor">ord_cons</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R4</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Alt.well_founded_rpo" class="idref" href="#Alt.well_founded_rpo"><span class="id" title="lemma">well_founded_rpo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span>  <a id="Alt.well_founded" class="idref" href="#Alt.well_founded"><span class="id" title="section">well_founded</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Alt.well_founded.R" class="idref" href="#Alt.well_founded.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <span class="id" title="var">R</span> : <span class="id" title="var">rpo</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.R_inc_rpo" class="idref" href="#Alt.R_inc_rpo"><span class="id" title="lemma">R_inc_rpo</span></a> (<a id="o:63" class="idref" href="#o:63"><span class="id" title="binder">o</span></a> <a id="o':64" class="idref" href="#o':64"><span class="id" title="binder">o'</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:63"><span class="id" title="variable">o</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':64"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term"><span class="id" title="definition">T1_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o:63"><span class="id" title="variable">o</span></a>) (<a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term"><span class="id" title="definition">T1_2_term</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#o':64"><span class="id" title="variable">o'</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.nf_Wf" class="idref" href="#Alt.nf_Wf"><span class="id" title="lemma">nf_Wf</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For compatibility with T1 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Alt.nf_Acc" class="idref" href="#Alt.nf_Acc"><span class="id" title="lemma">nf_Acc</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:69" class="idref" href="#alpha:69"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#alpha:69"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#alpha:69"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded"><span class="id" title="section">well_founded</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Alt.transfinite_recursor_lt" class="idref" href="#Alt.transfinite_recursor_lt"><span class="id" title="definition">transfinite_recursor_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:70" class="idref" href="#P:70"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:71" class="idref" href="#x:71"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:71"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:72" class="idref" href="#y:72"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#y:72"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#y:72"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:71"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:70"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#y:72"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:70"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:71"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:73" class="idref" href="#a:73"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:73"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:70"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:73"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Alt.transfinite_recursor" class="idref" href="#Alt.transfinite_recursor"><span class="id" title="definition">transfinite_recursor</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:74" class="idref" href="#P:74"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:76" class="idref" href="#y:76"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#y:76"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:74"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#y:76"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:74"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#x:75"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:77" class="idref" href="#a:77"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#P:74"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#a:77"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">transfinite_induction_lt</span> <span class="id" title="var">a</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">pattern</span> <span class="id" title="var">a</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor_lt"><span class="id" title="definition">transfinite_recursor_lt</span></a> ;[ | <span class="id" title="tactic">try</span> <span class="id" title="tactic">assumption</span>].<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">transfinite_induction</span> <span class="id" title="var">a</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">pattern</span> <span class="id" title="var">a</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor"><span class="id" title="definition">transfinite_recursor</span></a>;[ | <span class="id" title="tactic">try</span> <span class="id" title="tactic">assumption</span>].<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Epsilon0rpo.html#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.F_alpha.html b/theories/html/hydras.Epsilon0.F_alpha.html
new file mode 100644
index 00000000..9bb762c9
--- /dev/null
+++ b/theories/html/hydras.Epsilon0.F_alpha.html
@@ -0,0 +1,497 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.F_alpha</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.F_alpha</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab93"></a><h1 class="section">The Wainer hierarchy of rapidly growing functions (variant)</h1>
+
+
+<div class="paragraph"> </div>
+
+    After Wainer, Ketonen, Solovay, etc .
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a>  <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#"><span class="id" title="library">Hprime</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="module">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Equations</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For masking primRec's iterate 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="module">Prelude.Iterates</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab94"></a><h2 class="section">Definition, using <span class="inlinecode"><span class="id" title="var">coq</span>-<span class="id" title="var">equations</span></span></h2>
+
+
+<div class="paragraph"> </div>
+
+The following definition is not accepted by the <span class="inlinecode"><span class="id" title="var">equations</span></span> plug-in.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Olt" class="idref" href="#Olt"><span class="id" title="instance">Olt</span></a> : <span class="id" title="class">WellFounded</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+    Indeed, we define the -th iterate of <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> by well-founded
+    recursion  on the pair (alpha,n), then <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> as the first iterate 
+    of the defined function.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="call_lt" class="idref" href="#call_lt"><span class="id" title="definition">call_lt</span></a> (<a id="c:1" class="idref" href="#c:1"><span class="id" title="binder">c</span></a> <a id="c':2" class="idref" href="#c':2"><span class="id" title="binder">c'</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#c:1"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#c':2"><span class="id" title="variable">c'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="call_lt_wf" class="idref" href="#call_lt_wf"><span class="id" title="lemma">call_lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#call_lt"><span class="id" title="definition">call_lt</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="WF" class="idref" href="#WF"><span class="id" title="instance">WF</span></a> : <span class="id" title="class">WellFounded</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#call_lt"><span class="id" title="definition">call_lt</span></a> := <a class="idref" href="hydras.Epsilon0.F_alpha.html#call_lt_wf"><span class="id" title="lemma">call_lt_wf</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Equations</span>  <a id="F_star" class="idref" href="#F_star"><span class="id" title="definition">F_star</span></a> (<a id="c:3" class="idref" href="#c:3"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span>  <span class="id" title="var">c</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#call_lt"><span class="id" title="definition">call_lt</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:8"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:6"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:7"><span class="id" title="variable">i</span></a> := <span class="id" title="var">i</span>;<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:11"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:9"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:10"><span class="id" title="variable">i</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_dec"><span class="id" title="lemma">E0_eq_dec</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_zero:14"><span class="id" title="variable">_zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_nonzero:19"><span class="id" title="variable">_nonzero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Utils.html#dec"><span class="id" title="abbreviation">Utils.dec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <span class="id" title="var">alpha</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_limit:25"><span class="id" title="variable">_limit</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:5"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_notlimit:33"><span class="id" title="variable">_notlimit</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:5"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  <span class="id" title="var">i</span>}};<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:39"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:36"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:38"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:37"><span class="id" title="variable">i</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:5"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star:5"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <span class="id" title="var">i</span>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="F_" class="idref" href="#F_"><span class="id" title="definition">F_</span></a> <a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a> <a id="i:41" class="idref" href="#i:41"><span class="id" title="binder">i</span></a> := <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:40"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:41"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab95"></a><h2 class="section">We get the "usual" equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab96"></a><h3 class="section">Relations between <span class="inlinecode"><span class="id" title="var">F_star</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_star_zero_eqn" class="idref" href="#F_star_zero_eqn"><span class="id" title="lemma">F_star_zero_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a> <a id="i:43" class="idref" href="#i:43"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:42"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:43"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:43"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Fstar_S" class="idref" href="#Fstar_S"><span class="id" title="lemma">Fstar_S</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a> <a id="n:45" class="idref" href="#n:45"><span class="id" title="binder">n</span></a> <a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:44"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:45"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:46"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:44"><span class="id" title="variable">alpha</span></a>  (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:44"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:45"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:46"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_eq2" class="idref" href="#F_eq2"><span class="id" title="lemma">F_eq2</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:47" class="idref" href="#alpha:47"><span class="id" title="binder">alpha</span></a> <a id="i:48" class="idref" href="#i:48"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:48"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:47"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:48"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:48"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_star_Succ" class="idref" href="#F_star_Succ"><span class="id" title="lemma">F_star_Succ</span></a>:  <span class="id" title="keyword">∀</span> <a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a> <a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:50"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:50"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:51"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_star_iterate" class="idref" href="#F_star_iterate"><span class="id" title="lemma">F_star_iterate</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:52" class="idref" href="#alpha:52"><span class="id" title="binder">alpha</span></a> <a id="n:53" class="idref" href="#n:53"><span class="id" title="binder">n</span></a> <a id="i:54" class="idref" href="#i:54"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_star"><span class="id" title="definition">F_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:52"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:53"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:54"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:52"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:54"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab97"></a><h3 class="section">Usual equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_zero_eqn" class="idref" href="#F_zero_eqn"><span class="id" title="lemma">F_zero_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:55"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:55"><span class="id" title="variable">i</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_lim_eqn" class="idref" href="#F_lim_eqn"><span class="id" title="lemma">F_lim_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:56" class="idref" href="#alpha:56"><span class="id" title="binder">alpha</span></a> <a id="i:57" class="idref" href="#i:57"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:57"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:57"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:57"><span class="id" title="variable">i</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_succ_eqn" class="idref" href="#F_succ_eqn"><span class="id" title="lemma">F_succ_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:58" class="idref" href="#alpha:58"><span class="id" title="binder">alpha</span></a> <a id="i:59" class="idref" href="#i:59"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:58"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:59"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:58"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:59"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:59"><span class="id" title="variable">i</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab98"></a><h2 class="section">First steps of the hierarchy</h2>
+
+<div class="paragraph"> </div>
+
+ performs an induction only on the occ1-th and occ2_th occurrences of n 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> "undiag2" <span class="id" title="keyword">constr</span>(<span class="id" title="var">n</span>) <span class="id" title="var">integer</span>(<span class="id" title="var">occ1</span>) <span class="id" title="var">integer</span>(<span class="id" title="var">occ2</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">n'</span> := <span class="id" title="tactic">fresh</span> "n" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="var">occ1</span> <span class="id" title="var">occ2</span>; <span class="id" title="tactic">intro</span> <span class="id" title="var">n'</span>; <span class="id" title="tactic">induction</span> <span class="id" title="var">n'</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF1" class="idref" href="#LF1"><span class="id" title="lemma">LF1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 1 <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:60"><span class="id" title="variable">i</span></a>). 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF2" class="idref" href="#LF2"><span class="id" title="lemma">LF2</span></a> : <span class="id" title="keyword">∀</span> <a id="i:61" class="idref" href="#i:61"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:61"><span class="id" title="variable">i</span></a>. 
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="LF2'" class="idref" href="#LF2'"><span class="id" title="lemma">LF2'</span></a> : <span class="id" title="keyword">∀</span> <a id="i:62" class="idref" href="#i:62"><span class="id" title="binder">i</span></a>,  1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:62"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_0_eq" class="idref" href="#F_alpha_0_eq"><span class="id" title="lemma">F_alpha_0_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:63" class="idref" href="#alpha:63"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:63"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Search</span> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> ?<span class="id" title="var">a</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> ?<span class="id" title="var">a</span>).<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties of <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_alpha_mono" class="idref" href="#F_alpha_mono"><span class="id" title="lemma">F_alpha_mono</span></a> <a id="alpha:88" class="idref" href="#alpha:88"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:88"><span class="id" title="variable">alpha</span></a>).  
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_alpha_gt" class="idref" href="#F_alpha_gt"><span class="id" title="lemma">F_alpha_gt</span></a> <a id="alpha:89" class="idref" href="#alpha:89"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="n:90" class="idref" href="#n:90"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:90"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:89"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:90"><span class="id" title="variable">n</span></a>.  
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="F_alpha_positive" class="idref" href="#F_alpha_positive"><span class="id" title="lemma">F_alpha_positive</span></a> <a id="alpha:91" class="idref" href="#alpha:91"><span class="id" title="binder">alpha</span></a> :  <span class="id" title="keyword">∀</span> <a id="n:92" class="idref" href="#n:92"><span class="id" title="binder">n</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:91"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:92"><span class="id" title="variable">n</span></a>. 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_alpha_Succ_le" class="idref" href="#F_alpha_Succ_le"><span class="id" title="lemma">F_alpha_Succ_le</span></a> <a id="alpha:93" class="idref" href="#alpha:93"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:93"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:93"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_alpha_dom" class="idref" href="#F_alpha_dom"><span class="id" title="lemma">F_alpha_dom</span></a> <a id="alpha:94" class="idref" href="#alpha:94"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:94"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:94"><span class="id" title="variable">alpha</span></a>).  
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_restricted_mono_l" class="idref" href="#F_restricted_mono_l"><span class="id" title="lemma">F_restricted_mono_l</span></a> <a id="alpha:95" class="idref" href="#alpha:95"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:96" class="idref" href="#beta:96"><span class="id" title="binder">beta</span></a> <a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:96"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:96"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:97"><span class="id" title="variable">n</span></a>.  
+<br/>
+&nbsp;#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use F_alpha_gt")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="F_alpha_ge_S" class="idref" href="#F_alpha_ge_S"><span class="id" title="abbreviation">F_alpha_ge_S</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder_Transitive"><span class="id" title="method">StrictOrder_Transitive</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF2_0" class="idref" href="#LF2_0"><span class="id" title="lemma">LF2_0</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 0 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2) (<span class="id" title="keyword">fun</span> <a id="i:98" class="idref" href="#i:98"><span class="id" title="binder">i</span></a> ⇒ <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:98"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:98"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF3_2" class="idref" href="#LF3_2"><span class="id" title="lemma">LF3_2</span></a>  : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 2  (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 3) (<span class="id" title="keyword">fun</span>  <a id="n:99" class="idref" href="#n:99"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:99"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:99"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+From Ketonen and Solovay, page 284, op. cit. 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="F_monotony_l" class="idref" href="#F_monotony_l"><span class="id" title="section">F_monotony_l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="F_monotony_l.alpha" class="idref" href="#F_monotony_l.alpha"><span class="id" title="variable">alpha</span></a> <a id="F_monotony_l.beta" class="idref" href="#F_monotony_l.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="F_monotony_l.H'_beta_alpha" class="idref" href="#F_monotony_l.H'_beta_alpha"><span class="id" title="variable">H'_beta_alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="F_mono_l_0" class="idref" href="#F_mono_l_0"><span class="id" title="lemma">F_mono_l_0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:112" class="idref" href="#n:112"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:112"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:113" class="idref" href="#i:113"><span class="id" title="binder">i</span></a>, (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:112"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:113"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:113"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:113"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="F_mono_l" class="idref" href="#F_mono_l"><span class="id" title="lemma">F_mono_l</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta"><span class="id" title="variable">beta</span></a>). 
+<br/>
+<span class="id" title="keyword">End</span>  <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_monotony_l"><span class="id" title="section">F_monotony_l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab99"></a><h1 class="section">Comparison with the Hardy hierarchy</h1>
+
+
+<div class="paragraph"> </div>
+
+      <span class="inlinecode">(<span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" title="var">H'_</span></span> <span class="inlinecode">(<span class="id" title="var">Phi0</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span>)</span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>))</span>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="H'_F" class="idref" href="#H'_F"><span class="id" title="section">H'_F</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="H'_F.P" class="idref" href="#H'_F.P"><span class="id" title="variable">P</span></a> (<a id="alpha:114" class="idref" href="#alpha:114"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:115" class="idref" href="#n:115"><span class="id" title="binder">n</span></a>,  (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:114"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:115"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:114"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:115"><span class="id" title="variable">n</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="H'_F.alpha" class="idref" href="#H'_F.alpha"><span class="id" title="variable">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="H'_F.IHalpha" class="idref" href="#H'_F.IHalpha"><span class="id" title="variable">IHalpha</span></a> : <span class="id" title="keyword">∀</span>  <a id="beta:117" class="idref" href="#beta:117"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:117"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:117"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HF0" class="idref" href="#HF0"><span class="id" title="lemma">HF0</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HFsucc" class="idref" href="#HFsucc"><span class="id" title="lemma">HFsucc</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The following proof is far from being trivial.
+      It uses some lemmas from the Ketonen-Solovay machinery 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HFLim" class="idref" href="#HFLim"><span class="id" title="lemma">HFLim</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#H'_F"><span class="id" title="section">H'_F</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_F" class="idref" href="#H'_F"><span class="id" title="lemma">H'_F</span></a> <a id="alpha:119" class="idref" href="#alpha:119"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="n:120" class="idref" href="#n:120"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:119"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:120"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:119"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:120"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab100"></a><h1 class="section">A variant (Lob-Wainer hierarchy)</h1>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Equations</span>  <a id="f_star" class="idref" href="#f_star"><span class="id" title="definition">f_star</span></a> (<a id="c:121" class="idref" href="#c:121"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="i:122" class="idref" href="#i:122"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span> <span class="id" title="var">c</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#call_lt"><span class="id" title="definition">call_lt</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:126"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:124"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:125"><span class="id" title="variable">i</span></a> := <span class="id" title="var">i</span>;<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:129"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:127"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:128"><span class="id" title="variable">i</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_dec"><span class="id" title="lemma">E0_eq_dec</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_zero:132"><span class="id" title="variable">_zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_nonzero:137"><span class="id" title="variable">_nonzero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Utils.html#dec"><span class="id" title="abbreviation">Utils.dec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <span class="id" title="var">alpha</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_limit:143"><span class="id" title="variable">_limit</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:123"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#_successor:151"><span class="id" title="variable">_successor</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:123"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  <span class="id" title="var">i</span>}};<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:157"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:154"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:156"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:155"><span class="id" title="variable">i</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:123"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star:123"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">alpha</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <span class="id" title="var">i</span>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Finally, <span class="inlinecode"><span class="id" title="var">f_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> is defined as its first iterate  ! 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="f_" class="idref" href="#f_"><span class="id" title="definition">f_</span></a> <a id="alpha:158" class="idref" href="#alpha:158"><span class="id" title="binder">alpha</span></a> <a id="i:159" class="idref" href="#i:159"><span class="id" title="binder">i</span></a> := <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:158"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:159"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab101"></a><h2 class="section">We get the "usual" equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab102"></a><h3 class="section">Relations between <span class="inlinecode"><span class="id" title="var">F_star</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_star_zero_eqn" class="idref" href="#f_star_zero_eqn"><span class="id" title="lemma">f_star_zero_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:160" class="idref" href="#alpha:160"><span class="id" title="binder">alpha</span></a> <a id="i:161" class="idref" href="#i:161"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:160"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:161"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:161"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fstar_S" class="idref" href="#fstar_S"><span class="id" title="lemma">fstar_S</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:162" class="idref" href="#alpha:162"><span class="id" title="binder">alpha</span></a> <a id="n:163" class="idref" href="#n:163"><span class="id" title="binder">n</span></a> <a id="i:164" class="idref" href="#i:164"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:162"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:163"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:164"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:162"><span class="id" title="variable">alpha</span></a>  (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:162"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:163"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:164"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_eq2" class="idref" href="#f_eq2"><span class="id" title="lemma">f_eq2</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:165" class="idref" href="#alpha:165"><span class="id" title="binder">alpha</span></a> <a id="i:166" class="idref" href="#i:166"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:165"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:165"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:166"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:165"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:166"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:166"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_star_Succ" class="idref" href="#f_star_Succ"><span class="id" title="lemma">f_star_Succ</span></a>:  <span class="id" title="keyword">∀</span> <a id="alpha:167" class="idref" href="#alpha:167"><span class="id" title="binder">alpha</span></a> <a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a> <a id="i:169" class="idref" href="#i:169"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:167"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:168"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:169"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:167"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:167"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:168"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:169"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_star_iterate" class="idref" href="#f_star_iterate"><span class="id" title="lemma">f_star_iterate</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:170" class="idref" href="#alpha:170"><span class="id" title="binder">alpha</span></a> <a id="n:171" class="idref" href="#n:171"><span class="id" title="binder">n</span></a> <a id="i:172" class="idref" href="#i:172"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_star"><span class="id" title="definition">f_star</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:170"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:171"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:172"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:170"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:171"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:172"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab103"></a><h3 class="section">Usual equations for <span class="inlinecode"><span class="id" title="var">f_</span></span></h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_zero_eqn" class="idref" href="#f_zero_eqn"><span class="id" title="lemma">f_zero_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="i:173" class="idref" href="#i:173"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:173"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:173"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_lim_eqn" class="idref" href="#f_lim_eqn"><span class="id" title="lemma">f_lim_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:174" class="idref" href="#alpha:174"><span class="id" title="binder">alpha</span></a> <a id="i:175" class="idref" href="#i:175"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:174"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:174"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:175"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:174"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:175"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:175"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="f_succ_eqn" class="idref" href="#f_succ_eqn"><span class="id" title="lemma">f_succ_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:176" class="idref" href="#alpha:176"><span class="id" title="binder">alpha</span></a> <a id="i:177" class="idref" href="#i:177"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:176"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:177"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:176"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:177"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:177"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="id_le_f_alpha" class="idref" href="#id_le_f_alpha"><span class="id" title="lemma">id_le_f_alpha</span></a> (<a id="alpha:178" class="idref" href="#alpha:178"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <span class="id" title="keyword">∀</span> <a id="i:179" class="idref" href="#i:179"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:179"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:178"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#i:179"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Properties_of_f_alpha" class="idref" href="#Properties_of_f_alpha"><span class="id" title="section">Properties_of_f_alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span>  <a id="Q" class="idref" href="#Q"><span class="id" title="record">Q</span></a> (<a id="alpha:180" class="idref" href="#alpha:180"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="mkQ" class="idref" href="#mkQ"><span class="id" title="constructor">mkQ</span></a> {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="QA" class="idref" href="#QA"><span class="id" title="projection">QA</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:180"><span class="id" title="variable">alpha</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="QD" class="idref" href="#QD"><span class="id" title="projection">QD</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 2 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:180"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:180"><span class="id" title="variable">alpha</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="QE" class="idref" href="#QE"><span class="id" title="projection">QE</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:184" class="idref" href="#beta:184"><span class="id" title="binder">beta</span></a> <a id="n:185" class="idref" href="#n:185"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:180"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:184"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#beta:184"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#alpha:180"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#n:185"><span class="id" title="variable">n</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Properties_of_f_alpha.The_induction" class="idref" href="#Properties_of_f_alpha.The_induction"><span class="id" title="section">The_induction</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="QA0" class="idref" href="#QA0"><span class="id" title="lemma">QA0</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="QD0" class="idref" href="#QD0"><span class="id" title="lemma">QD0</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 2 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>)) (<a class="idref" href="hydras.Epsilon0.F_alpha.html#f_"><span class="id" title="definition">f_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+TODO : Study the equality F alpha i = Nat.pred (f alpha (S i)) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha.The_induction"><span class="id" title="section">The_induction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha"><span class="id" title="section">Properties_of_f_alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <span class="id" title="keyword">Assumptions</span> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_zero_eqn"><span class="id" title="lemma">F_zero_eqn</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.F_omega</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.F_omega</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ About <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">omega</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#"><span class="id" title="library">F_alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#"><span class="id" title="library">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#"><span class="id" title="library">AckNotPR</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="module">Exp2</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab104"></a><h2 class="section">Rewriting lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_iterate_rw" class="idref" href="#Ack_iterate_rw"><span class="id" title="lemma">Ack_iterate_rw</span></a> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> <a id="p:2" class="idref" href="#p:2"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:1"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:2"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:1"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:2"><span class="id" title="variable">p</span></a>)) 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_iterate_rw" class="idref" href="#F_iterate_rw"><span class="id" title="lemma">F_iterate_rw</span></a> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> <a id="p:4" class="idref" href="#p:4"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:3"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:4"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:3"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:4"><span class="id" title="variable">p</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:4"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab105"></a><h2 class="section">Comparison between <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L02" class="idref" href="#L02"><span class="id" title="lemma">L02</span></a> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a>: 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:5"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:5"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="inductive_step" class="idref" href="#inductive_step"><span class="id" title="section">inductive_step</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="inductive_step.n" class="idref" href="#inductive_step.n"><span class="id" title="variable">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="inductive_step.Hn" class="idref" href="#inductive_step.Hn"><span class="id" title="variable">Hn</span></a> : <span class="id" title="keyword">∀</span> <a id="p:7" class="idref" href="#p:7"><span class="id" title="binder">p</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:7"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#inductive_step.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:7"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#inductive_step.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:7"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L" class="idref" href="#L"><span class="id" title="lemma">L</span></a>: <span class="id" title="keyword">∀</span> <a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:9"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#inductive_step.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.F_omega.html#p:9"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#inductive_step.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.F_omega.html#p:9"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#inductive_step"><span class="id" title="section">inductive_step</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <span class="id" title="keyword">∀</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#p:11"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_vs_Ack" class="idref" href="#F_vs_Ack"><span class="id" title="lemma">F_vs_Ack</span></a> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:12"><span class="id" title="variable">n</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab106"></a><h2 class="section"><span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">omega</span></span> is not primitive recursive</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="module">primRec</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="module">extEqualNat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="F_omega_notPR" class="idref" href="#F_omega_notPR"><span class="id" title="section">F_omega_notPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="F_omega_PR:13" class="idref" href="#F_omega_PR:13"><span class="id" title="binder">F_omega_PR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="F_omega_not_PR" class="idref" href="#F_omega_not_PR"><span class="id" title="lemma">F_omega_not_PR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_omega_notPR"><span class="id" title="section">F_omega_notPR</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab107"></a><h2 class="section"><span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> is not primitive recursive, for <span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" title="var">alpha</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="F_alpha_notPR" class="idref" href="#F_alpha_notPR"><span class="id" title="section">F_alpha_notPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="F_alpha_notPR.alpha" class="idref" href="#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="F_alpha_notPR.case_lt" class="idref" href="#F_alpha_notPR.case_lt"><span class="id" title="section">case_lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="F_alpha_notPR.case_lt.Halpha" class="idref" href="#F_alpha_notPR.case_lt.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="N:16" class="idref" href="#N:16"><span class="id" title="binder">N</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <a id="k:17" class="idref" href="#k:17"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.F_omega.html#N:16"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#k:17"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#k:17"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#k:17"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="H:18" class="idref" href="#H:18"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R00" class="idref" href="#R00"><span class="id" title="lemma">R00</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#:::x_'&gt;&gt;'_x"><span class="id" title="notation">&gt;&gt;</span></a> <span class="id" title="keyword">fun</span> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#n:19"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="FF" class="idref" href="#FF"><span class="id" title="lemma">FF</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.case_lt"><span class="id" title="section">case_lt</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="F_alpha_notPR.H" class="idref" href="#F_alpha_notPR.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="H0:21" class="idref" href="#H0:21"><span class="id" title="binder">H0</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha"><span class="id" title="variable">alpha</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="F_alpha_not_PR" class="idref" href="#F_alpha_not_PR"><span class="id" title="lemma">F_alpha_not_PR</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_notPR"><span class="id" title="section">F_alpha_notPR</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab108"></a><h2 class="section">On the other hand, <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is PR for any <span class="inlinecode"><span class="id" title="var">n</span>:<span class="id" title="var">nat</span></span></h2>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="F_0_isPR" class="idref" href="#F_0_isPR"><span class="id" title="instance">F_0_isPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 0).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="step" class="idref" href="#step"><span class="id" title="section">step</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="step.n" class="idref" href="#step.n"><span class="id" title="variable">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="step.Hn" class="idref" href="#step.Hn"><span class="id" title="variable">Hn</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#step.n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="step.F" class="idref" href="#step.F"><span class="id" title="variable">F</span></a> := <span class="id" title="keyword">fun</span> <a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.Epsilon0.F_omega.html#a:24"><span class="id" title="variable">a</span></a> (<span class="id" title="keyword">fun</span> <a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> <a id="y:27" class="idref" href="#y:27"><span class="id" title="binder">y</span></a>  ⇒ <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#step.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#y:27"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.Epsilon0.F_omega.html#b:25"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="L00" class="idref" href="#L00"><span class="id" title="lemma">L00</span></a>: <span class="id" title="keyword">∀</span> <a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#step.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.F_omega.html#i:28"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#step.F"><span class="id" title="variable">F</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#i:28"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#i:28"><span class="id" title="variable">i</span></a>) .<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="R01" class="idref" href="#R01"><span class="id" title="instance">R01</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.Epsilon0.F_omega.html#step.F"><span class="id" title="variable">F</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="R02" class="idref" href="#R02"><span class="id" title="instance">R02</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a>  ⇒  <a class="idref" href="hydras.Epsilon0.F_omega.html#step.F"><span class="id" title="variable">F</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#i:37"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#i:37"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="R03" class="idref" href="#R03"><span class="id" title="instance">R03</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#step.n"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#step"><span class="id" title="section">step</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="F_n_PR" class="idref" href="#F_n_PR"><span class="id" title="instance">F_n_PR</span></a> (<a id="n:38" class="idref" href="#n:38"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a>  <a class="idref" href="hydras.Epsilon0.F_omega.html#n:38"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Keep in mind that <span class="inlinecode"><span class="id" title="var">isPR</span></span> is of sort <span class="inlinecode"><span class="id" title="keyword">Set</span></span>, so <span class="inlinecode"><span class="id" title="var">not</span></span> and <span class="inlinecode"><span class="id" title="var">iff</span></span>
+     cannot be used 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_PR_inv" class="idref" href="#F_alpha_PR_inv"><span class="id" title="lemma">F_alpha_PR_inv</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#alpha:39"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_alpha_notPR_inv" class="idref" href="#F_alpha_notPR_inv"><span class="id" title="lemma">F_alpha_notPR_inv</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#alpha:40"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#alpha:40"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_not_PR"><span class="id" title="lemma">F_alpha_not_PR</span></a>.<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.F_omega.html#F_alpha_PR_inv"><span class="id" title="lemma">F_alpha_PR_inv</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Hessenberg</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Hessenberg</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab17"></a><h1 class="section">Hessenberg sum of ordinals terms  (commutative and strictly monotonous)</h1>
+
+
+<div class="paragraph"> </div>
+
+Pierre Castéran, Labri, University of  Bordeaux 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">Prelude.More_Arith</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#"><span class="id" title="library">Prelude.Merge_Sort</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">Epsilon0.T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="oplus" class="idref" href="#oplus"><span class="id" title="definition">oplus</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> <a id="beta:2" class="idref" href="#beta:2"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="oplus_aux:8" class="idref" href="#oplus_aux:8"><span class="id" title="binder">oplus_aux</span></a> <a id="beta:4" class="idref" href="#beta:4"><span class="id" title="binder">beta</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="keyword">beta</span>} :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:1"><span class="id" title="variable">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:4"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:4"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:1"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> <span class="id" title="var">b1</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> <span class="id" title="var">b2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus:3"><span class="id" title="definition">oplus</span></a> <span class="id" title="var">b1</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:4"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus_aux:5"><span class="id" title="variable">oplus_aux</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n2</span>)%<span class="id" title="var">nat</span>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus:3"><span class="id" title="definition">oplus</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus_aux:8"><span class="id" title="variable">oplus_aux</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:2"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="3cf9144baff8c6bf36bff5a2c3dc2aab" class="idref" href="#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">&quot;</span></a>o+" := <a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a>  (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="o_finite_mult" class="idref" href="#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="alpha:10" class="idref" href="#alpha:10"><span class="id" title="binder">alpha</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:10"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult:11"><span class="id" title="definition">o_finite_mult</span></a> <span class="id" title="var">p</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:10"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_0_r" class="idref" href="#oplus_0_r"><span class="id" title="lemma">oplus_0_r</span></a> (<a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:13"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_0_l" class="idref" href="#oplus_0_l"><span class="id" title="lemma">oplus_0_l</span></a> (<a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:14"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:14"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_compare_Lt" class="idref" href="#oplus_compare_Lt"><span class="id" title="lemma">oplus_compare_Lt</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> <a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a> <a id="a':18" class="idref" href="#a':18"><span class="id" title="binder">a'</span></a> <a id="n':19" class="idref" href="#n':19"><span class="id" title="binder">n'</span></a> <a id="b':20" class="idref" href="#b':20"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:15"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':18"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:17"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':18"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':19"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':20"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':18"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':19"><span class="id" title="variable">n'</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:17"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':20"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt_rw" class="idref" href="#oplus_lt_rw"><span class="id" title="lemma">oplus_lt_rw</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> <a id="b:23" class="idref" href="#b:23"><span class="id" title="binder">b</span></a> <a id="a':24" class="idref" href="#a':24"><span class="id" title="binder">a'</span></a> <a id="n':25" class="idref" href="#n':25"><span class="id" title="binder">n'</span></a> <a id="b':26" class="idref" href="#b':26"><span class="id" title="binder">b'</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:21"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':24"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':24"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':25"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':26"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':24"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':25"><span class="id" title="variable">n'</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:21"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':26"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_eq_rw" class="idref" href="#oplus_eq_rw"><span class="id" title="lemma">oplus_eq_rw</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a>  <a id="n':30" class="idref" href="#n':30"><span class="id" title="binder">n'</span></a> <a id="b':31" class="idref" href="#b':31"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':30"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':31"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:27"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:28"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':30"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span>)  (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':31"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_gt_rw" class="idref" href="#oplus_gt_rw"><span class="id" title="lemma">oplus_gt_rw</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a> <a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a> <a id="b:34" class="idref" href="#b:34"><span class="id" title="binder">b</span></a> <a id="a':35" class="idref" href="#a':35"><span class="id" title="binder">a'</span></a> <a id="n':36" class="idref" href="#n':36"><span class="id" title="binder">n'</span></a> <a id="b':37" class="idref" href="#b':37"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':35"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:33"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':35"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':36"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':37"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:33"><span class="id" title="variable">n</span></a>  (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':35"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':36"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':37"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_compare_Gt" class="idref" href="#oplus_compare_Gt"><span class="id" title="lemma">oplus_compare_Gt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a> <a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> <a id="b:40" class="idref" href="#b:40"><span class="id" title="binder">b</span></a> <a id="a':41" class="idref" href="#a':41"><span class="id" title="binder">a'</span></a> <a id="n':42" class="idref" href="#n':42"><span class="id" title="binder">n'</span></a> <a id="b':43" class="idref" href="#b':43"><span class="id" title="binder">b'</span></a>, <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':41"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:40"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':41"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':42"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':43"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:39"><span class="id" title="variable">n</span></a>  (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:40"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':41"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':42"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':43"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_eqn" class="idref" href="#oplus_eqn"><span class="id" title="lemma">oplus_eqn</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> <a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:44"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:45"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:45"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:44"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> <span class="id" title="var">b1</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> <span class="id" title="var">b2</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>  <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">n1</span> (<span class="id" title="var">b1</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:45"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n2</span>)) (<span class="id" title="var">b1</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">n2</span> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <span class="id" title="var">b2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_cons_cons" class="idref" href="#oplus_cons_cons"><span class="id" title="lemma">oplus_cons_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a> <a id="b:50" class="idref" href="#b:50"><span class="id" title="binder">b</span></a> <a id="a':51" class="idref" href="#a':51"><span class="id" title="binder">a'</span></a> <a id="n':52" class="idref" href="#n':52"><span class="id" title="binder">n'</span></a> <a id="b':53" class="idref" href="#b':53"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:49"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:50"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':51"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':52"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':53"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':51"><span class="id" title="variable">a'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:49"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:50"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':51"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':52"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':53"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> )<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:48"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:49"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':52"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:50"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':53"><span class="id" title="variable">b'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a':51"><span class="id" title="variable">a'</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n':52"><span class="id" title="variable">n'</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:49"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:50"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b':53"><span class="id" title="variable">b'</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_oplus" class="idref" href="#lt_a_phi0_b_oplus"><span class="id" title="lemma">lt_a_phi0_b_oplus</span></a> : <span class="id" title="keyword">∀</span> <a id="gamma:54" class="idref" href="#gamma:54"><span class="id" title="binder">gamma</span></a> <a id="alpha:55" class="idref" href="#alpha:55"><span class="id" title="binder">alpha</span></a> <a id="beta:56" class="idref" href="#beta:56"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:54"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:56"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:54"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:56"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:54"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_bounded_phi0" class="idref" href="#oplus_bounded_phi0"><span class="id" title="lemma">oplus_bounded_phi0</span></a> <a id="alpha:57" class="idref" href="#alpha:57"><span class="id" title="binder">alpha</span></a> <a id="beta:58" class="idref" href="#beta:58"><span class="id" title="binder">beta</span></a> <a id="gamma:59" class="idref" href="#gamma:59"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:57"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:58"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:59"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:57"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:59"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:58"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:59"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:57"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:58"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:59"><span class="id" title="variable">gamma</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_plus_nf" class="idref" href="#Proof_of_plus_nf"><span class="id" title="section">Proof_of_plus_nf</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_nf_0" class="idref" href="#oplus_nf_0"><span class="id" title="lemma">oplus_nf_0</span></a> (<a id="gamma:60" class="idref" href="#gamma:60"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:60"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="keyword">∀</span> <a id="alpha:61" class="idref" href="#alpha:61"><span class="id" title="binder">alpha</span></a> <a id="beta:62" class="idref" href="#beta:62"><span class="id" title="binder">beta</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:62"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:60"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:62"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:60"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:62"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_nf" class="idref" href="#oplus_nf"><span class="id" title="lemma">oplus_nf</span></a> (<a id="alpha:63" class="idref" href="#alpha:63"><span class="id" title="binder">alpha</span></a>  <a id="beta:64" class="idref" href="#beta:64"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:64"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:64"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_plus_nf"><span class="id" title="section">Proof_of_plus_nf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="o_finite_mult_nf" class="idref" href="#o_finite_mult_nf"><span class="id" title="lemma">o_finite_mult_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="a:65" class="idref" href="#a:65"><span class="id" title="binder">a</span></a> <a id="n:66" class="idref" href="#n:66"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:65"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:66"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:65"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_oplus_comm" class="idref" href="#Proof_of_oplus_comm"><span class="id" title="section">Proof_of_oplus_comm</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_comm_0" class="idref" href="#oplus_comm_0"><span class="id" title="lemma">oplus_comm_0</span></a> (<a id="gamma:67" class="idref" href="#gamma:67"><span class="id" title="binder">gamma</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:67"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:68" class="idref" href="#alpha:68"><span class="id" title="binder">alpha</span></a> <a id="beta:69" class="idref" href="#beta:69"><span class="id" title="binder">beta</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:68"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:69"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:68"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:67"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:69"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:67"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:68"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:69"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:69"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:68"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_comm" class="idref" href="#oplus_comm"><span class="id" title="lemma">oplus_comm</span></a> (<a id="alpha:70" class="idref" href="#alpha:70"><span class="id" title="binder">alpha</span></a> <a id="beta:71" class="idref" href="#beta:71"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:71"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:71"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:71"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:70"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_comm"><span class="id" title="section">Proof_of_oplus_comm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt_rw2" class="idref" href="#oplus_lt_rw2"><span class="id" title="lemma">oplus_lt_rw2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> <a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> <a id="b:74" class="idref" href="#b:74"><span class="id" title="binder">b</span></a> <a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:74"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:74"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#x:75"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:73"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:74"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#x:75"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_oplus_assoc" class="idref" href="#Proof_of_oplus_assoc"><span class="id" title="section">Proof_of_oplus_assoc</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">ass_rw</span> <span class="id" title="var">Hrec</span> <span class="id" title="var">alpha</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> |- <span class="id" title="keyword">context</span> <span class="id" title="var">Gamma</span> [<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a> ?<span class="id" title="var">a</span> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a> ?<span class="id" title="var">b</span>  ?<span class="id" title="var">c</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">erewrite</span> <span class="id" title="var">Hrec</span> <span class="id" title="keyword">with</span> <span class="id" title="var">alpha</span>  <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">ass_rw_rev</span> <span class="id" title="var">Hrec</span> <span class="id" title="var">alpha</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> |- <span class="id" title="keyword">context</span> <span class="id" title="var">Gamma</span> [<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#oplus"><span class="id" title="definition">oplus</span></a> ?<span class="id" title="var">a</span>  ?<span class="id" title="var">b</span>)  ?<span class="id" title="var">c</span>] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">erewrite</span> &lt;- <span class="id" title="var">Hrec</span> <span class="id" title="keyword">with</span> <span class="id" title="var">alpha</span>  <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_assoc_0" class="idref" href="#oplus_assoc_0"><span class="id" title="lemma">oplus_assoc_0</span></a> (<a id="alpha:76" class="idref" href="#alpha:76"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:76"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:77" class="idref" href="#a:77"><span class="id" title="binder">a</span></a> <a id="b:78" class="idref" href="#b:78"><span class="id" title="binder">b</span></a> <a id="c:79" class="idref" href="#c:79"><span class="id" title="binder">c</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:78"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:79"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:76"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:78"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:76"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:79"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:76"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:78"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:79"><span class="id" title="variable">c</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:78"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:79"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_assoc" class="idref" href="#oplus_assoc"><span class="id" title="lemma">oplus_assoc</span></a> (<a id="alpha:80" class="idref" href="#alpha:80"><span class="id" title="binder">alpha</span></a> <a id="beta:81" class="idref" href="#beta:81"><span class="id" title="binder">beta</span></a> <a id="gamma:82" class="idref" href="#gamma:82"><span class="id" title="binder">gamma</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:82"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:82"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:82"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_assoc"><span class="id" title="section">Proof_of_oplus_assoc</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_oplus_lt1" class="idref" href="#Proof_of_oplus_lt1"><span class="id" title="section">Proof_of_oplus_lt1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Proof_of_oplus_lt1.a1" class="idref" href="#Proof_of_oplus_lt1.a1"><span class="id" title="variable">a1</span></a> <a id="Proof_of_oplus_lt1.a2" class="idref" href="#Proof_of_oplus_lt1.a2"><span class="id" title="variable">a2</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_oplus_lt1.n" class="idref" href="#Proof_of_oplus_lt1.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_oplus_lt1.H0" class="idref" href="#Proof_of_oplus_lt1.H0"><span class="id" title="variable">H0</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a2"><span class="id" title="variable">a2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="oplus_lt_0" class="idref" href="#oplus_lt_0"><span class="id" title="lemma">oplus_lt_0</span></a> : <span class="id" title="keyword">∀</span> <a id="b:87" class="idref" href="#b:87"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:87"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1"><span class="id" title="section">Proof_of_oplus_lt1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt1" class="idref" href="#oplus_lt1"><span class="id" title="lemma">oplus_lt1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:88" class="idref" href="#a:88"><span class="id" title="binder">a</span></a> <a id="b:89" class="idref" href="#b:89"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:89"><span class="id" title="variable">b</span></a>  (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:88"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt2" class="idref" href="#oplus_lt2"><span class="id" title="lemma">oplus_lt2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:90" class="idref" href="#a:90"><span class="id" title="binder">a</span></a> <a id="b:91" class="idref" href="#b:91"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:90"><span class="id" title="variable">a</span></a>  (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:90"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_le1" class="idref" href="#oplus_le1"><span class="id" title="lemma">oplus_le1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:92" class="idref" href="#a:92"><span class="id" title="binder">a</span></a> <a id="b:93" class="idref" href="#b:93"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:93"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:92"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:93"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_le2" class="idref" href="#oplus_le2"><span class="id" title="lemma">oplus_le2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:94" class="idref" href="#a:94"><span class="id" title="binder">a</span></a> <a id="b:95" class="idref" href="#b:95"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:94"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:95"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:95"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:94"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:95"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_0" class="idref" href="#oplus_strict_mono_0"><span class="id" title="lemma">oplus_strict_mono_0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:96" class="idref" href="#alpha:96"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:97" class="idref" href="#a:97"><span class="id" title="binder">a</span></a> (<a id="Ha:98" class="idref" href="#Ha:98"><span class="id" title="binder">Ha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:97"><span class="id" title="variable">a</span></a>) <a id="b:99" class="idref" href="#b:99"><span class="id" title="binder">b</span></a> (<a id="Hb:100" class="idref" href="#Hb:100"><span class="id" title="binder">Hb</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:99"><span class="id" title="variable">b</span></a>) <a id="c:101" class="idref" href="#c:101"><span class="id" title="binder">c</span></a> (<a id="Hc:102" class="idref" href="#Hc:102"><span class="id" title="binder">Hc</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:101"><span class="id" title="variable">c</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:101"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:99"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:101"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:99"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:101"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_r" class="idref" href="#oplus_strict_mono_r"><span class="id" title="lemma">oplus_strict_mono_r</span></a> : <span class="id" title="keyword">∀</span> <a id="a:103" class="idref" href="#a:103"><span class="id" title="binder">a</span></a> <a id="b:104" class="idref" href="#b:104"><span class="id" title="binder">b</span></a> <a id="c:105" class="idref" href="#c:105"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:103"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:104"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:105"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:104"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:105"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:103"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:104"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:103"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:105"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_l" class="idref" href="#oplus_strict_mono_l"><span class="id" title="lemma">oplus_strict_mono_l</span></a> : <span class="id" title="keyword">∀</span> <a id="a:106" class="idref" href="#a:106"><span class="id" title="binder">a</span></a> <a id="b:107" class="idref" href="#b:107"><span class="id" title="binder">b</span></a> <a id="c:108" class="idref" href="#c:108"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:106"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:107"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:108"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:106"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:107"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:106"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:108"><span class="id" title="variable">c</span></a>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:107"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:108"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_LT_l" class="idref" href="#oplus_strict_mono_LT_l"><span class="id" title="lemma">oplus_strict_mono_LT_l</span></a> (<a id="alpha:109" class="idref" href="#alpha:109"><span class="id" title="binder">alpha</span></a> <a id="beta:110" class="idref" href="#beta:110"><span class="id" title="binder">beta</span></a> <a id="gamma:111" class="idref" href="#gamma:111"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:111"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:109"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:110"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:111"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:110"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:111"><span class="id" title="variable">gamma</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_LT_r" class="idref" href="#oplus_strict_mono_LT_r"><span class="id" title="lemma">oplus_strict_mono_LT_r</span></a> (<a id="alpha:112" class="idref" href="#alpha:112"><span class="id" title="binder">alpha</span></a> <a id="beta:113" class="idref" href="#beta:113"><span class="id" title="binder">beta</span></a> <a id="gamma:114" class="idref" href="#gamma:114"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:112"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:113"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:114"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:112"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#beta:113"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#alpha:112"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#gamma:114"><span class="id" title="variable">gamma</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_strict_mono_bi" class="idref" href="#oplus_strict_mono_bi"><span class="id" title="lemma">oplus_strict_mono_bi</span></a> : <span class="id" title="keyword">∀</span> <a id="a:115" class="idref" href="#a:115"><span class="id" title="binder">a</span></a> <a id="b:116" class="idref" href="#b:116"><span class="id" title="binder">b</span></a> <a id="c:117" class="idref" href="#c:117"><span class="id" title="binder">c</span></a> <a id="d:118" class="idref" href="#d:118"><span class="id" title="binder">d</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:115"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:116"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:117"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#d:118"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:115"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:116"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:117"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#d:118"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:115"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:117"><span class="id" title="variable">c</span></a>) (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:116"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#d:118"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_of_phi0_0" class="idref" href="#oplus_of_phi0_0"><span class="id" title="lemma">oplus_of_phi0_0</span></a> : <span class="id" title="keyword">∀</span> <a id="a:119" class="idref" href="#a:119"><span class="id" title="binder">a</span></a> <a id="b:120" class="idref" href="#b:120"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:119"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:120"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:119"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:120"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:120"><span class="id" title="variable">b</span></a> 0 (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:119"><span class="id" title="variable">a</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:119"><span class="id" title="variable">a</span></a> 1 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:119"><span class="id" title="variable">a</span></a> 0 (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:120"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_of_phi0" class="idref" href="#oplus_of_phi0"><span class="id" title="lemma">oplus_of_phi0</span></a> : <span class="id" title="keyword">∀</span> <a id="a:121" class="idref" href="#a:121"><span class="id" title="binder">a</span></a> <a id="b:122" class="idref" href="#b:122"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:121"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:122"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:121"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:122"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:122"><span class="id" title="variable">b</span></a> 0 (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:121"><span class="id" title="variable">a</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:121"><span class="id" title="variable">a</span></a> 1 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:121"><span class="id" title="variable">a</span></a> 0 (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:122"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="o_finite_mult_rw" class="idref" href="#o_finite_mult_rw"><span class="id" title="lemma">o_finite_mult_rw</span></a> : <span class="id" title="keyword">∀</span> <a id="a:123" class="idref" href="#a:123"><span class="id" title="binder">a</span></a> <a id="n:124" class="idref" href="#n:124"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:124"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:123"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:123"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:124"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="o_finite_mult_lt_phi0_1" class="idref" href="#o_finite_mult_lt_phi0_1"><span class="id" title="lemma">o_finite_mult_lt_phi0_1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:125" class="idref" href="#a:125"><span class="id" title="binder">a</span></a> <a id="b:126" class="idref" href="#b:126"><span class="id" title="binder">b</span></a> <a id="n:127" class="idref" href="#n:127"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:125"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:126"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:127"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:125"><span class="id" title="variable">a</span></a>)) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:126"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="o_finite_mult_mono" class="idref" href="#o_finite_mult_mono"><span class="id" title="lemma">o_finite_mult_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="a:128" class="idref" href="#a:128"><span class="id" title="binder">a</span></a> <a id="b:129" class="idref" href="#b:129"><span class="id" title="binder">b</span></a> <a id="n:130" class="idref" href="#n:130"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:128"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:129"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:128"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:129"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:130"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:128"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#n:130"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:129"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="oplus_lt_phi0" class="idref" href="#oplus_lt_phi0"><span class="id" title="lemma">oplus_lt_phi0</span></a> : <span class="id" title="keyword">∀</span> <a id="a:131" class="idref" href="#a:131"><span class="id" title="binder">a</span></a> <a id="b:132" class="idref" href="#b:132"><span class="id" title="binder">b</span></a> <a id="c:133" class="idref" href="#c:133"><span class="id" title="binder">c</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:131"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:132"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:133"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:131"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:133"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:132"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:133"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#a:131"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#b:132"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#c:133"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Hprime</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Hprime</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab31"></a><h1 class="section">A hierarchy of rapidly growing functions</h1>
+
+    After Hardy, Wainer, Ketonen, Solovay, etc .
+
+<div class="paragraph"> </div>
+
+    Pierre Casteran, LaBRI, University of  Bordeaux 
+<div class="paragraph"> </div>
+
+ Remark
+
+<div class="paragraph"> </div>
+
+Some notations (mainly names of fast-growing functions) of our development 
+ may differ slightly from the litterature. Although this fact does not affect 
+our proofs, we are preparing a future version where the names
+ <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>, <span class="inlinecode"><span class="id" title="var">f_alpha</span></span>, <span class="inlinecode"><span class="id" title="var">H</span></span> <span class="inlinecode"><span class="id" title="var">_alpha</span></span>, etc., are fully consistent with the 
+ cited articles. 
+ This change will be provisionally made in a branch called <span class="inlinecode"><span class="id" title="var">Hardy</span></span>
+   (in reference to the Hardy hierarchy).
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>    <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="module">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Equations</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab32"></a><h2 class="section">A variant of the Hardy Wainer hierarchy</h2>
+
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Olt" class="idref" href="#Olt"><span class="id" title="instance">Olt</span></a> : <span class="id" title="class">WellFounded</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Equations</span> <a id="H'_" class="idref" href="#H'_"><span class="id" title="definition">H'_</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>  <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span>  <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_:6"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:4"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Hprime.html#i:5"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_dec"><span class="id" title="lemma">E0_eq_dec</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#_zero:9"><span class="id" title="variable">_zero</span></a> ⇒  <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#_nonzero:14"><span class="id" title="variable">_nonzero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Utils.html#dec"><span class="id" title="abbreviation">Utils.dec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <span class="id" title="var">alpha</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#_limit:20"><span class="id" title="variable">_limit</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_:3"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <span class="id" title="var">alpha</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>))  <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#_successor:28"><span class="id" title="variable">_successor</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_:3"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <span class="id" title="var">alpha</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>)}}.<br/>
+
+<br/>
+<span class="id" title="keyword">Solve</span> <span class="id" title="keyword">All</span> <span class="id" title="keyword">Obligations</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Paraphrases of the equations for H'</i> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq1" class="idref" href="#H'_eq1"><span class="id" title="lemma">H'_eq1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:31" class="idref" href="#i:31"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:31"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:31"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq2_0" class="idref" href="#H'_eq2_0"><span class="id" title="lemma">H'_eq2_0</span></a> <a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> <a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:32"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:33"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq3" class="idref" href="#H'_eq3"><span class="id" title="lemma">H'_eq3</span></a> <a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a> <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:34"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:34"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:34"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:35"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#i:35"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_eq2" class="idref" href="#H'_eq2"><span class="id" title="lemma">H'_eq2</span></a>  <a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:36"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#i:37"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:36"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:37"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_eq1"><span class="id" title="lemma">H'_eq1</span></a>  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_eq2"><span class="id" title="lemma">H'_eq2</span></a> : <span class="id" title="var">H'_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">lim_rw</span> <span class="id" title="var">alpha</span> := (<span class="id" title="tactic">assert</span> (<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a>  <span class="id" title="var">alpha</span>) <span class="id" title="tactic">by</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">E0</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_eq3"><span class="id" title="lemma">H'_eq3</span></a> <span class="id" title="var">alpha</span>); <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Examples : First steps of the hierarchy 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Fin" class="idref" href="#H'_Fin"><span class="id" title="lemma">H'_Fin</span></a> : <span class="id" title="keyword">∀</span> <a id="i:38" class="idref" href="#i:38"><span class="id" title="binder">i</span></a> <a id="k:39" class="idref" href="#k:39"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a>  (<a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:38"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:39"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#i:38"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Hprime.html#k:39"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega" class="idref" href="#H'_omega"><span class="id" title="lemma">H'_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="k:40" class="idref" href="#k:40"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:40"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:40"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Plus_Fin" class="idref" href="#H'_Plus_Fin"><span class="id" title="lemma">H'_Plus_Fin</span></a> <a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="i:42" class="idref" href="#i:42"><span class="id" title="binder">i</span></a> <a id="k:43" class="idref" href="#k:43"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:41"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:42"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:43"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:41"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#i:42"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:43"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_double" class="idref" href="#H'_omega_double"><span class="id" title="lemma">H'_omega_double</span></a> <a id="k:44" class="idref" href="#k:44"><span class="id" title="binder">k</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> 2)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:44"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  (4 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:44"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 3)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_3" class="idref" href="#H'_omega_3"><span class="id" title="lemma">H'_omega_3</span></a> <a id="k:45" class="idref" href="#k:45"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> 3)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:45"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (8 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:45"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 7)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_4" class="idref" href="#H'_omega_4"><span class="id" title="lemma">H'_omega_4</span></a> <a id="k:46" class="idref" href="#k:46"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> 4)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:46"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (16 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:46"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 15)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_i" class="idref" href="#H'_omega_i"><span class="id" title="lemma">H'_omega_i</span></a> (<a id="i:47" class="idref" href="#i:47"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  : <span class="id" title="keyword">∀</span> <a id="k:48" class="idref" href="#k:48"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:47"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:48"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:47"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:48"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:47"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us note that the square of omega can be expressed through the
+    Phi0 function 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Phi0_def" class="idref" href="#Phi0_def"><span class="id" title="lemma">Phi0_def</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">ltac</span>:(<span class="id" title="var">mko</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>)%<span class="id" title="var">t1</span>).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_sqr" class="idref" href="#H'_omega_sqr"><span class="id" title="lemma">H'_omega_sqr</span></a> : <span class="id" title="keyword">∀</span> <a id="k:49" class="idref" href="#k:49"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a>  2)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:49"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:49"><span class="id" title="variable">k</span></a> ) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:49"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>. 
+<br/>
+</div>
+
+<div class="doc">
+Composition lemma 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="H'_cons" class="idref" href="#H'_cons"><span class="id" title="section">H'_cons</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="H'_cons.alpha" class="idref" href="#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="H'_cons.i" class="idref" href="#H'_cons.i"><span class="id" title="variable">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_cons" class="idref" href="#H'_cons"><span class="id" title="lemma">H'_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:52" class="idref" href="#beta:52"><span class="id" title="binder">beta</span></a>,  (<a class="idref" href="hydras.Epsilon0.Hprime.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:53" class="idref" href="#k:53"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Cons"><span class="id" title="instance">Cons</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.i"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:52"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:53"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:53"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Omega_term_1" class="idref" href="#H'_Omega_term_1"><span class="id" title="lemma">H'_Omega_term_1</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span>  <a id="k:54" class="idref" href="#k:54"><span class="id" title="binder">k</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.i"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:54"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:54"><span class="id" title="variable">k</span></a>). 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_cons"><span class="id" title="section">H'_cons</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Omega_term_0" class="idref" href="#H'_Omega_term_0"><span class="id" title="lemma">H'_Omega_term_0</span></a> (<a id="alpha:55" class="idref" href="#alpha:55"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>)  :<br/>
+<a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="keyword">∀</span> <a id="i:56" class="idref" href="#i:56"><span class="id" title="binder">i</span></a> <a id="k:57" class="idref" href="#k:57"><span class="id" title="binder">k</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:56"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:57"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a>  (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:55"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:56"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:57"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Fin_iterate" class="idref" href="#H'_Fin_iterate"><span class="id" title="lemma">H'_Fin_iterate</span></a> : <span class="id" title="keyword">∀</span> <a id="i:58" class="idref" href="#i:58"><span class="id" title="binder">i</span></a> <a id="k:59" class="idref" href="#k:59"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:58"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:59"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> 1)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:58"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:59"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Omega_term" class="idref" href="#H'_Omega_term"><span class="id" title="lemma">H'_Omega_term</span></a> (<a id="alpha:60" class="idref" href="#alpha:60"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>)  :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:61" class="idref" href="#i:61"><span class="id" title="binder">i</span></a> <a id="k:62" class="idref" href="#k:62"><span class="id" title="binder">k</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:60"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:61"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:62"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a>  (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:60"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:61"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:62"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="H'_succ_fun" class="idref" href="#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a> <a id="f:63" class="idref" href="#f:63"><span class="id" title="binder">f</span></a> <a id="k:64" class="idref" href="#k:64"><span class="id" title="binder">k</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#f:63"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:64"><span class="id" title="variable">k</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:64"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_succ_1" class="idref" href="#H'_Phi0_succ_1"><span class="id" title="lemma">H'_Phi0_succ_1</span></a> <a id="alpha:65" class="idref" href="#alpha:65"><span class="id" title="binder">alpha</span></a>  : <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="k:66" class="idref" href="#k:66"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:65"><span class="id" title="variable">alpha</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:66"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:65"><span class="id" title="variable">alpha</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:66"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_succ_0" class="idref" href="#H'_Phi0_succ_0"><span class="id" title="lemma">H'_Phi0_succ_0</span></a> : <span class="id" title="keyword">∀</span> <a id="k:67" class="idref" href="#k:67"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:67"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:67"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_succ" class="idref" href="#H'_Phi0_succ"><span class="id" title="lemma">H'_Phi0_succ</span></a> <a id="alpha:68" class="idref" href="#alpha:68"><span class="id" title="binder">alpha</span></a>  : <span class="id" title="keyword">∀</span> <a id="k:69" class="idref" href="#k:69"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:68"><span class="id" title="variable">alpha</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:69"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:68"><span class="id" title="variable">alpha</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:69"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_Si" class="idref" href="#H'_Phi0_Si"><span class="id" title="lemma">H'_Phi0_Si</span></a> : <span class="id" title="keyword">∀</span> <a id="i:70" class="idref" href="#i:70"><span class="id" title="binder">i</span></a> <a id="k:71" class="idref" href="#k:71"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:70"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:71"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:70"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:71"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_cube" class="idref" href="#H'_omega_cube"><span class="id" title="lemma">H'_omega_cube</span></a> : <span class="id" title="keyword">∀</span> <a id="k:72" class="idref" href="#k:72"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:72"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 2)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:72"><span class="id" title="variable">k</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:72"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="H'_omega_cube_3" class="idref" href="#H'_omega_cube_3"><span class="id" title="section">H'_omega_cube_3</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="H'_omega_cube_3.f" class="idref" href="#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> <a id="k:73" class="idref" href="#k:73"><span class="id" title="binder">k</span></a> :=   (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:73"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:73"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R0" class="idref" href="#R0"><span class="id" title="lemma">R0</span></a> <a id="k:74" class="idref" href="#k:74"><span class="id" title="binder">k</span></a> :  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3)%<span class="id" title="var">e0</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:74"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:74"><span class="id" title="variable">k</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:74"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F0" class="idref" href="#F0"><span class="id" title="lemma">F0</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3) 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> 3))).<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+f (f 3)) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="H'_omega_cube_3.N" class="idref" href="#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a> := (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> 64 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 64 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="N_simpl" class="idref" href="#N_simpl"><span class="id" title="lemma">N_simpl</span></a>: <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> 70 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F1" class="idref" href="#F1"><span class="id" title="lemma">F1</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3) 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span>  <a id="F1_simpl" class="idref" href="#F1_simpl"><span class="id" title="lemma">F1_simpl</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3) 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F2" class="idref" href="#F2"><span class="id" title="lemma">F2</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3 <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> 3) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="f_minoration" class="idref" href="#f_minoration"><span class="id" title="lemma">f_minoration</span></a> <a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a> <a id="p:82" class="idref" href="#p:82"><span class="id" title="binder">p</span></a> : 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:82"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:82"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F3" class="idref" href="#F3"><span class="id" title="lemma">F3</span></a> : (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N"><span class="id" title="variable">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3 <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> 3) 0).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3"><span class="id" title="section">H'_omega_cube_3</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_omega" class="idref" href="#H'_Phi0_omega"><span class="id" title="lemma">H'_Phi0_omega</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:83" class="idref" href="#k:83"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:83"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_succ_fun"><span class="id" title="definition">H'_succ_fun</span></a>  <a class="idref" href="hydras.Epsilon0.Hprime.html#k:83"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:83"><span class="id" title="variable">k</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_Phi0_omega_exact_formula" class="idref" href="#H'_Phi0_omega_exact_formula"><span class="id" title="lemma">H'_Phi0_omega_exact_formula</span></a> <a id="k:84" class="idref" href="#k:84"><span class="id" title="binder">k</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:84"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="F:87" class="idref" href="#F:87"><span class="id" title="binder">F</span></a> <a id="f:85" class="idref" href="#f:85"><span class="id" title="binder">f</span></a> <a id="i:86" class="idref" href="#i:86"><span class="id" title="binder">i</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#f:85"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#i:86"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Hprime.html#i:86"><span class="id" title="variable">i</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <a id="g:89" class="idref" href="#g:89"><span class="id" title="binder">g</span></a> <a id="k:88" class="idref" href="#k:88"><span class="id" title="binder">k</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:88"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#F:87"><span class="id" title="variable">F</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:84"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#g:89"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:84"><span class="id" title="variable">k</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_sqr_min" class="idref" href="#H'_omega_sqr_min"><span class="id" title="lemma">H'_omega_sqr_min</span></a> : <span class="id" title="keyword">∀</span> <a id="k:90" class="idref" href="#k:90"><span class="id" title="binder">k</span></a>,  0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:90"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:90"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 2) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:90"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_omega_cube_min" class="idref" href="#H'_omega_cube_min"><span class="id" title="lemma">H'_omega_cube_min</span></a> <a id="k:91" class="idref" href="#k:91"><span class="id" title="binder">k</span></a> :<br/>
+&nbsp;&nbsp;0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:91"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:91"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> 3) <a class="idref" href="hydras.Epsilon0.Hprime.html#k:91"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="H'_non_mono1" class="idref" href="#H'_non_mono1"><span class="id" title="lemma">H'_non_mono1</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="alpha:92" class="idref" href="#alpha:92"><span class="id" title="binder">alpha</span></a> <a id="beta:93" class="idref" href="#beta:93"><span class="id" title="binder">beta</span></a> <a id="k:94" class="idref" href="#k:94"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:92"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:93"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:92"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:94"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:93"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:94"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_Abstract_Properties" class="idref" href="#Proof_of_Abstract_Properties"><span class="id" title="section">Proof_of_Abstract_Properties</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Record</span> <a id="P" class="idref" href="#P"><span class="id" title="record">P</span></a> (<a id="alpha:95" class="idref" href="#alpha:95"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="mkP" class="idref" href="#mkP"><span class="id" title="constructor">mkP</span></a> {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="PA" class="idref" href="#PA"><span class="id" title="projection">PA</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="PB" class="idref" href="#PB"><span class="id" title="projection">PB</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:98" class="idref" href="#n:98"><span class="id" title="binder">n</span></a>,  (<a class="idref" href="hydras.Epsilon0.Hprime.html#n:98"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:98"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="PC" class="idref" href="#PC"><span class="id" title="projection">PC</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="PD" class="idref" href="#PD"><span class="id" title="projection">PD</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="PE" class="idref" href="#PE"><span class="id" title="projection">PE</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:102" class="idref" href="#beta:102"><span class="id" title="binder">beta</span></a> <a id="n:103" class="idref" href="#n:103"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:103"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:102"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:102"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:103"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:95"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:103"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Abstract_Properties.The_induction" class="idref" href="#Proof_of_Abstract_Properties.The_induction"><span class="id" title="section">The_induction</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PA_Zero" class="idref" href="#PA_Zero"><span class="id" title="lemma">PA_Zero</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PD_Zero" class="idref" href="#PD_Zero"><span class="id" title="lemma">PD_Zero</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>)) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">PD_Zero</span> <span class="id" title="var">PA_Zero</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PC_Zero" class="idref" href="#PC_Zero"><span class="id" title="lemma">PC_Zero</span></a> :  <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span>  <span class="id" title="var">PC_Zero</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="PZero" class="idref" href="#PZero"><span class="id" title="lemma">PZero</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#P"><span class="id" title="record">P</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Abstract_Properties.The_induction.Halpha" class="idref" href="#Proof_of_Abstract_Properties.The_induction.Halpha"><span class="id" title="variable">Halpha</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:106" class="idref" href="#beta:106"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:106"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#P"><span class="id" title="record">P</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:106"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha_Succ" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha_Succ"><span class="id" title="section">alpha_Succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta"><span class="id" title="variable">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def"><span class="id" title="variable">alpha_def</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="PA_Succ" class="idref" href="#PA_Succ"><span class="id" title="lemma">PA_Succ</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RB" class="idref" href="#RB"><span class="id" title="lemma">RB</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">∀</span> <a id="n:110" class="idref" href="#n:110"><span class="id" title="binder">n</span></a>, (<a class="idref" href="hydras.Epsilon0.Hprime.html#n:110"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:110"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RD" class="idref" href="#RD"><span class="id" title="lemma">RD</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RE" class="idref" href="#RE"><span class="id" title="lemma">RE</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:111" class="idref" href="#beta:111"><span class="id" title="binder">beta</span></a> <a id="n:112" class="idref" href="#n:112"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:112"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:112"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:112"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RC" class="idref" href="#RC"><span class="id" title="lemma">RC</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RP" class="idref" href="#RP"><span class="id" title="lemma">RP</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#P"><span class="id" title="record">P</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ"><span class="id" title="section">alpha_Succ</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha_limit" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha_limit"><span class="id" title="section">alpha_limit</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim" class="idref" href="#Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim"><span class="id" title="variable">Hlim</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RBlim" class="idref" href="#RBlim"><span class="id" title="lemma">RBlim</span></a> : <span class="id" title="keyword">∀</span> <a id="n:114" class="idref" href="#n:114"><span class="id" title="binder">n</span></a>, (<a class="idref" href="hydras.Epsilon0.Hprime.html#n:114"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:114"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RAlim" class="idref" href="#RAlim"><span class="id" title="lemma">RAlim</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RClim" class="idref" href="#RClim"><span class="id" title="lemma">RClim</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RDlim" class="idref" href="#RDlim"><span class="id" title="lemma">RDlim</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="RElim" class="idref" href="#RElim"><span class="id" title="lemma">RElim</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:115" class="idref" href="#beta:115"><span class="id" title="binder">beta</span></a> <a id="n:116" class="idref" href="#n:116"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:115"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:115"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:116"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_limit"><span class="id" title="section">alpha_limit</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="P_alpha_0" class="idref" href="#P_alpha_0"><span class="id" title="lemma">P_alpha_0</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#P"><span class="id" title="record">P</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction"><span class="id" title="section">The_induction</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="P_alpha" class="idref" href="#P_alpha"><span class="id" title="lemma">P_alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:117" class="idref" href="#alpha:117"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#P"><span class="id" title="record">P</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:117"><span class="id" title="variable">alpha</span></a>.   
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties"><span class="id" title="section">Proof_of_Abstract_Properties</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Abstract_Properties" class="idref" href="#Abstract_Properties"><span class="id" title="section">Abstract_Properties</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Abstract_Properties.alpha" class="idref" href="#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_alpha_mono" class="idref" href="#H'_alpha_mono"><span class="id" title="lemma">H'_alpha_mono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_alpha_gt" class="idref" href="#H'_alpha_gt"><span class="id" title="lemma">H'_alpha_gt</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:119" class="idref" href="#n:119"><span class="id" title="binder">n</span></a>, (<a class="idref" href="hydras.Epsilon0.Hprime.html#n:119"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:119"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_alpha_Succ_le" class="idref" href="#H'_alpha_Succ_le"><span class="id" title="lemma">H'_alpha_Succ_le</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_alpha_dom" class="idref" href="#H'_alpha_dom"><span class="id" title="lemma">H'_alpha_dom</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">H'_</span></span> is not mononotonous in <span class="inlinecode"><span class="id" title="var">alpha</span></span> in general. 
+      Nevertheless, this lemma may help (from <span class="inlinecode"><span class="id" title="var">KS</span></span>) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_restricted_mono_l" class="idref" href="#H'_restricted_mono_l"><span class="id" title="lemma">H'_restricted_mono_l</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:120" class="idref" href="#beta:120"><span class="id" title="binder">beta</span></a> <a id="n:121" class="idref" href="#n:121"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:121"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:120"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:120"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:121"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#n:121"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="H'_alpha_ge_id" class="idref" href="#H'_alpha_ge_id"><span class="id" title="lemma">H'_alpha_ge_id</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="H'_alpha_mono_weak" class="idref" href="#H'_alpha_mono_weak"><span class="id" title="lemma">H'_alpha_mono_weak</span></a> : <span class="id" title="keyword">∀</span> <a id="k:122" class="idref" href="#k:122"><span class="id" title="binder">k</span></a> <a id="l:123" class="idref" href="#l:123"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#k:122"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#l:123"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#k:122"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#l:123"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Abstract_Properties"><span class="id" title="section">Abstract_Properties</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab33"></a><h2 class="section">Monotony of <span class="inlinecode"><span class="id" title="var">H'</span></span> w.r.t. its first argument</h2>
+
+
+<div class="paragraph"> </div>
+
+   Although Lemma <span class="inlinecode"><span class="id" title="var">H'_non_mono1</span></span> tells us that <span class="inlinecode"><span class="id" title="var">H'</span></span> is not monotonous
+   with respect to its argument <span class="inlinecode"><span class="id" title="var">alpha</span></span>, we prove that, if <span class="inlinecode"><span class="id" title="var">alpha</span>&lt;<span class="id" title="keyword">beta</span></span>, then
+   <span class="inlinecode"><span class="id" title="var">H'_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode"><span class="id" title="var">k</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">H'_</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span> <span class="inlinecode"><span class="id" title="var">k</span></span> for large enough <span class="inlinecode"><span class="id" title="var">k</span></span>.
+  For that purpose, we apply a few lemmas from the Ketonen-Solovay article.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_mono_l_0" class="idref" href="#H'_mono_l_0"><span class="id" title="lemma">H'_mono_l_0</span></a> <a id="alpha:124" class="idref" href="#alpha:124"><span class="id" title="binder">alpha</span></a> <a id="beta:125" class="idref" href="#beta:125"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:124"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:125"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:126" class="idref" href="#n:126"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:127" class="idref" href="#p:127"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:126"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:127"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:124"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:127"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:125"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:127"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="H'_mono_l_1" class="idref" href="#H'_mono_l_1"><span class="id" title="lemma">H'_mono_l_1</span></a> <a id="alpha:128" class="idref" href="#alpha:128"><span class="id" title="binder">alpha</span></a> <a id="beta:129" class="idref" href="#beta:129"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:128"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:129"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:130" class="idref" href="#n:130"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:131" class="idref" href="#p:131"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:130"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:131"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#alpha:128"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:131"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#beta:129"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:131"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_H'_mono_l" class="idref" href="#Proof_of_H'_mono_l"><span class="id" title="section">Proof_of_H'_mono_l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Proof_of_H'_mono_l.alpha" class="idref" href="#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> <a id="Proof_of_H'_mono_l.beta" class="idref" href="#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_H'_mono_l.H_alpha_beta" class="idref" href="#Proof_of_H'_mono_l.H_alpha_beta"><span class="id" title="variable">H_alpha_beta</span></a>: <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_H'_mono_l.Succ_case" class="idref" href="#Proof_of_H'_mono_l.Succ_case"><span class="id" title="section">Succ_case</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_H'_mono_l.Succ_case.gamma" class="idref" href="#Proof_of_H'_mono_l.Succ_case.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_H'_mono_l.Succ_case.Hgamma" class="idref" href="#Proof_of_H'_mono_l.Succ_case.Hgamma"><span class="id" title="variable">Hgamma</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:137" class="idref" href="#n:137"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:138" class="idref" href="#p:138"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:137"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:138"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:138"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.gamma"><span class="id" title="variable">gamma</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:138"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:139" class="idref" href="#n:139"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:140" class="idref" href="#p:140"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:139"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:140"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:140"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:140"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case"><span class="id" title="section">Succ_case</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_H'_mono_l.Limit_case" class="idref" href="#Proof_of_H'_mono_l.Limit_case"><span class="id" title="section">Limit_case</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_H'_mono_l.Limit_case.Hbeta" class="idref" href="#Proof_of_H'_mono_l.Limit_case.Hbeta"><span class="id" title="variable">Hbeta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4" class="idref" href="#R4"><span class="id" title="lemma">R4</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:142" class="idref" href="#n:142"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:143" class="idref" href="#p:143"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:142"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:143"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:143"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:143"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Limit_case"><span class="id" title="section">Limit_case</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="H'_mono_l" class="idref" href="#H'_mono_l"><span class="id" title="lemma">H'_mono_l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:144" class="idref" href="#n:144"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <a id="p:145" class="idref" href="#p:145"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.Hprime.html#n:144"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:145"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:145"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#p:145"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="H'_dom" class="idref" href="#H'_dom"><span class="id" title="lemma">H'_dom</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha"><span class="id" title="variable">alpha</span></a>). 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l"><span class="id" title="section">Proof_of_H'_mono_l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Epsilon0.Hprime.html#H'_dom"><span class="id" title="lemma">H'_dom</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <br/>
+&nbsp;&nbsp;<a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">&lt;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">H'_</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">ltac</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">:(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">mko</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">×</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1omega</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">×</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">T1omega</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">t1</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">))</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">12)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">nat</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.L_alpha.html b/theories/html/hydras.Epsilon0.L_alpha.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.L_alpha</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.L_alpha</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab92"></a><h1 class="section">Computing the length of paths from <span class="inlinecode"><span class="id" title="var">alpha</span></span> to <span class="inlinecode"><span class="id" title="var">zero</span></span></h1>
+
+
+<div class="paragraph"> </div>
+
+ After Wainer, Ketonen, Solovay, etc .
+
+<div class="paragraph"> </div>
+
+    Pierre Casteran, LaBRI, University of  Bordeaux 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Epsilon0.Hprime.html#"><span class="id" title="library">Hprime</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>  <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="library">Large_Sets</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> .<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Equations</span>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="module">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Olt" class="idref" href="#Olt"><span class="id" title="instance">Olt</span></a> : <span class="id" title="class">WellFounded</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Olt</span> : <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Using Coq-Equations for building a function which satisfies 
+    <span class="inlinecode"><span class="id" title="var">Large_sets.L_spec</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Equations</span>  <a id="L_" class="idref" href="#L_"><span class="id" title="definition">L_</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_:6"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:4"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:5"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0_eq_dec"><span class="id" title="lemma">E0_eq_dec</span></a> <span class="id" title="var">alpha</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#_zero:9"><span class="id" title="variable">_zero</span></a> ⇒ <span class="id" title="var">i</span> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#_nonzero:14"><span class="id" title="variable">_nonzero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Utils.html#dec"><span class="id" title="abbreviation">Utils.dec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <span class="id" title="var">alpha</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{ | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#_limit:20"><span class="id" title="variable">_limit</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_:3"><span class="id" title="definition">L_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">i</span>)  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>) ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#_successor:28"><span class="id" title="variable">_successor</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_:3"><span class="id" title="definition">L_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <span class="id" title="var">alpha</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>)}}.<br/>
+
+<br/>
+<span class="id" title="keyword">Solve</span> <span class="id" title="keyword">All</span> <span class="id" title="keyword">Obligations</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">E0</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L__equation_1"><span class="id" title="definition">L__equation_1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Paraphrase of equations generated by coq-equations 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_zero_eqn" class="idref" href="#L_zero_eqn"><span class="id" title="lemma">L_zero_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="i:31" class="idref" href="#i:31"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:31"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:31"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_eq2" class="idref" href="#L_eq2"><span class="id" title="lemma">L_eq2</span></a> <a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> <a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0is_succ"><span class="id" title="definition">E0is_succ</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_pred"><span class="id" title="definition">E0_pred</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:32"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:33"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_succ_eqn" class="idref" href="#L_succ_eqn"><span class="id" title="lemma">L_succ_eqn</span></a> <a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a> <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:34"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:34"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:35"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Rewrite</span> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_zero_eqn"><span class="id" title="lemma">L_zero_eqn</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_succ_eqn"><span class="id" title="lemma">L_succ_eqn</span></a> : <span class="id" title="var">L_rw</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_lim_eqn" class="idref" href="#L_lim_eqn"><span class="id" title="lemma">L_lim_eqn</span></a> <a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:37"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:37"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:37"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_finite" class="idref" href="#L_finite"><span class="id" title="lemma">L_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="i:38" class="idref" href="#i:38"><span class="id" title="binder">i</span></a> <a id="k:39" class="idref" href="#k:39"><span class="id" title="binder">k</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:38"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#k:39"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Epsilon0.L_alpha.html#i:38"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.L_alpha.html#k:39"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega" class="idref" href="#L_omega"><span class="id" title="lemma">L_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="k:40" class="idref" href="#k:40"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#k:40"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#k:40"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_ge_id" class="idref" href="#L_ge_id"><span class="id" title="lemma">L_ge_id</span></a> <a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="i:42" class="idref" href="#i:42"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:42"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:41"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:42"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_ge_S" class="idref" href="#L_ge_S"><span class="id" title="lemma">L_ge_S</span></a> <a id="alpha:43" class="idref" href="#alpha:43"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:43"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:43"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_succ_ok" class="idref" href="#L_succ_ok"><span class="id" title="lemma">L_succ_ok</span></a>  <a id="beta:44" class="idref" href="#beta:44"><span class="id" title="binder">beta</span></a> <a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:44"><span class="id" title="variable">beta</span></a>)  (<span class="id" title="keyword">fun</span> <a id="k:46" class="idref" href="#k:46"><span class="id" title="binder">k</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.L_alpha.html#f:45"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#k:46"><span class="id" title="variable">k</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">L_</span></span> is correct w.r.t. its specification 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="L_correct_proof" class="idref" href="#L_correct_proof"><span class="id" title="section">L_correct_proof</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="L_correct_proof.P" class="idref" href="#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a id="alpha:47" class="idref" href="#alpha:47"><span class="id" title="binder">alpha</span></a> :=  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:47"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:47"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_ok0" class="idref" href="#L_ok0"><span class="id" title="lemma">L_ok0</span></a> : <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_ok_succ" class="idref" href="#L_ok_succ"><span class="id" title="lemma">L_ok_succ</span></a> <a id="beta:48" class="idref" href="#beta:48"><span class="id" title="binder">beta</span></a>  : <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:48"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:48"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_ok_lim" class="idref" href="#L_ok_lim"><span class="id" title="lemma">L_ok_lim</span></a>  <a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a>  :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:50" class="idref" href="#beta:50"><span class="id" title="binder">beta</span></a>,  (<a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:50"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#beta:50"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:49"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_ok" class="idref" href="#L_ok"><span class="id" title="lemma">L_ok</span></a> (<a id="alpha:57" class="idref" href="#alpha:57"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) : <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:57"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_correct_proof"><span class="id" title="section">L_correct_proof</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="L_correct" class="idref" href="#L_correct"><span class="id" title="lemma">L_correct</span></a> <a id="alpha:58" class="idref" href="#alpha:58"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:58"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:58"><span class="id" title="variable">alpha</span></a>).  
+<br/>
+</div>
+
+<div class="doc">
+Comparison with Hardy's function H  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="H'_L_" class="idref" href="#H'_L_"><span class="id" title="lemma">H'_L_</span></a> <a id="alpha:59" class="idref" href="#alpha:59"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  (<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#alpha:59"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#i:60"><span class="id" title="variable">i</span></a>))%<span class="id" title="var">nat</span>. 
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.extraction.Extraction.html#"><span class="id" title="library">Extraction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Recursive Extraction</span> <span class="id" title="var">L_</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.Large_Sets.html b/theories/html/hydras.Epsilon0.Large_Sets.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Epsilon0.Large_Sets.html
@@ -0,0 +1,729 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Large_Sets</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Large_Sets</h1>
+
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="library">MoreLists</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="module">Relation_Operators</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="module">Sorted</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Operators_Properties.html#"><span class="id" title="module">Operators_Properties</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab18"></a><h2 class="section">minimal large sequences</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mlarge" class="idref" href="#mlarge"><span class="id" title="definition">mlarge</span></a> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> (<a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:2"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:1"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mlargeS" class="idref" href="#mlargeS"><span class="id" title="definition">mlargeS</span></a> <a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> <a id="s:4" class="idref" href="#s:4"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:4"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:3"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab19"></a><h2 class="section">Last step of a minimal path</h2>
+
+<div class="paragraph"> </div>
+
+ TODO : compare with K &amp; S's H_alpha (p. 297) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="L_spec" class="idref" href="#L_spec"><span class="id" title="inductive">L_spec</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="L_spec0" class="idref" href="#L_spec0"><span class="id" title="constructor">L_spec0</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:7" class="idref" href="#f:7"><span class="id" title="binder">f</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="k:8" class="idref" href="#k:8"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:7"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:8"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:8"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec:5"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:7"><span class="id" title="variable">f</span></a><br/>
+| <a id="L_spec1" class="idref" href="#L_spec1"><span class="id" title="constructor">L_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:9"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="k:11" class="idref" href="#k:11"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:9"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:11"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:10"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:11"><span class="id" title="variable">k</span></a>))))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec:5"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:9"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:10"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Test functions 
+<ul class="doclist">
+<li> If f is correct w.r.t. <span class="inlinecode"><span class="id" title="var">L_spec</span></span>, "Compute L_test alpha f k" should return (one=one).
+
+</li>
+<li> If f k is too small, returns (alpha = one) (with one &lt; alpha)
+
+</li>
+<li> If f k is too big, returns (zero = one) 
+
+</li>
+</ul>
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_test" class="idref" href="#L_test"><span class="id" title="definition">L_test</span></a> (<a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) <a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> <a id="k:14" class="idref" href="#k:14"><span class="id" title="binder">k</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:12"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:14"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:14"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 2)%<span class="id" title="var">nat</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab20"></a><h2 class="section">Paths starting with a finite ordinal (<span class="inlinecode"><span class="id" title="var">fin</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>)</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_finite_1_iota" class="idref" href="#gnaw_finite_1_iota"><span class="id" title="lemma">gnaw_finite_1_iota</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> <a id="i:17" class="idref" href="#i:17"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:16"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:17"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:16"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_finite_1" class="idref" href="#gnaw_finite_1"><span class="id" title="lemma">gnaw_finite_1</span></a> : <span class="id" title="keyword">∀</span> <a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a> <a id="i:19" class="idref" href="#i:19"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:18"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:19"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:19"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_finite" class="idref" href="#gnaw_finite"><span class="id" title="lemma">gnaw_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> <a id="i:21" class="idref" href="#i:21"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:20"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:21"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:20"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_n_R" class="idref" href="#gnaw_n_R"><span class="id" title="lemma">gnaw_n_R</span></a> : <span class="id" title="keyword">∀</span> <a id="s:22" class="idref" href="#s:22"><span class="id" title="binder">s</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:22"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:23"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:22"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:23"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:22"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab21"></a><h2 class="section">Properties of m-largeness and L-functions</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_unicity" class="idref" href="#mlarge_unicity"><span class="id" title="lemma">mlarge_unicity</span></a> <a id="alpha:24" class="idref" href="#alpha:24"><span class="id" title="binder">alpha</span></a> <a id="k:25" class="idref" href="#k:25"><span class="id" title="binder">k</span></a> <a id="l:26" class="idref" href="#l:26"><span class="id" title="binder">l</span></a> <a id="l':27" class="idref" href="#l':27"><span class="id" title="binder">l'</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:24"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:25"><span class="id" title="variable">k</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#l:26"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:24"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:25"><span class="id" title="variable">k</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#l':27"><span class="id" title="variable">l'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#l:26"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#l':27"><span class="id" title="variable">l'</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlargeS_iff" class="idref" href="#mlargeS_iff"><span class="id" title="lemma">mlargeS_iff</span></a> <a id="alpha:28" class="idref" href="#alpha:28"><span class="id" title="binder">alpha</span></a> <a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a> <a id="s:30" class="idref" href="#s:30"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:30"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlargeS"><span class="id" title="definition">mlargeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:28"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:29"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:30"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:28"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:29"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:30"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_unshift" class="idref" href="#mlarge_unshift"><span class="id" title="lemma">mlarge_unshift</span></a> <a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a> <a id="s:32" class="idref" href="#s:32"><span class="id" title="binder">s</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:32"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:32"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlargeS"><span class="id" title="definition">mlargeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:31"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:32"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_iff" class="idref" href="#mlarge_iff"><span class="id" title="lemma">mlarge_iff</span></a> <a id="alpha:33" class="idref" href="#alpha:33"><span class="id" title="binder">alpha</span></a> <a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a> (<a id="s:35" class="idref" href="#s:35"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:35"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:34"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:35"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:33"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:34"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:35"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:33"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:35"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab22"></a><h2 class="section">About the length of mlarge intervals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_spec_inv2" class="idref" href="#L_spec_inv2"><span class="id" title="lemma">L_spec_inv2</span></a> <a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> <a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="k:38" class="idref" href="#k:38"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:36"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:38"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:37"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:38"><span class="id" title="variable">k</span></a>))))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_spec_compat" class="idref" href="#L_spec_compat"><span class="id" title="lemma">L_spec_compat</span></a> <a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a> (<a id="f:40" class="idref" href="#f:40"><span class="id" title="binder">f</span></a> <a id="g:41" class="idref" href="#g:41"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:40"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:40"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:42"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#g:41"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:42"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#g:41"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab23"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">L_spec</span></span></h2>
+
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_zero_inv" class="idref" href="#L_zero_inv"><span class="id" title="lemma">L_zero_inv</span></a> <a id="f:43" class="idref" href="#f:43"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:43"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="k:44" class="idref" href="#k:44"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:43"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:44"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:44"><span class="id" title="variable">k</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_pos_inv" class="idref" href="#L_pos_inv"><span class="id" title="lemma">L_pos_inv</span></a> <a id="alpha:45" class="idref" href="#alpha:45"><span class="id" title="binder">alpha</span></a>  <a id="f:46" class="idref" href="#f:46"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:45"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:45"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:47" class="idref" href="#k:47"><span class="id" title="binder">k</span></a>, (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:47"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:46"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:47"><span class="id" title="variable">k</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_spec_unicity" class="idref" href="#L_spec_unicity"><span class="id" title="lemma">L_spec_unicity</span></a> <a id="alpha:48" class="idref" href="#alpha:48"><span class="id" title="binder">alpha</span></a> <a id="f:49" class="idref" href="#f:49"><span class="id" title="binder">f</span></a> <a id="g:50" class="idref" href="#g:50"><span class="id" title="binder">g</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:48"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:48"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#g:50"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="k:51" class="idref" href="#k:51"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:49"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:51"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#g:50"><span class="id" title="variable">g</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:51"><span class="id" title="variable">k</span></a>). 
+<br/>
+</div>
+
+<div class="doc">
+Composition lemmas for computing <span class="inlinecode"><span class="id" title="var">L_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="succ" class="idref" href="#succ"><span class="id" title="section">succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="succ.beta" class="idref" href="#succ.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="succ.f" class="idref" href="#succ.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypotheses</span> (<a id="succ.Hbeta" class="idref" href="#succ.Hbeta"><span class="id" title="variable">Hbeta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.beta"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="succ.f_mono" class="idref" href="#succ.f_mono"><span class="id" title="variable">f_mono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.f"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="succ.f_Sle" class="idref" href="#succ.f_Sle"><span class="id" title="variable">f_Sle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.f"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="succ.f_ok" class="idref" href="#succ.f_ok"><span class="id" title="variable">f_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.f"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="L_succ" class="idref" href="#L_succ"><span class="id" title="definition">L_succ</span></a> := <span class="id" title="keyword">fun</span> <a id="k:58" class="idref" href="#k:58"><span class="id" title="binder">k</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:58"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_succ_mono" class="idref" href="#L_succ_mono"><span class="id" title="lemma">L_succ_mono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_succ"><span class="id" title="definition">L_succ</span></a>. 
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_succ_Sle" class="idref" href="#L_succ_Sle"><span class="id" title="lemma">L_succ_Sle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_succ"><span class="id" title="definition">L_succ</span></a>. 
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_succ_ok" class="idref" href="#L_succ_ok"><span class="id" title="lemma">L_succ_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ.beta"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_succ"><span class="id" title="definition">L_succ</span></a>. <span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#succ"><span class="id" title="section">succ</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="lim" class="idref" href="#lim"><span class="id" title="section">lim</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="lim.lambda" class="idref" href="#lim.lambda"><span class="id" title="variable">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lim.Hnf" class="idref" href="#lim.Hnf"><span class="id" title="variable">Hnf</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lambda:59"><span class="id" title="variable">lambda</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lim.Hlim" class="idref" href="#lim.Hlim"><span class="id" title="variable">Hlim</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lambda:59"><span class="id" title="variable">lambda</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lim.f" class="idref" href="#lim.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lim.H" class="idref" href="#lim.H"><span class="id" title="variable">H</span></a> : <span class="id" title="keyword">∀</span> <a id="k:63" class="idref" href="#k:63"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lambda:59"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:63"><span class="id" title="variable">k</span></a>)) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#f:62"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:63"><span class="id" title="variable">k</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="canon_not_null" class="idref" href="#canon_not_null"><span class="id" title="lemma">canon_not_null</span></a> : <span class="id" title="keyword">∀</span> <a id="k:65" class="idref" href="#k:65"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lim.lambda"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:65"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span>  <a id="L_lim" class="idref" href="#L_lim"><span class="id" title="definition">L_lim</span></a> <a id="k:66" class="idref" href="#k:66"><span class="id" title="binder">k</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:66"><span class="id" title="variable">k</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:66"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L_lim_ok" class="idref" href="#L_lim_ok"><span class="id" title="lemma">L_lim_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lim.lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_lim"><span class="id" title="definition">L_lim</span></a>. 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#lim"><span class="id" title="section">lim</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab24"></a><h2 class="section">Finite ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_fin" class="idref" href="#L_fin"><span class="id" title="definition">L_fin</span></a> <a id="i:67" class="idref" href="#i:67"><span class="id" title="binder">i</span></a> := <span class="id" title="keyword">fun</span> <a id="k:68" class="idref" href="#k:68"><span class="id" title="binder">k</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:67"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:68"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_finS_S_le" class="idref" href="#L_finS_S_le"><span class="id" title="lemma">L_finS_S_le</span></a> <a id="i:69" class="idref" href="#i:69"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:69"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_fin_smono" class="idref" href="#L_fin_smono"><span class="id" title="lemma">L_fin_smono</span></a> <a id="i:70" class="idref" href="#i:70"><span class="id" title="binder">i</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:70"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_S_succ_rw" class="idref" href="#L_S_succ_rw"><span class="id" title="lemma">L_S_succ_rw</span></a> <a id="i:71" class="idref" href="#i:71"><span class="id" title="binder">i</span></a> : <span class="id" title="keyword">∀</span> <a id="k:72" class="idref" href="#k:72"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:71"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:72"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_succ"><span class="id" title="definition">L_succ</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:71"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:72"><span class="id" title="variable">k</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_fin_ok" class="idref" href="#L_fin_ok"><span class="id" title="lemma">L_fin_ok</span></a> <a id="i:73" class="idref" href="#i:73"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:73"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_fin"><span class="id" title="definition">L_fin</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:73"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_FS" class="idref" href="#mlarge_FS"><span class="id" title="lemma">mlarge_FS</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:74" class="idref" href="#i:74"><span class="id" title="binder">i</span></a> <a id="k:75" class="idref" href="#k:75"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:74"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:75"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:74"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:75"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>)).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_omega" class="idref" href="#L_omega"><span class="id" title="definition">L_omega</span></a> <a id="k:76" class="idref" href="#k:76"><span class="id" title="binder">k</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:76"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_Sle" class="idref" href="#L_omega_Sle"><span class="id" title="lemma">L_omega_Sle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega"><span class="id" title="definition">L_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_smono" class="idref" href="#L_omega_smono"><span class="id" title="lemma">L_omega_smono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega"><span class="id" title="definition">L_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mlarge_omega" class="idref" href="#mlarge_omega"><span class="id" title="lemma">mlarge_omega</span></a> <a id="k:77" class="idref" href="#k:77"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:77"><span class="id" title="variable">k</span></a>) (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:77"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nat</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_ok" class="idref" href="#L_omega_ok"><span class="id" title="lemma">L_omega_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega"><span class="id" title="definition">L_omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_omega_mult" class="idref" href="#path_to_omega_mult"><span class="id" title="lemma">path_to_omega_mult</span></a> (<a id="i:82" class="idref" href="#i:82"><span class="id" title="binder">i</span></a> <a id="k:83" class="idref" href="#k:83"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:82"><span class="id" title="variable">i</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:83"><span class="id" title="variable">k</span></a>) (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:83"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nat</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:82"><span class="id" title="variable">i</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>). 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_mult_mlarge_0" class="idref" href="#omega_mult_mlarge_0"><span class="id" title="lemma">omega_mult_mlarge_0</span></a> <a id="i:84" class="idref" href="#i:84"><span class="id" title="binder">i</span></a>  : <span class="id" title="keyword">∀</span> <a id="k:85" class="idref" href="#k:85"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:84"><span class="id" title="variable">i</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:85"><span class="id" title="variable">k</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<span class="id" title="keyword">fun</span> <a id="p:86" class="idref" href="#p:86"><span class="id" title="binder">p</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#p:86"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:84"><span class="id" title="variable">i</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:85"><span class="id" title="variable">k</span></a>)))). 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_omega_mult" class="idref" href="#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> <a id="i:87" class="idref" href="#i:87"><span class="id" title="binder">i</span></a> (<a id="x:88" class="idref" href="#x:88"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega"><span class="id" title="definition">L_omega</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:87"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#x:88"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex" class="idref" href="#Ex"><span class="id" title="definition">Ex</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> 8 5 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1535.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_mult_Sle" class="idref" href="#L_omega_mult_Sle"><span class="id" title="lemma">L_omega_mult_Sle</span></a> <a id="i:89" class="idref" href="#i:89"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:89"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_mult_ok" class="idref" href="#L_omega_mult_ok"><span class="id" title="lemma">L_omega_mult_ok</span></a> (<a id="i:90" class="idref" href="#i:90"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:90"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:90"><span class="id" title="variable">i</span></a>). 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_mult_eqn" class="idref" href="#L_omega_mult_eqn"><span class="id" title="lemma">L_omega_mult_eqn</span></a> (<a id="i:91" class="idref" href="#i:91"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="k:92" class="idref" href="#k:92"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:92"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:91"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:92"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:91"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:92"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab25"></a><h2 class="section"><span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">omega</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_omega_square" class="idref" href="#L_omega_square"><span class="id" title="definition">L_omega_square</span></a> <a id="k:93" class="idref" href="#k:93"><span class="id" title="binder">k</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<span class="id" title="keyword">fun</span> <a id="z:94" class="idref" href="#z:94"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#z:94"><span class="id" title="variable">z</span></a>)%<span class="id" title="var">nat</span>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:93"><span class="id" title="variable">k</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:93"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="L_omega_square_eqn1" class="idref" href="#L_omega_square_eqn1"><span class="id" title="lemma">L_omega_square_eqn1</span></a> <a id="k:95" class="idref" href="#k:95"><span class="id" title="binder">k</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:95"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_mult"><span class="id" title="definition">L_omega_mult</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:95"><span class="id" title="variable">k</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:95"><span class="id" title="variable">k</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab26"></a><h3 class="section">Closed formula</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_square_eqn" class="idref" href="#L_omega_square_eqn"><span class="id" title="lemma">L_omega_square_eqn</span></a> <a id="k:96" class="idref" href="#k:96"><span class="id" title="binder">k</span></a> :<br/>
+&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:96"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:96"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:96"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:96"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_square_Sle" class="idref" href="#L_omega_square_Sle"><span class="id" title="lemma">L_omega_square_Sle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_square_smono" class="idref" href="#L_omega_square_smono"><span class="id" title="lemma">L_omega_square_smono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_square_ok" class="idref" href="#L_omega_square_ok"><span class="id" title="lemma">L_omega_square_ok</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="phi0_mult" class="idref" href="#phi0_mult"><span class="id" title="section">phi0_mult</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="phi0_mult.alpha" class="idref" href="#phi0_mult.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="phi0_mult.f" class="idref" href="#phi0_mult.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypotheses</span> (<a id="phi0_mult.Halpha" class="idref" href="#phi0_mult.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="phi0_mult.f_mono" class="idref" href="#phi0_mult.f_mono"><span class="id" title="variable">f_mono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="phi0_mult.f_Sle" class="idref" href="#phi0_mult.f_Sle"><span class="id" title="variable">f_Sle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="phi0_mult.f_ok" class="idref" href="#phi0_mult.f_ok"><span class="id" title="variable">f_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="L_phi0_mult" class="idref" href="#L_phi0_mult"><span class="id" title="definition">L_phi0_mult</span></a> <a id="i:109" class="idref" href="#i:109"><span class="id" title="binder">i</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:109"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Remark</span> <a id="f_ok_inv" class="idref" href="#f_ok_inv"><span class="id" title="lemma">f_ok_inv</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:110" class="idref" href="#k:110"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:110"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:110"><span class="id" title="variable">k</span></a>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_phi0_mult_ok" class="idref" href="#L_phi0_mult_ok"><span class="id" title="lemma">L_phi0_mult_ok</span></a> <a id="i:111" class="idref" href="#i:111"><span class="id" title="binder">i</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:111"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult"><span class="id" title="definition">L_phi0_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:111"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_phi0_mult_smono" class="idref" href="#L_phi0_mult_smono"><span class="id" title="lemma">L_phi0_mult_smono</span></a> <a id="i:112" class="idref" href="#i:112"><span class="id" title="binder">i</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult"><span class="id" title="definition">L_phi0_mult</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:112"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_phi0_mult_Sle" class="idref" href="#L_phi0_mult_Sle"><span class="id" title="lemma">L_phi0_mult_Sle</span></a> <a id="i:113" class="idref" href="#i:113"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult"><span class="id" title="definition">L_phi0_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:113"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#phi0_mult"><span class="id" title="section">phi0_mult</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_omega_square_times" class="idref" href="#L_omega_square_times"><span class="id" title="definition">L_omega_square_times</span></a> <a id="i:114" class="idref" href="#i:114"><span class="id" title="binder">i</span></a> :=  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square"><span class="id" title="definition">L_omega_square</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:114"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_square_times_ok" class="idref" href="#L_omega_square_times_ok"><span class="id" title="lemma">L_omega_square_times_ok</span></a> <a id="i:115" class="idref" href="#i:115"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> 2 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:115"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times"><span class="id" title="definition">L_omega_square_times</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:115"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="L_omega_cube" class="idref" href="#L_omega_cube"><span class="id" title="definition">L_omega_cube</span></a>  := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_lim"><span class="id" title="definition">L_lim</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times"><span class="id" title="definition">L_omega_square_times</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_cube_ok" class="idref" href="#L_omega_cube_ok"><span class="id" title="lemma">L_omega_cube_ok</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_spec"><span class="id" title="inductive">L_spec</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> 3) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_cube"><span class="id" title="definition">L_omega_cube</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_cube_eqn" class="idref" href="#L_omega_cube_eqn"><span class="id" title="lemma">L_omega_cube_eqn</span></a> <a id="i:116" class="idref" href="#i:116"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_cube"><span class="id" title="definition">L_omega_cube</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:116"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times"><span class="id" title="definition">L_omega_square_times</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:116"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:116"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_k_mult_pos" class="idref" href="#exp2_k_mult_pos"><span class="id" title="lemma">exp2_k_mult_pos</span></a> <a id="k:117" class="idref" href="#k:117"><span class="id" title="binder">k</span></a>:<br/>
+&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:117"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 4 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:117"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:117"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="omega_square_thrice_eqn" class="idref" href="#omega_square_thrice_eqn"><span class="id" title="definition">omega_square_thrice_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="k:118" class="idref" href="#k:118"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:118"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="M:119" class="idref" href="#M:119"><span class="id" title="binder">M</span></a> := (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:118"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:118"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="N:120" class="idref" href="#N:120"><span class="id" title="binder">N</span></a> := <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#M:119"><span class="id" title="variable">M</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="P:121" class="idref" href="#P:121"><span class="id" title="binder">P</span></a> := (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#N:120"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#M:119"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times"><span class="id" title="definition">L_omega_square_times</span></a> 3 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#k:118"><span class="id" title="variable">k</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#P:121"><span class="id" title="variable">P</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#P:121"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_omega_cube_3_eq" class="idref" href="#L_omega_cube_3_eq"><span class="id" title="lemma">L_omega_cube_3_eq</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="N:123" class="idref" href="#N:123"><span class="id" title="binder">N</span></a> := <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> 95 <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="P:124" class="idref" href="#P:124"><span class="id" title="binder">P</span></a> := (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#N:123"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 97 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#L_omega_cube"><span class="id" title="definition">L_omega_cube</span></a> 3  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#P:124"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#P:124"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">nat</span>. 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab27"></a><h1 class="section">Plain large sequences (not necessarily minimal)</h1>
+
+
+<div class="paragraph"> </div>
+
+Note : Some names of lemmas and theorem like <span class="inlinecode"><span class="id" title="var">Theorem_4_5</span></span> or
+<span class="inlinecode"><span class="id" title="var">Lemma_4_5_2</span></span> refer to sections in Ketonen and Solovay's article.
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab28"></a><h2 class="section">large sequences (as in KS ) (may contain zeros)</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="largeb" class="idref" href="#largeb"><span class="id" title="definition">largeb</span></a> (<a id="alpha:125" class="idref" href="#alpha:125"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:126" class="idref" href="#s:126"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:125"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:126"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="large" class="idref" href="#large"><span class="id" title="definition">large</span></a> (<a id="alpha:127" class="idref" href="#alpha:127"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:128" class="idref" href="#s:128"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:127"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:128"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="largeSb" class="idref" href="#largeSb"><span class="id" title="definition">largeSb</span></a> (<a id="alpha:129" class="idref" href="#alpha:129"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:130" class="idref" href="#s:130"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:129"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:130"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="largeS" class="idref" href="#largeS"><span class="id" title="definition">largeS</span></a> (<a id="alpha:131" class="idref" href="#alpha:131"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:132" class="idref" href="#s:132"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeSb"><span class="id" title="definition">largeSb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:131"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:132"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Large" class="idref" href="#Large"><span class="id" title="definition">Large</span></a> <a id="alpha:133" class="idref" href="#alpha:133"><span class="id" title="binder">alpha</span></a> <a id="s:134" class="idref" href="#s:134"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:133"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:134"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="large_iff" class="idref" href="#large_iff"><span class="id" title="lemma">large_iff</span></a> (<a id="alpha:135" class="idref" href="#alpha:135"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:136" class="idref" href="#s:136"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:135"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:136"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:135"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:136"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeSb_b" class="idref" href="#largeSb_b"><span class="id" title="lemma">largeSb_b</span></a> (<a id="alpha:137" class="idref" href="#alpha:137"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:138" class="idref" href="#s:138"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeSb"><span class="id" title="definition">largeSb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:137"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:138"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:137"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:138"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeb_Sb" class="idref" href="#largeb_Sb"><span class="id" title="lemma">largeb_Sb</span></a> <a id="alpha:139" class="idref" href="#alpha:139"><span class="id" title="binder">alpha</span></a> <a id="s:140" class="idref" href="#s:140"><span class="id" title="binder">s</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:139"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:140"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeSb"><span class="id" title="definition">largeSb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:139"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:140"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeS_iff" class="idref" href="#largeS_iff"><span class="id" title="lemma">largeS_iff</span></a> (<a id="alpha:141" class="idref" href="#alpha:141"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:142" class="idref" href="#s:142"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:141"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:142"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:141"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:142"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Lemma_4_4_Proof" class="idref" href="#Lemma_4_4_Proof"><span class="id" title="section">Lemma_4_4_Proof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Lemma_4_4_Proof.alpha" class="idref" href="#Lemma_4_4_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a id="Lemma_4_4_Proof.beta" class="idref" href="#Lemma_4_4_Proof.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_4_Proof.Halpha" class="idref" href="#Lemma_4_4_Proof.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_4_Proof.Hbeta" class="idref" href="#Lemma_4_4_Proof.Hbeta"><span class="id" title="variable">Hbeta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Lemma_4_4_Proof.n" class="idref" href="#Lemma_4_4_Proof.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Lemma_4_4_Proof.s" class="idref" href="#Lemma_4_4_Proof.s"><span class="id" title="variable">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_4_Proof.Hs" class="idref" href="#Lemma_4_4_Proof.Hs"><span class="id" title="variable">Hs</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_4_Proof.H" class="idref" href="#Lemma_4_4_Proof.H"><span class="id" title="variable">H</span></a> :  <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_4_Proof.H0" class="idref" href="#Lemma_4_4_Proof.H0"><span class="id" title="variable">H0</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.s"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma4_4" class="idref" href="#Lemma4_4"><span class="id" title="lemma">Lemma4_4</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.s"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof"><span class="id" title="section">Lemma_4_4_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_4_2" class="idref" href="#Lemma_4_4_2"><span class="id" title="lemma">Lemma_4_4_2</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:152" class="idref" href="#s:152"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="n:153" class="idref" href="#n:153"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="alpha:154" class="idref" href="#alpha:154"><span class="id" title="binder">alpha</span></a> <a id="beta:155" class="idref" href="#beta:155"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:154"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#beta:155"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:153"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:152"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:153"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:154"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#beta:155"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#beta:155"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:152"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:154"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:152"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_5_1" class="idref" href="#Lemma_4_5_1"><span class="id" title="lemma">Lemma_4_5_1</span></a>  <a id="n:156" class="idref" href="#n:156"><span class="id" title="binder">n</span></a> <a id="alpha:157" class="idref" href="#alpha:157"><span class="id" title="binder">alpha</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:158" class="idref" href="#s:158"><span class="id" title="binder">s</span></a> <a id="t:159" class="idref" href="#t:159"><span class="id" title="binder">t</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#ptwise_le"><span class="id" title="definition">ptwise_le</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:158"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#t:159"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:156"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:158"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:156"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#t:159"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:156"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#t:159"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#t:159"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:158"><span class="id" title="variable">s</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_4_5_2" class="idref" href="#Proof_of_4_5_2"><span class="id" title="section">Proof_of_4_5_2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Proof_of_4_5_2.A" class="idref" href="#Proof_of_4_5_2.A"><span class="id" title="variable">A</span></a> <a id="Proof_of_4_5_2.B1" class="idref" href="#Proof_of_4_5_2.B1"><span class="id" title="variable">B1</span></a> <a id="Proof_of_4_5_2.B2" class="idref" href="#Proof_of_4_5_2.B2"><span class="id" title="variable">B2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_4_5_2.alpha" class="idref" href="#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_4_5_2.Halpha" class="idref" href="#Proof_of_4_5_2.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_4_5_2.HA" class="idref" href="#Proof_of_4_5_2.HA"><span class="id" title="variable">HA</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_4_5_2.HB" class="idref" href="#Proof_of_4_5_2.HB"><span class="id" title="variable">HB</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> 0 (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B2"><span class="id" title="variable">B2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_4_5_2.HAB1" class="idref" href="#Proof_of_4_5_2.HAB1"><span class="id" title="variable">HAB1</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#ptwise_le"><span class="id" title="definition">ptwise_le</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_4_5_2.HlargeA" class="idref" href="#Proof_of_4_5_2.HlargeA"><span class="id" title="variable">HlargeA</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A"><span class="id" title="variable">A</span></a> .<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a>  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_5_2" class="idref" href="#Lemma_4_5_2"><span class="id" title="lemma">Lemma_4_5_2</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B2"><span class="id" title="variable">B2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2"><span class="id" title="section">Proof_of_4_5_2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Theorem_4_5" class="idref" href="#Theorem_4_5"><span class="id" title="lemma">Theorem_4_5</span></a> (<a id="alpha:169" class="idref" href="#alpha:169"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)(<a id="Halpha:170" class="idref" href="#Halpha:170"><span class="id" title="binder">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:169"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:171" class="idref" href="#A:171"><span class="id" title="binder">A</span></a> <a id="B:172" class="idref" href="#B:172"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HA:173" class="idref" href="#HA:173"><span class="id" title="binder">HA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#A:171"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HB:174" class="idref" href="#HB:174"><span class="id" title="binder">HB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#B:172"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HAB:175" class="idref" href="#HAB:175"><span class="id" title="binder">HAB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#incl"><span class="id" title="definition">incl</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#A:171"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#B:172"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:169"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#A:171"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:169"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#B:172"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_last_step" class="idref" href="#gnaw_last_step"><span class="id" title="lemma">gnaw_last_step</span></a> <a id="alpha:176" class="idref" href="#alpha:176"><span class="id" title="binder">alpha</span></a> <a id="s:177" class="idref" href="#s:177"><span class="id" title="binder">s</span></a> <a id="i:178" class="idref" href="#i:178"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:176"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:177"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:176"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:177"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:178"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ For alpha in class E0 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LargeS" class="idref" href="#LargeS"><span class="id" title="definition">LargeS</span></a> (<a id="alpha:179" class="idref" href="#alpha:179"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) <a id="s:180" class="idref" href="#s:180"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeS"><span class="id" title="definition">largeS</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:179"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:180"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gnaw" class="idref" href="#Gnaw"><span class="id" title="definition">Gnaw</span></a> <a id="alpha:181" class="idref" href="#alpha:181"><span class="id" title="binder">alpha</span></a> <a id="s:182" class="idref" href="#s:182"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:181"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:182"><span class="id" title="variable">s</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="GnawS" class="idref" href="#GnawS"><span class="id" title="definition">GnawS</span></a> <a id="alpha:183" class="idref" href="#alpha:183"><span class="id" title="binder">alpha</span></a> <a id="s:184" class="idref" href="#s:184"><span class="id" title="binder">s</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:183"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:184"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gnaw_GnawS" class="idref" href="#Gnaw_GnawS"><span class="id" title="lemma">Gnaw_GnawS</span></a> <a id="s:185" class="idref" href="#s:185"><span class="id" title="binder">s</span></a> <a id="alpha:186" class="idref" href="#alpha:186"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#GnawS"><span class="id" title="definition">GnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:186"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:185"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Gnaw"><span class="id" title="definition">Gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:186"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:185"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="GnawS_Gnaw" class="idref" href="#GnawS_Gnaw"><span class="id" title="lemma">GnawS_Gnaw</span></a> (<a id="s:187" class="idref" href="#s:187"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:188" class="idref" href="#alpha:188"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Gnaw"><span class="id" title="definition">Gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:188"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:187"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#GnawS"><span class="id" title="definition">GnawS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:188"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:187"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="GnawS_omega" class="idref" href="#GnawS_omega"><span class="id" title="lemma">GnawS_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="i:189" class="idref" href="#i:189"><span class="id" title="binder">i</span></a> <a id="s:190" class="idref" href="#s:190"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#GnawS"><span class="id" title="definition">GnawS</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:189"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:190"><span class="id" title="variable">s</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#GnawS"><span class="id" title="definition">GnawS</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0finS"><span class="id" title="instance">E0finS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:189"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:190"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Gnaw_omega" class="idref" href="#Gnaw_omega"><span class="id" title="lemma">Gnaw_omega</span></a> <a id="i:191" class="idref" href="#i:191"><span class="id" title="binder">i</span></a> <a id="s:192" class="idref" href="#s:192"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Gnaw"><span class="id" title="definition">Gnaw</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:191"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:192"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Gnaw"><span class="id" title="definition">Gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0finS"><span class="id" title="instance">E0finS</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:191"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:192"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Largeb" class="idref" href="#Largeb"><span class="id" title="definition">Largeb</span></a> (<a id="alpha:193" class="idref" href="#alpha:193"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="s:194" class="idref" href="#s:194"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:193"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:194"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LargeSb" class="idref" href="#LargeSb"><span class="id" title="definition">LargeSb</span></a> (<a id="alpha:195" class="idref" href="#alpha:195"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="s:196" class="idref" href="#s:196"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeSb"><span class="id" title="definition">largeSb</span></a> (@<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:195"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:196"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LargeSb_b" class="idref" href="#LargeSb_b"><span class="id" title="lemma">LargeSb_b</span></a> (<a id="alpha:197" class="idref" href="#alpha:197"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) (<a id="s:198" class="idref" href="#s:198"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#LargeSb"><span class="id" title="definition">LargeSb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:197"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:198"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#Largeb"><span class="id" title="definition">Largeb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:197"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:198"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Largeb_Sb" class="idref" href="#Largeb_Sb"><span class="id" title="lemma">Largeb_Sb</span></a> <a id="alpha:199" class="idref" href="#alpha:199"><span class="id" title="binder">alpha</span></a> <a id="s:200" class="idref" href="#s:200"><span class="id" title="binder">s</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#Largeb"><span class="id" title="definition">Largeb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:199"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:200"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#LargeSb"><span class="id" title="definition">LargeSb</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#alpha:199"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:200"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeb_finite" class="idref" href="#largeb_finite"><span class="id" title="lemma">largeb_finite</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:201" class="idref" href="#n:201"><span class="id" title="binder">n</span></a> <a id="i:202" class="idref" href="#i:202"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:201"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#i:202"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:201"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeb_n" class="idref" href="#largeb_n"><span class="id" title="lemma">largeb_n</span></a> (<a id="n:203" class="idref" href="#n:203"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <span class="id" title="keyword">∀</span> <a id="s:204" class="idref" href="#s:204"><span class="id" title="binder">s</span></a>,   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:204"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:203"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:204"><span class="id" title="variable">s</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:203"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:204"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="largeb_n_R" class="idref" href="#largeb_n_R"><span class="id" title="lemma">largeb_n_R</span></a> : <span class="id" title="keyword">∀</span> <a id="s:205" class="idref" href="#s:205"><span class="id" title="binder">s</span></a> <a id="n:206" class="idref" href="#n:206"><span class="id" title="binder">n</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:205"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:206"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:205"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#largeb"><span class="id" title="definition">largeb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:206"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:205"><span class="id" title="variable">s</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="large_n_iff" class="idref" href="#large_n_iff"><span class="id" title="lemma">large_n_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="s:207" class="idref" href="#s:207"><span class="id" title="binder">s</span></a> (<a id="n:208" class="idref" href="#n:208"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:207"><span class="id" title="variable">s</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:208"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:207"><span class="id" title="variable">s</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:208"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:207"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex3" class="idref" href="#ex3"><span class="id" title="definition">ex3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> 156 (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 100 254).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Gnawing omega  omega-large intervals 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_omega_n_SSn" class="idref" href="#gnaw_omega_n_SSn"><span class="id" title="lemma">gnaw_omega_n_SSn</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:209" class="idref" href="#n:209"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:209"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:209"><span class="id" title="variable">n</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_omega_1" class="idref" href="#gnaw_omega_1"><span class="id" title="lemma">gnaw_omega_1</span></a> (<a id="n:210" class="idref" href="#n:210"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:210"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:210"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:210"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="omega_ex1" class="idref" href="#omega_ex1"><span class="id" title="definition">omega_ex1</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 7 13) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="omega_ex2" class="idref" href="#omega_ex2"><span class="id" title="definition">omega_ex2</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a>  1000 1999) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="large_omega_1" class="idref" href="#large_omega_1"><span class="id" title="lemma">large_omega_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:211" class="idref" href="#s:211"><span class="id" title="binder">s</span></a> <a id="n:212" class="idref" href="#n:212"><span class="id" title="binder">n</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:212"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:211"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:212"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:211"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:212"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:211"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="large_omega_2" class="idref" href="#large_omega_2"><span class="id" title="lemma">large_omega_2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:213" class="idref" href="#s:213"><span class="id" title="binder">s</span></a> <a id="n:214" class="idref" href="#n:214"><span class="id" title="binder">n</span></a>,   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:214"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:213"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:214"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:213"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:214"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:213"><span class="id" title="variable">s</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="large_omega_iff" class="idref" href="#large_omega_iff"><span class="id" title="lemma">large_omega_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="s:215" class="idref" href="#s:215"><span class="id" title="binder">s</span></a> <a id="n:216" class="idref" href="#n:216"><span class="id" title="binder">n</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:216"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:215"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:216"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:215"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Large_Sets.html#n:216"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#s:215"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.Large_Sets_Examples.html b/theories/html/hydras.Epsilon0.Large_Sets_Examples.html
new file mode 100644
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+++ b/theories/html/hydras.Epsilon0.Large_Sets_Examples.html
@@ -0,0 +1,172 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Large_Sets_Examples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Large_Sets_Examples</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="library">MoreLists</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="library">Large_Sets</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_pathS1" class="idref" href="#ex_pathS1"><span class="id" title="definition">ex_pathS1</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#pathS"><span class="id" title="definition">pathS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>4<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_pathS2" class="idref" href="#ex_pathS2"><span class="id" title="definition">ex_pathS2</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#pathS"><span class="id" title="definition">pathS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 3 6) 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="omega_omega_1_4" class="idref" href="#omega_omega_1_4"><span class="id" title="definition">omega_omega_1_4</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 1 4) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="omega_omega_1_3" class="idref" href="#omega_omega_1_3"><span class="id" title="definition">omega_omega_1_3</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 1 3) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="answer" class="idref" href="#answer"><span class="id" title="inductive">answer</span></a> : <span class="id" title="keyword">Set</span> := <a id="Ok" class="idref" href="#Ok"><span class="id" title="constructor">Ok</span></a> | <a id="Too_far" class="idref" href="#Too_far"><span class="id" title="constructor">Too_far</span></a> | <a id="Remaining" class="idref" href="#Remaining"><span class="id" title="constructor">Remaining</span></a> (<a id="rest:3" class="idref" href="#rest:3"><span class="id" title="binder">rest</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="large_set_check" class="idref" href="#large_set_check"><span class="id" title="definition">large_set_check</span></a> <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> <a id="j:6" class="idref" href="#j:6"><span class="id" title="binder">j</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="beta:7" class="idref" href="#beta:7"><span class="id" title="binder">beta</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#alpha:4"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#i:5"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#j:6"><span class="id" title="variable">j</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#beta:7"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#Ok"><span class="id" title="constructor">Ok</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#Too_far"><span class="id" title="constructor">Too_far</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#Remaining"><span class="id" title="constructor">Remaining</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#pp"><span class="id" title="definition">pp</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#beta:7"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets_Examples.html#j:6"><span class="id" title="variable">j</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">iota_from_lt_not_In</span>: <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 6 510).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#large"><span class="id" title="definition">large</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 6 700).<br/>
+ 
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+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.Paths</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.Paths</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+*
+
+<div class="paragraph"> </div>
+
+   Transition systems associated with canonical sequences
+ 
+<div class="paragraph"> </div>
+
+  P. Casteran, University of Bordeaux and Labri
+
+<div class="paragraph"> </div>
+
+       After J. Ketonen and R. Solovay's paper
+  " Rapidly Growing Ramsey Functions" <i>in</i>
+    Annals of mathematics, Mar. 1981 
+
+<div class="paragraph"> </div>
+
+ 
+<div class="paragraph"> </div>
+
+ TODO: Check wether the predicates ...PathS... are still useful 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="library">MoreLists</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#"><span class="id" title="library">First_toggle</span></a> <a class="idref" href="hydras.Prelude.OrdNotations.html#"><span class="id" title="library">OrdNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="module">Relation_Operators</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab75"></a><h2 class="section">relations  associated with canonical sequences</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transition_S" class="idref" href="#transition_S"><span class="id" title="definition">transition_S</span></a> <a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> <a id="beta:3" class="idref" href="#beta:3"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:3"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:2"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:1"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transition" class="idref" href="#transition"><span class="id" title="definition">transition</span></a> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Paths.html#i:4"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#transition_S"><span class="id" title="definition">transition_S</span></a> <span class="id" title="var">j</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bounded_transitionS" class="idref" href="#bounded_transitionS"><span class="id" title="definition">bounded_transitionS</span></a> (<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> <a id="beta:8" class="idref" href="#beta:8"><span class="id" title="binder">beta</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:9" class="idref" href="#i:9"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:9"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:6"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#transition_S"><span class="id" title="definition">transition_S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:9"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:8"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab76"></a><h2 class="section">Paths inside epsilon_0</h2>
+
+
+<div class="paragraph"> </div>
+
+  In this module, we study paths; i.e. sequences generated by the relations 
+associated with   the <span class="inlinecode"><span class="id" title="var">canonS</span></span> <span class="inlinecode"><span class="id" title="var">i</span></span> functions. In module <span class="inlinecode"><span class="id" title="var">O2H</span></span> we show how pathes are 
+ related to hydra battles. Thus, the various classes of battles are described by predicates on pathes.
+
+<div class="paragraph"> </div>
+
+ path_toS beta s alpha :
+    pathS alpha s beta :
+
+<div class="paragraph"> </div>
+
+   beta is reachable from alpha through the sequences s of indices
+   ( zeros are ignored) 
+
+<div class="paragraph"> </div>
+
+   Note that only beta can be equal to zero
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="path_to" class="idref" href="#path_to"><span class="id" title="inductive">path_to</span></a> (<a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="path_to_1" class="idref" href="#path_to_1"><span class="id" title="constructor">path_to_1</span></a> : <span class="id" title="keyword">∀</span> (<a id="i:13" class="idref" href="#i:13"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:13"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#transition"><span class="id" title="definition">transition</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:13"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:10"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to:11"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:10"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:13"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:14"><span class="id" title="variable">alpha</span></a><br/>
+| <a id="path_to_cons" class="idref" href="#path_to_cons"><span class="id" title="constructor">path_to_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:15" class="idref" href="#i:15"><span class="id" title="binder">i</span></a> <a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a> <a id="s:17" class="idref" href="#s:17"><span class="id" title="binder">s</span></a> <a id="gamma:18" class="idref" href="#gamma:18"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:15"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#transition"><span class="id" title="definition">transition</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:15"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:16"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:18"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to:11"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:10"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:18"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to:11"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:10"><span class="id" title="variable">beta</span></a>  (<a class="idref" href="hydras.Epsilon0.Paths.html#i:15"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:17"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:16"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="path" class="idref" href="#path"><span class="id" title="definition">path</span></a> <a id="alpha:19" class="idref" href="#alpha:19"><span class="id" title="binder">alpha</span></a> <a id="s:20" class="idref" href="#s:20"><span class="id" title="binder">s</span></a> <a id="beta:21" class="idref" href="#beta:21"><span class="id" title="binder">beta</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:21"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:20"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:19"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+tries to solve a goal of the form (path_to beta s alpha) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">path_tac</span> := <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> (<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| [ |- <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> ?<span class="id" title="var">b</span> (?<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> ?<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> ?<span class="id" title="var">s</span>) ?<span class="id" title="var">a</span> ] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">eright</span>; [<span class="id" title="tactic">discriminate</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="tactic">split</span>;[<span class="id" title="tactic">discriminate</span> | <span class="id" title="tactic">reflexivity</span>]<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">cbn</span>])<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  [ |- <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> ?<span class="id" title="var">b</span> (?<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) ?<span class="id" title="var">a</span> ] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">eleft</span>; [<span class="id" title="tactic">discriminate</span> |<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>;<span class="id" title="var">cbn</span> ; [<span class="id" title="tactic">discriminate</span> | <span class="id" title="tactic">reflexivity</span>]])<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>).<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path1" class="idref" href="#ex_path1"><span class="id" title="definition">ex_path1</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 2).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path2" class="idref" href="#ex_path2"><span class="id" title="definition">ex_path2</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>4<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>5<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a>6<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 2).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path3" class="idref" href="#ex_path3"><span class="id" title="definition">ex_path3</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> 3 14) (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 2).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex_path4" class="idref" href="#ex_path4"><span class="id" title="definition">ex_path4</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#repeat"><span class="id" title="definition">List.repeat</span></a> 3 8) (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 2).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+path_toS beta s alpha :
+    pathS alpha s beta :
+
+<div class="paragraph"> </div>
+
+   beta is reachable from alpha through the sequence of indices
+   (shift s)
+
+<div class="paragraph"> </div>
+
+   Note that only beta can be equal to zero
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_not_nil" class="idref" href="#path_to_not_nil"><span class="id" title="lemma">path_to_not_nil</span></a>  <a id="alpha:22" class="idref" href="#alpha:22"><span class="id" title="binder">alpha</span></a> <a id="s:23" class="idref" href="#s:23"><span class="id" title="binder">s</span></a> <a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:24"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:23"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:22"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:23"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="path_toS" class="idref" href="#path_toS"><span class="id" title="inductive">path_toS</span></a> (<a id="beta:25" class="idref" href="#beta:25"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+|  <a id="path_toS_1" class="idref" href="#path_toS_1"><span class="id" title="constructor">path_toS_1</span></a> : <span class="id" title="keyword">∀</span> (<a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="alpha:29" class="idref" href="#alpha:29"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#transition_S"><span class="id" title="definition">transition_S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:28"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:25"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS:26"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:25"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:28"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:29"><span class="id" title="variable">alpha</span></a><br/>
+| <a id="path_toS_cons" class="idref" href="#path_toS_cons"><span class="id" title="constructor">path_toS_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:30" class="idref" href="#i:30"><span class="id" title="binder">i</span></a> <a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a> <a id="s:32" class="idref" href="#s:32"><span class="id" title="binder">s</span></a> <a id="gamma:33" class="idref" href="#gamma:33"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#transition_S"><span class="id" title="definition">transition_S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:30"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:33"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS:26"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:25"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#s:32"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:33"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS:26"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:25"><span class="id" title="variable">beta</span></a>  (<a class="idref" href="hydras.Epsilon0.Paths.html#i:30"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:32"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:31"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pathS" class="idref" href="#pathS"><span class="id" title="definition">pathS</span></a> <a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a> <a id="s:35" class="idref" href="#s:35"><span class="id" title="binder">s</span></a> <a id="beta:36" class="idref" href="#beta:36"><span class="id" title="binder">beta</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:36"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:35"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:34"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab77"></a><h2 class="section">Accessibility (without traces)</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="acc_from" class="idref" href="#acc_from"><span class="id" title="definition">acc_from</span></a> <a id="alpha:37" class="idref" href="#alpha:37"><span class="id" title="binder">alpha</span></a> <a id="beta:38" class="idref" href="#beta:38"><span class="id" title="binder">beta</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s:39" class="idref" href="#s:39"><span class="id" title="binder">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:38"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:39"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:37"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab78"></a><h2 class="section">following a path</h2>
+
+<div class="paragraph"> </div>
+
+  useful helper ? 
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="gnawS" class="idref" href="#gnawS"><span class="id" title="definition">gnawS</span></a> (<a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:41" class="idref" href="#s:41"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span>  <a class="idref" href="hydras.Epsilon0.Paths.html#s:41"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:40"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (<span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s'</span>) ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS:42"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:40"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>)) <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="gnaw" class="idref" href="#gnaw"><span class="id" title="definition">gnaw</span></a> (<a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="s:45" class="idref" href="#s:45"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span>  <a class="idref" href="hydras.Epsilon0.Paths.html#s:45"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:44"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s'</span>) ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw:46"><span class="id" title="definition">gnaw</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:44"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span>)  ⇒  <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw:46"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:44"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span>)) <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="standard_gnaw" class="idref" href="#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a id="i:48" class="idref" href="#i:48"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)(<a id="l:50" class="idref" href="#l:50"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):  <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>  :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Paths.html#l:50"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:49"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw:51"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:48"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:48"><span class="id" title="variable">i</span></a>) <span class="id" title="var">m</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+alpha ---&gt; beta in KP 
+             n
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="KP_arrowS" class="idref" href="#KP_arrowS"><span class="id" title="definition">KP_arrowS</span></a> <a id="n:53" class="idref" href="#n:53"><span class="id" title="binder">n</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_trans_1n"><span class="id" title="inductive">clos_trans_1n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#bounded_transitionS"><span class="id" title="definition">bounded_transitionS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:53"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab79"></a><h2 class="section">Paths with constant index</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="const_pathS" class="idref" href="#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a id="i:54" class="idref" href="#i:54"><span class="id" title="binder">i</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_trans_1n"><span class="id" title="inductive">clos_trans_1n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> (<span class="id" title="keyword">fun</span> <a id="alpha:55" class="idref" href="#alpha:55"><span class="id" title="binder">alpha</span></a> <a id="beta:56" class="idref" href="#beta:56"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:56"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:55"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:54"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="const_path" class="idref" href="#const_path"><span class="id" title="definition">const_path</span></a> <a id="i:57" class="idref" href="#i:57"><span class="id" title="binder">i</span></a> <a id="alpha:58" class="idref" href="#alpha:58"><span class="id" title="binder">alpha</span></a> <a id="beta:59" class="idref" href="#beta:59"><span class="id" title="binder">beta</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.Paths.html#i:57"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <span class="id" title="var">j</span> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:58"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:59"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathSE" class="idref" href="#const_pathSE"><span class="id" title="lemma">const_pathSE</span></a> <a id="i:61" class="idref" href="#i:61"><span class="id" title="binder">i</span></a> <a id="alpha:62" class="idref" href="#alpha:62"><span class="id" title="binder">alpha</span></a> <a id="beta:63" class="idref" href="#beta:63"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:62"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:63"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:61"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:62"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:63"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="const_pathS_eps" class="idref" href="#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a id="i:64" class="idref" href="#i:64"><span class="id" title="binder">i</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:64"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab80"></a><h2 class="section">standard paths</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="standard_path_toS" class="idref" href="#standard_path_toS"><span class="id" title="inductive">standard_path_toS</span></a> (<a id="j:65" class="idref" href="#j:65"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)( <a id="beta:66" class="idref" href="#beta:66"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="stdS_1" class="idref" href="#stdS_1"><span class="id" title="constructor">stdS_1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:69" class="idref" href="#i:69"><span class="id" title="binder">i</span></a> <a id="alpha:70" class="idref" href="#alpha:70"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:66"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:70"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:69"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:65"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:69"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS:67"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:65"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:66"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:69"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:70"><span class="id" title="variable">alpha</span></a><br/>
+| <a id="stdS_S" class="idref" href="#stdS_S"><span class="id" title="constructor">stdS_S</span></a> : <span class="id" title="keyword">∀</span> <a id="i:71" class="idref" href="#i:71"><span class="id" title="binder">i</span></a> <a id="alpha:72" class="idref" href="#alpha:72"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS:67"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:65"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:66"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:71"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:72"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:71"><span class="id" title="variable">i</span></a>))  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS:67"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:65"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:66"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:71"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:72"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="standard_pathS" class="idref" href="#standard_pathS"><span class="id" title="definition">standard_pathS</span></a>  <a id="i:73" class="idref" href="#i:73"><span class="id" title="binder">i</span></a> <a id="alpha:74" class="idref" href="#alpha:74"><span class="id" title="binder">alpha</span></a> <a id="j:75" class="idref" href="#j:75"><span class="id" title="binder">j</span></a> <a id="beta:76" class="idref" href="#beta:76"><span class="id" title="binder">beta</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:75"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:76"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:73"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:74"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="standard_path_to" class="idref" href="#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a> (<a id="j:77" class="idref" href="#j:77"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="beta:78" class="idref" href="#beta:78"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="std_1" class="idref" href="#std_1"><span class="id" title="constructor">std_1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:81" class="idref" href="#i:81"><span class="id" title="binder">i</span></a> <a id="alpha:82" class="idref" href="#alpha:82"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:82"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:78"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:82"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:77"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to:79"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:77"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:78"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:81"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:82"><span class="id" title="variable">alpha</span></a><br/>
+| <a id="std_S" class="idref" href="#std_S"><span class="id" title="constructor">std_S</span></a> : <span class="id" title="keyword">∀</span> <a id="i:83" class="idref" href="#i:83"><span class="id" title="binder">i</span></a> <a id="alpha:84" class="idref" href="#alpha:84"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to:79"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:77"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:78"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:83"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:84"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:83"><span class="id" title="variable">i</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to:79"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:77"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:78"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:83"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:84"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="standard_path" class="idref" href="#standard_path"><span class="id" title="definition">standard_path</span></a>  <a id="i:85" class="idref" href="#i:85"><span class="id" title="binder">i</span></a> <a id="alpha:86" class="idref" href="#alpha:86"><span class="id" title="binder">alpha</span></a> <a id="j:87" class="idref" href="#j:87"><span class="id" title="binder">j</span></a> <a id="beta:88" class="idref" href="#beta:88"><span class="id" title="binder">beta</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:87"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:88"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:85"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:86"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_interval_inv_le" class="idref" href="#path_to_interval_inv_le"><span class="id" title="lemma">path_to_interval_inv_le</span></a> <a id="alpha:89" class="idref" href="#alpha:89"><span class="id" title="binder">alpha</span></a> <a id="beta:90" class="idref" href="#beta:90"><span class="id" title="binder">beta</span></a> <a id="i:91" class="idref" href="#i:91"><span class="id" title="binder">i</span></a> <a id="j:92" class="idref" href="#j:92"><span class="id" title="binder">j</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:90"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:91"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:92"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:89"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:91"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:92"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab81"></a><h2 class="section">Bridge Lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_gnaw" class="idref" href="#gnawS_gnaw"><span class="id" title="lemma">gnawS_gnaw</span></a> <a id="s:93" class="idref" href="#s:93"><span class="id" title="binder">s</span></a> <a id="alpha:94" class="idref" href="#alpha:94"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:94"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:93"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:94"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:93"><span class="id" title="variable">s</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_gnawS" class="idref" href="#gnaw_gnawS"><span class="id" title="lemma">gnaw_gnawS</span></a> <a id="s:95" class="idref" href="#s:95"><span class="id" title="binder">s</span></a> <a id="alpha:96" class="idref" href="#alpha:96"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:95"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:96"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:95"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_path_to" class="idref" href="#path_toS_path_to"><span class="id" title="lemma">path_toS_path_to</span></a>  <a id="alpha:97" class="idref" href="#alpha:97"><span class="id" title="binder">alpha</span></a> <a id="s:98" class="idref" href="#s:98"><span class="id" title="binder">s</span></a> <a id="beta:99" class="idref" href="#beta:99"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:99"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:98"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:97"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:99"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:98"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:97"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_path_toS" class="idref" href="#path_to_path_toS"><span class="id" title="lemma">path_to_path_toS</span></a>  :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:100" class="idref" href="#alpha:100"><span class="id" title="binder">alpha</span></a> <a id="s:101" class="idref" href="#s:101"><span class="id" title="binder">s</span></a> <a id="beta:102" class="idref" href="#beta:102"><span class="id" title="binder">beta</span></a>,   <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:102"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:101"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:100"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:102"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:101"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:100"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_path_toS_iff" class="idref" href="#path_to_path_toS_iff"><span class="id" title="lemma">path_to_path_toS_iff</span></a>  :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:103" class="idref" href="#alpha:103"><span class="id" title="binder">alpha</span></a> <a id="s:104" class="idref" href="#s:104"><span class="id" title="binder">s</span></a> <a id="beta:105" class="idref" href="#beta:105"><span class="id" title="binder">beta</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#s:104"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:105"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:104"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:103"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:105"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:104"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:103"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_nf" class="idref" href="#path_toS_nf"><span class="id" title="lemma">path_toS_nf</span></a> <a id="beta:106" class="idref" href="#beta:106"><span class="id" title="binder">beta</span></a> <a id="s:107" class="idref" href="#s:107"><span class="id" title="binder">s</span></a> <a id="alpha:108" class="idref" href="#alpha:108"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:106"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:107"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:106"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_acc_from" class="idref" href="#path_acc_from"><span class="id" title="lemma">path_acc_from</span></a> <a id="alpha:109" class="idref" href="#alpha:109"><span class="id" title="binder">alpha</span></a> <a id="s:110" class="idref" href="#s:110"><span class="id" title="binder">s</span></a> <a id="beta:111" class="idref" href="#beta:111"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:110"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:111"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_gnawS" class="idref" href="#path_toS_gnawS"><span class="id" title="lemma">path_toS_gnawS</span></a> : <span class="id" title="keyword">∀</span> <a id="s:112" class="idref" href="#s:112"><span class="id" title="binder">s</span></a> <a id="alpha:113" class="idref" href="#alpha:113"><span class="id" title="binder">alpha</span></a> <a id="beta:114" class="idref" href="#beta:114"><span class="id" title="binder">beta</span></a>,   <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:112"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:112"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab82"></a><h2 class="section">Composition/decomposition of paths</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_app" class="idref" href="#path_toS_app"><span class="id" title="lemma">path_toS_app</span></a> <a id="beta:115" class="idref" href="#beta:115"><span class="id" title="binder">beta</span></a> <a id="t:116" class="idref" href="#t:116"><span class="id" title="binder">t</span></a> <a id="gamma:117" class="idref" href="#gamma:117"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:117"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:116"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:115"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="alpha:118" class="idref" href="#alpha:118"><span class="id" title="binder">alpha</span></a> <a id="s:119" class="idref" href="#s:119"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:115"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:119"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:118"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:117"><span class="id" title="variable">gamma</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#s:119"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#t:116"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">list</span> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:118"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_app" class="idref" href="#path_to_app"><span class="id" title="lemma">path_to_app</span></a> <a id="beta:120" class="idref" href="#beta:120"><span class="id" title="binder">beta</span></a> <a id="t:121" class="idref" href="#t:121"><span class="id" title="binder">t</span></a> <a id="gamma:122" class="idref" href="#gamma:122"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:122"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:121"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:120"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="alpha:123" class="idref" href="#alpha:123"><span class="id" title="binder">alpha</span></a> <a id="s:124" class="idref" href="#s:124"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:120"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:124"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:123"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:122"><span class="id" title="variable">gamma</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#s:124"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#t:121"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">list</span> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:123"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_decompose" class="idref" href="#path_toS_decompose"><span class="id" title="lemma">path_toS_decompose</span></a> <a id="gamma:125" class="idref" href="#gamma:125"><span class="id" title="binder">gamma</span></a> <a id="s:126" class="idref" href="#s:126"><span class="id" title="binder">s</span></a> <a id="alpha:127" class="idref" href="#alpha:127"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:128" class="idref" href="#beta:128"><span class="id" title="binder">beta</span></a> <a id="t:129" class="idref" href="#t:129"><span class="id" title="binder">t</span></a> <a id="u:130" class="idref" href="#u:130"><span class="id" title="binder">u</span></a>,  <a class="idref" href="hydras.Epsilon0.Paths.html#s:126"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:129"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#u:130"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:128"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:129"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:127"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:125"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:130"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:128"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:125"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:126"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:127"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_decompose" class="idref" href="#path_to_decompose"><span class="id" title="lemma">path_to_decompose</span></a> <a id="gamma:131" class="idref" href="#gamma:131"><span class="id" title="binder">gamma</span></a>  <a id="alpha:132" class="idref" href="#alpha:132"><span class="id" title="binder">alpha</span></a> <a id="beta:133" class="idref" href="#beta:133"><span class="id" title="binder">beta</span></a> <a id="s:134" class="idref" href="#s:134"><span class="id" title="binder">s</span></a> <a id="t:135" class="idref" href="#t:135"><span class="id" title="binder">t</span></a> <a id="u:136" class="idref" href="#u:136"><span class="id" title="binder">u</span></a>  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#s:134"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:135"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:136"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:133"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:135"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:132"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:131"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:136"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:133"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:131"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:134"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:132"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">path_decompose</span> <span class="id" title="var">x</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> |- <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> ?<span class="id" title="var">a</span> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> ?<span class="id" title="var">from</span> ?<span class="id" title="var">to</span>) ?<span class="id" title="var">b</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval_app"><span class="id" title="lemma">interval_app</span></a> <span class="id" title="var">from</span> <span class="id" title="var">x</span> <span class="id" title="var">to</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to_app"><span class="id" title="lemma">path_to_app</span></a> | <span class="id" title="tactic">try</span> <span class="id" title="var">lia</span> | <span class="id" title="tactic">try</span> <span class="id" title="var">lia</span>]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_appR" class="idref" href="#path_toS_appR"><span class="id" title="lemma">path_toS_appR</span></a>     : <span class="id" title="keyword">∀</span> <a id="u:137" class="idref" href="#u:137"><span class="id" title="binder">u</span></a>  <a id="gamma:138" class="idref" href="#gamma:138"><span class="id" title="binder">gamma</span></a>  <a id="alpha:139" class="idref" href="#alpha:139"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:138"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:137"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:139"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:140" class="idref" href="#s:140"><span class="id" title="binder">s</span></a> <a id="t:141" class="idref" href="#t:141"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#s:140"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:141"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:137"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:140"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:141"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:142" class="idref" href="#beta:142"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:142"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:140"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:139"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:138"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:141"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:142"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_appR" class="idref" href="#path_to_appR"><span class="id" title="lemma">path_to_appR</span></a>     : <span class="id" title="keyword">∀</span> <a id="u:143" class="idref" href="#u:143"><span class="id" title="binder">u</span></a>  <a id="gamma:144" class="idref" href="#gamma:144"><span class="id" title="binder">gamma</span></a>  <a id="alpha:145" class="idref" href="#alpha:145"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:144"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:143"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:145"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:146" class="idref" href="#s:146"><span class="id" title="binder">s</span></a> <a id="t:147" class="idref" href="#t:147"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#s:146"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:147"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:143"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:146"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:147"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#u:143"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:148" class="idref" href="#beta:148"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:148"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:146"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:145"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:144"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:147"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:148"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_zero_but_last" class="idref" href="#path_toS_zero_but_last"><span class="id" title="lemma">path_toS_zero_but_last</span></a> : <span class="id" title="keyword">∀</span> <a id="x:149" class="idref" href="#x:149"><span class="id" title="binder">x</span></a> <a id="u:150" class="idref" href="#u:150"><span class="id" title="binder">u</span></a> <a id="alpha:151" class="idref" href="#alpha:151"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#x:149"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#u:150"><span class="id" title="variable">u</span></a>)  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:151"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:150"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#x:149"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:150"><span class="id" title="variable">u</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:151"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_zero_one" class="idref" href="#path_toS_zero_one"><span class="id" title="lemma">path_toS_zero_one</span></a> : <span class="id" title="keyword">∀</span> <a id="x:152" class="idref" href="#x:152"><span class="id" title="binder">x</span></a>  <a id="alpha:153" class="idref" href="#alpha:153"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#x:152"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:153"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:153"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_zero_inv" class="idref" href="#path_toS_zero_inv"><span class="id" title="lemma">path_toS_zero_inv</span></a> <a id="x:154" class="idref" href="#x:154"><span class="id" title="binder">x</span></a> <a id="u:155" class="idref" href="#u:155"><span class="id" title="binder">u</span></a> <a id="alpha:156" class="idref" href="#alpha:156"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#x:154"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#u:155"><span class="id" title="variable">u</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:156"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#alpha:156"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:155"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#u:155"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#x:154"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#u:155"><span class="id" title="variable">u</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:156"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_zero" class="idref" href="#path_toS_zero"><span class="id" title="lemma">path_toS_zero</span></a> <a id="alpha:157" class="idref" href="#alpha:157"><span class="id" title="binder">alpha</span></a> <a id="s:158" class="idref" href="#s:158"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:158"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_zero" class="idref" href="#path_to_zero"><span class="id" title="lemma">path_to_zero</span></a> <a id="alpha:159" class="idref" href="#alpha:159"><span class="id" title="binder">alpha</span></a> <a id="s:160" class="idref" href="#s:160"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:159"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:160"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+todo : use this lemma to remove useless hypotheses 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_not_In_zero" class="idref" href="#path_to_not_In_zero"><span class="id" title="lemma">path_to_not_In_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:161" class="idref" href="#alpha:161"><span class="id" title="binder">alpha</span></a> <a id="s:162" class="idref" href="#s:162"><span class="id" title="binder">s</span></a> <a id="beta:163" class="idref" href="#beta:163"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:163"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:162"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:161"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#s:162"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_tail" class="idref" href="#path_toS_tail"><span class="id" title="lemma">path_toS_tail</span></a>  <a id="alpha:164" class="idref" href="#alpha:164"><span class="id" title="binder">alpha</span></a> <a id="s:165" class="idref" href="#s:165"><span class="id" title="binder">s</span></a> <a id="beta:166" class="idref" href="#beta:166"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:166"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:165"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:164"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="gamma:167" class="idref" href="#gamma:167"><span class="id" title="binder">gamma</span></a> <a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:167"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:164"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:167"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:166"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:165"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:167"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:164"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_tail" class="idref" href="#path_to_tail"><span class="id" title="lemma">path_to_tail</span></a>  <a id="alpha:169" class="idref" href="#alpha:169"><span class="id" title="binder">alpha</span></a> <a id="s:170" class="idref" href="#s:170"><span class="id" title="binder">s</span></a> <a id="beta:171" class="idref" href="#beta:171"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:171"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:170"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:169"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="gamma:172" class="idref" href="#gamma:172"><span class="id" title="binder">gamma</span></a> <a id="n:173" class="idref" href="#n:173"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:172"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:169"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:172"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:171"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:170"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:172"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:169"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_mult" class="idref" href="#path_toS_mult"><span class="id" title="lemma">path_toS_mult</span></a>  <a id="alpha:174" class="idref" href="#alpha:174"><span class="id" title="binder">alpha</span></a> <a id="s:175" class="idref" href="#s:175"><span class="id" title="binder">s</span></a>  <a id="i:176" class="idref" href="#i:176"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:174"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:175"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:174"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:174"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:176"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:175"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:174"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:176"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_mult" class="idref" href="#path_to_mult"><span class="id" title="lemma">path_to_mult</span></a>  <a id="alpha:177" class="idref" href="#alpha:177"><span class="id" title="binder">alpha</span></a> <a id="s:178" class="idref" href="#s:178"><span class="id" title="binder">s</span></a> <a id="i:179" class="idref" href="#i:179"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:177"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#s:178"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:177"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:177"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:179"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:178"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:177"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:179"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab83"></a><h2 class="section">Properties of gnaw</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_nf" class="idref" href="#gnaw_nf"><span class="id" title="lemma">gnaw_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="s:180" class="idref" href="#s:180"><span class="id" title="binder">s</span></a> <a id="alpha:181" class="idref" href="#alpha:181"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:181"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:181"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:180"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_zero" class="idref" href="#gnaw_zero"><span class="id" title="lemma">gnaw_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="s:182" class="idref" href="#s:182"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#s:182"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_zero" class="idref" href="#gnawS_zero"><span class="id" title="lemma">gnawS_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="s:183" class="idref" href="#s:183"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:183"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_nf" class="idref" href="#gnawS_nf"><span class="id" title="lemma">gnawS_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="s:184" class="idref" href="#s:184"><span class="id" title="binder">s</span></a> <a id="alpha:185" class="idref" href="#alpha:185"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:185"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:185"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:184"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_succ" class="idref" href="#gnaw_succ"><span class="id" title="lemma">gnaw_succ</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:186" class="idref" href="#alpha:186"><span class="id" title="binder">alpha</span></a>  <a id="i:187" class="idref" href="#i:187"><span class="id" title="binder">i</span></a> <a id="s:188" class="idref" href="#s:188"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:186"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:186"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:187"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:188"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:186"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:188"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_rw" class="idref" href="#gnaw_rw"><span class="id" title="lemma">gnaw_rw</span></a> <a id="i:189" class="idref" href="#i:189"><span class="id" title="binder">i</span></a> <a id="s:190" class="idref" href="#s:190"><span class="id" title="binder">s</span></a> <a id="alpha:191" class="idref" href="#alpha:191"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:191"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:189"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:190"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:191"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:189"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#s:190"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_to_path_toS" class="idref" href="#gnawS_to_path_toS"><span class="id" title="lemma">gnawS_to_path_toS</span></a> : <span class="id" title="keyword">∀</span> <a id="s:192" class="idref" href="#s:192"><span class="id" title="binder">s</span></a> <a id="alpha:193" class="idref" href="#alpha:193"><span class="id" title="binder">alpha</span></a> <a id="beta:194" class="idref" href="#beta:194"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:194"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:193"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:194"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:194"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:193"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_to_path_to" class="idref" href="#gnaw_to_path_to"><span class="id" title="lemma">gnaw_to_path_to</span></a> : <span class="id" title="keyword">∀</span> <a id="s:195" class="idref" href="#s:195"><span class="id" title="binder">s</span></a> <a id="alpha:196" class="idref" href="#alpha:196"><span class="id" title="binder">alpha</span></a> <a id="beta:197" class="idref" href="#beta:197"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:197"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:196"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:197"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:197"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:196"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_app" class="idref" href="#gnaw_app"><span class="id" title="lemma">gnaw_app</span></a> : <span class="id" title="keyword">∀</span>  <a id="s:198" class="idref" href="#s:198"><span class="id" title="binder">s</span></a> <a id="s':199" class="idref" href="#s':199"><span class="id" title="binder">s'</span></a> <a id="alpha:200" class="idref" href="#alpha:200"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:200"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#s:198"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s':199"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:200"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:198"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s':199"><span class="id" title="variable">s'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_app" class="idref" href="#gnawS_app"><span class="id" title="lemma">gnawS_app</span></a> : <span class="id" title="keyword">∀</span>  <a id="s:201" class="idref" href="#s:201"><span class="id" title="binder">s</span></a> <a id="s':202" class="idref" href="#s':202"><span class="id" title="binder">s'</span></a> <a id="alpha:203" class="idref" href="#alpha:203"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:203"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#s:201"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s':202"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:203"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:201"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s':202"><span class="id" title="variable">s'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaws_rw" class="idref" href="#gnaws_rw"><span class="id" title="lemma">gnaws_rw</span></a> <a id="i:204" class="idref" href="#i:204"><span class="id" title="binder">i</span></a> <a id="s:205" class="idref" href="#s:205"><span class="id" title="binder">s</span></a> <a id="alpha:206" class="idref" href="#alpha:206"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:206"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:204"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:205"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:206"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:204"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#s:205"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_lim1" class="idref" href="#gnawS_lim1"><span class="id" title="lemma">gnawS_lim1</span></a> (<a id="i:207" class="idref" href="#i:207"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="s:208" class="idref" href="#s:208"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="lambda:209" class="idref" href="#lambda:209"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:209"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:209"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:209"><span class="id" title="variable">lambda</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:207"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:208"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:209"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:207"><span class="id" title="variable">i</span></a>)) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:208"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_lim2" class="idref" href="#gnawS_lim2"><span class="id" title="lemma">gnawS_lim2</span></a> (<a id="i:210" class="idref" href="#i:210"><span class="id" title="binder">i</span></a> <a id="n:211" class="idref" href="#n:211"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="s:212" class="idref" href="#s:212"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="lambda:213" class="idref" href="#lambda:213"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:213"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:213"><span class="id" title="variable">lambda</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:213"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:211"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:210"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:212"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:213"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:211"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:213"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:210"><span class="id" title="variable">i</span></a>)) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#s:212"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_succ_eqn1" class="idref" href="#gnawS_succ_eqn1"><span class="id" title="lemma">gnawS_succ_eqn1</span></a> <a id="i:214" class="idref" href="#i:214"><span class="id" title="binder">i</span></a> <a id="s:215" class="idref" href="#s:215"><span class="id" title="binder">s</span></a> <a id="gamma:216" class="idref" href="#gamma:216"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:216"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:216"><span class="id" title="variable">gamma</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:214"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:215"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:216"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:214"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:215"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_succ_eqn2" class="idref" href="#gnawS_succ_eqn2"><span class="id" title="lemma">gnawS_succ_eqn2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:217" class="idref" href="#i:217"><span class="id" title="binder">i</span></a> <a id="n:218" class="idref" href="#n:218"><span class="id" title="binder">n</span></a> <a id="s:219" class="idref" href="#s:219"><span class="id" title="binder">s</span></a> <a id="gamma:220" class="idref" href="#gamma:220"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:220"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:220"><span class="id" title="variable">gamma</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:218"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:217"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:219"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:220"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#n:218"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:220"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:217"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#s:219"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_tail" class="idref" href="#gnawS_tail"><span class="id" title="lemma">gnawS_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:221" class="idref" href="#i:221"><span class="id" title="binder">i</span></a>  <a id="s:222" class="idref" href="#s:222"><span class="id" title="binder">s</span></a> <a id="alpha:223" class="idref" href="#alpha:223"><span class="id" title="binder">alpha</span></a> <a id="n:224" class="idref" href="#n:224"><span class="id" title="binder">n</span></a> <a id="beta:225" class="idref" href="#beta:225"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:223"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:224"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:225"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:225"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:223"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:224"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:225"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:221"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:222"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:223"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:224"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:225"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:221"><span class="id" title="variable">i</span></a>))) <a class="idref" href="hydras.Epsilon0.Paths.html#s:222"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_SSn" class="idref" href="#gnawS_SSn"><span class="id" title="lemma">gnawS_SSn</span></a> (<a id="i:226" class="idref" href="#i:226"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="s:227" class="idref" href="#s:227"><span class="id" title="binder">s</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:228" class="idref" href="#alpha:228"><span class="id" title="binder">alpha</span></a> <a id="n:229" class="idref" href="#n:229"><span class="id" title="binder">n</span></a>  ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:228"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:228"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:229"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#i:226"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#s:227"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:228"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:229"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:228"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:226"><span class="id" title="variable">i</span></a>))) <a class="idref" href="hydras.Epsilon0.Paths.html#s:227"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_cut1" class="idref" href="#gnawS_cut1"><span class="id" title="lemma">gnawS_cut1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:230" class="idref" href="#s:230"><span class="id" title="binder">s</span></a> <a id="alpha:231" class="idref" href="#alpha:231"><span class="id" title="binder">alpha</span></a> <a id="n:232" class="idref" href="#n:232"><span class="id" title="binder">n</span></a> <a id="beta:233" class="idref" href="#beta:233"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:231"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:232"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:233"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:231"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:232"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:233"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:230"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s1:234" class="idref" href="#s1:234"><span class="id" title="binder">s1</span></a> <a id="s2:235" class="idref" href="#s2:235"><span class="id" title="binder">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:230"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s1:234"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s2:235"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:233"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s1:234"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:231"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:232"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s2:235"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_cut2" class="idref" href="#gnawS_cut2"><span class="id" title="lemma">gnawS_cut2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:236" class="idref" href="#s:236"><span class="id" title="binder">s</span></a> <a id="alpha:237" class="idref" href="#alpha:237"><span class="id" title="binder">alpha</span></a> <a id="n:238" class="idref" href="#n:238"><span class="id" title="binder">n</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:237"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:237"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:238"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:236"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s1:239" class="idref" href="#s1:239"><span class="id" title="binder">s1</span></a> <a id="s2:240" class="idref" href="#s2:240"><span class="id" title="binder">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:236"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s1:239"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s2:240"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:237"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s1:239"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:237"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:238"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s2:240"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_gnaw" class="idref" href="#path_to_gnaw"><span class="id" title="lemma">path_to_gnaw</span></a> : <span class="id" title="keyword">∀</span> <a id="s:241" class="idref" href="#s:241"><span class="id" title="binder">s</span></a> <a id="alpha:242" class="idref" href="#alpha:242"><span class="id" title="binder">alpha</span></a> <a id="beta:243" class="idref" href="#beta:243"><span class="id" title="binder">beta</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:243"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:241"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:242"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:243"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:242"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:241"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnawS_path_toS" class="idref" href="#gnawS_path_toS"><span class="id" title="lemma">gnawS_path_toS</span></a> : <span class="id" title="keyword">∀</span> <a id="s:244" class="idref" href="#s:244"><span class="id" title="binder">s</span></a> <a id="alpha:245" class="idref" href="#alpha:245"><span class="id" title="binder">alpha</span></a> , <a class="idref" href="hydras.Epsilon0.Paths.html#s:244"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:245"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:244"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:245"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:244"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:244"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:245"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="gnaw_path_to" class="idref" href="#gnaw_path_to"><span class="id" title="lemma">gnaw_path_to</span></a> : <span class="id" title="keyword">∀</span> <a id="s:246" class="idref" href="#s:246"><span class="id" title="binder">s</span></a> <a id="alpha:247" class="idref" href="#alpha:247"><span class="id" title="binder">alpha</span></a> , <a class="idref" href="hydras.Epsilon0.Paths.html#s:246"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#s:246"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:247"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:246"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:247"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:246"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#s:246"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:247"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab84"></a><h2 class="section">Properties of acc_from</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="acc_from_trans" class="idref" href="#acc_from_trans"><span class="id" title="lemma">acc_from_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:248" class="idref" href="#alpha:248"><span class="id" title="binder">alpha</span></a> <a id="beta:249" class="idref" href="#beta:249"><span class="id" title="binder">beta</span></a> <a id="gamma:250" class="idref" href="#gamma:250"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:248"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:249"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:249"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:250"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:248"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:250"><span class="id" title="variable">gamma</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_path_toS" class="idref" href="#LT_path_toS"><span class="id" title="lemma">LT_path_toS</span></a> (<a id="alpha:251" class="idref" href="#alpha:251"><span class="id" title="binder">alpha</span></a> <a id="beta:252" class="idref" href="#beta:252"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:252"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:251"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="s:253" class="idref" href="#s:253"><span class="id" title="binder">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:252"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:253"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:251"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_path_to" class="idref" href="#LT_path_to"><span class="id" title="lemma">LT_path_to</span></a> (<a id="alpha:254" class="idref" href="#alpha:254"><span class="id" title="binder">alpha</span></a> <a id="beta:255" class="idref" href="#beta:255"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:255"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:254"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="s:256" class="idref" href="#s:256"><span class="id" title="binder">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:255"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:256"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:254"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_acc_from" class="idref" href="#LT_acc_from"><span class="id" title="lemma">LT_acc_from</span></a> (<a id="alpha:257" class="idref" href="#alpha:257"><span class="id" title="binder">alpha</span></a> <a id="beta:258" class="idref" href="#beta:258"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:258"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:257"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:257"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:258"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_LT" class="idref" href="#path_toS_LT"><span class="id" title="lemma">path_toS_LT</span></a> <a id="beta:259" class="idref" href="#beta:259"><span class="id" title="binder">beta</span></a> <a id="s:260" class="idref" href="#s:260"><span class="id" title="binder">s</span></a> <a id="alpha:261" class="idref" href="#alpha:261"><span class="id" title="binder">alpha</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:259"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:260"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:261"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:261"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:259"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:261"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_LT" class="idref" href="#path_to_LT"><span class="id" title="lemma">path_to_LT</span></a> <a id="beta:262" class="idref" href="#beta:262"><span class="id" title="binder">beta</span></a> <a id="s:263" class="idref" href="#s:263"><span class="id" title="binder">s</span></a> <a id="alpha:264" class="idref" href="#alpha:264"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:262"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:263"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:264"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:264"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:262"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:264"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="acc_from_LT" class="idref" href="#acc_from_LT"><span class="id" title="lemma">acc_from_LT</span></a> (<a id="alpha:265" class="idref" href="#alpha:265"><span class="id" title="binder">alpha</span></a> <a id="beta:266" class="idref" href="#beta:266"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:265"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:266"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:265"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <a class="idref" href="hydras.Epsilon0.Paths.html#beta:266"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:265"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab85"></a><h2 class="section">Paths with constant index: lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_repeat" class="idref" href="#const_pathS_repeat"><span class="id" title="lemma">const_pathS_repeat</span></a> <a id="i:267" class="idref" href="#i:267"><span class="id" title="binder">i</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:268" class="idref" href="#alpha:268"><span class="id" title="binder">alpha</span></a>  <a id="beta:269" class="idref" href="#beta:269"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:267"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:268"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:269"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:270" class="idref" href="#n:270"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:269"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#repeat"><span class="id" title="definition">repeat</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:267"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:270"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:268"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_repeatR" class="idref" href="#const_pathS_repeatR"><span class="id" title="lemma">const_pathS_repeatR</span></a> <a id="i:271" class="idref" href="#i:271"><span class="id" title="binder">i</span></a> : <span class="id" title="keyword">∀</span> <a id="n:272" class="idref" href="#n:272"><span class="id" title="binder">n</span></a>  <a id="alpha:273" class="idref" href="#alpha:273"><span class="id" title="binder">alpha</span></a> <a id="beta:274" class="idref" href="#beta:274"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:274"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#repeat"><span class="id" title="definition">repeat</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:271"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:272"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:273"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:271"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:273"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:274"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_nf" class="idref" href="#const_pathS_nf"><span class="id" title="lemma">const_pathS_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="n:275" class="idref" href="#n:275"><span class="id" title="binder">n</span></a> <a id="alpha:276" class="idref" href="#alpha:276"><span class="id" title="binder">alpha</span></a> <a id="beta:277" class="idref" href="#beta:277"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:275"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:276"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:277"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:276"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:277"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_zero" class="idref" href="#const_pathS_zero"><span class="id" title="lemma">const_pathS_zero</span></a> <a id="n:278" class="idref" href="#n:278"><span class="id" title="binder">n</span></a> <a id="alpha:279" class="idref" href="#alpha:279"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:278"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:279"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_LT" class="idref" href="#const_pathS_LT"><span class="id" title="lemma">const_pathS_LT</span></a> <a id="i:280" class="idref" href="#i:280"><span class="id" title="binder">i</span></a>  <a id="alpha:281" class="idref" href="#alpha:281"><span class="id" title="binder">alpha</span></a> <a id="beta:282" class="idref" href="#beta:282"><span class="id" title="binder">beta</span></a>:  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:281"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:280"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:281"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:282"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:282"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:281"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_path_LT" class="idref" href="#const_path_LT"><span class="id" title="lemma">const_path_LT</span></a> <a id="i:283" class="idref" href="#i:283"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:284" class="idref" href="#alpha:284"><span class="id" title="binder">alpha</span></a> <a id="beta:285" class="idref" href="#beta:285"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:284"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:283"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:284"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:285"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:285"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:284"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_LE" class="idref" href="#const_pathS_LE"><span class="id" title="lemma">const_pathS_LE</span></a> <a id="i:286" class="idref" href="#i:286"><span class="id" title="binder">i</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:287" class="idref" href="#alpha:287"><span class="id" title="binder">alpha</span></a> <a id="beta:288" class="idref" href="#beta:288"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:287"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:286"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:287"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:288"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:288"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:287"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_inv" class="idref" href="#const_pathS_inv"><span class="id" title="lemma">const_pathS_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="n:289" class="idref" href="#n:289"><span class="id" title="binder">n</span></a> <a id="alpha:290" class="idref" href="#alpha:290"><span class="id" title="binder">alpha</span></a> <a id="beta:291" class="idref" href="#beta:291"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:289"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:290"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:291"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:291"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:290"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:289"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:289"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:290"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:289"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:291"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_inv_strong" class="idref" href="#const_pathS_inv_strong"><span class="id" title="lemma">const_pathS_inv_strong</span></a> : <span class="id" title="keyword">∀</span> <a id="n:292" class="idref" href="#n:292"><span class="id" title="binder">n</span></a> <a id="alpha:293" class="idref" href="#alpha:293"><span class="id" title="binder">alpha</span></a> <a id="beta:294" class="idref" href="#beta:294"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:292"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:293"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:294"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#beta:294"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:293"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:292"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:292"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:293"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:292"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:294"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_trans" class="idref" href="#const_pathS_trans"><span class="id" title="lemma">const_pathS_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="n:295" class="idref" href="#n:295"><span class="id" title="binder">n</span></a> <a id="alpha:296" class="idref" href="#alpha:296"><span class="id" title="binder">alpha</span></a> <a id="beta:297" class="idref" href="#beta:297"><span class="id" title="binder">beta</span></a> <a id="gamma:298" class="idref" href="#gamma:298"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:295"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:296"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:297"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:295"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:297"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:298"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:295"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:296"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:298"><span class="id" title="variable">gamma</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_eps_trans" class="idref" href="#const_pathS_eps_trans"><span class="id" title="lemma">const_pathS_eps_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="i:299" class="idref" href="#i:299"><span class="id" title="binder">i</span></a> <a id="alpha:300" class="idref" href="#alpha:300"><span class="id" title="binder">alpha</span></a> <a id="beta:301" class="idref" href="#beta:301"><span class="id" title="binder">beta</span></a> <a id="gamma:302" class="idref" href="#gamma:302"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:299"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:300"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:301"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:299"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:301"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:302"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:299"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:300"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:302"><span class="id" title="variable">gamma</span></a> .<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_eps_LE_2" class="idref" href="#const_pathS_eps_LE_2"><span class="id" title="lemma">const_pathS_eps_LE_2</span></a> : <span class="id" title="keyword">∀</span> <a id="n:303" class="idref" href="#n:303"><span class="id" title="binder">n</span></a> <a id="alpha:304" class="idref" href="#alpha:304"><span class="id" title="binder">alpha</span></a> <a id="beta:305" class="idref" href="#beta:305"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:304"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:305"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:303"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:304"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:305"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:305"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:304"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Proposition_2_3a" class="idref" href="#Proposition_2_3a"><span class="id" title="lemma">Proposition_2_3a</span></a>:  <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:306" class="idref" href="#n:306"><span class="id" title="binder">n</span></a> <a id="alpha:307" class="idref" href="#alpha:307"><span class="id" title="binder">alpha</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:308" class="idref" href="#beta:308"><span class="id" title="binder">beta</span></a> <a id="gamma:309" class="idref" href="#gamma:309"><span class="id" title="binder">gamma</span></a>,   <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:309"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#beta:308"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:306"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:309"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:306"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:308"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:306"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:308"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:309"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma1" class="idref" href="#KS_thm_2_4_lemma1"><span class="id" title="lemma">KS_thm_2_4_lemma1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:310" class="idref" href="#i:310"><span class="id" title="binder">i</span></a> <a id="alpha:311" class="idref" href="#alpha:311"><span class="id" title="binder">alpha</span></a> <a id="n:312" class="idref" href="#n:312"><span class="id" title="binder">n</span></a> <a id="beta:313" class="idref" href="#beta:313"><span class="id" title="binder">beta</span></a> <a id="beta':314" class="idref" href="#beta':314"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:311"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:312"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:313"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:313"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:310"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:313"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta':314"><span class="id" title="variable">beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:310"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:311"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:312"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:313"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:311"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:312"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta':314"><span class="id" title="variable">beta'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_first_step" class="idref" href="#const_pathS_first_step"><span class="id" title="lemma">const_pathS_first_step</span></a> : <span class="id" title="keyword">∀</span> <a id="i:315" class="idref" href="#i:315"><span class="id" title="binder">i</span></a> <a id="alpha:316" class="idref" href="#alpha:316"><span class="id" title="binder">alpha</span></a> <a id="beta:317" class="idref" href="#beta:317"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:315"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:316"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:317"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#beta:317"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:316"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:315"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:315"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:316"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:315"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:317"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma1'" class="idref" href="#KS_thm_2_4_lemma1'"><span class="id" title="lemma">KS_thm_2_4_lemma1'</span></a> : <span class="id" title="keyword">∀</span> <a id="i:318" class="idref" href="#i:318"><span class="id" title="binder">i</span></a> <a id="alpha:319" class="idref" href="#alpha:319"><span class="id" title="binder">alpha</span></a> <a id="n:320" class="idref" href="#n:320"><span class="id" title="binder">n</span></a> <a id="beta:321" class="idref" href="#beta:321"><span class="id" title="binder">beta</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:319"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:319"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:318"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:319"><span class="id" title="variable">alpha</span></a>)  <a class="idref" href="hydras.Epsilon0.Paths.html#beta:321"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:318"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:319"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:320"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:319"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:320"><span class="id" title="variable">n</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#beta:321"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma2" class="idref" href="#KS_thm_2_4_lemma2"><span class="id" title="lemma">KS_thm_2_4_lemma2</span></a> (<a id="n:322" class="idref" href="#n:322"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="alpha:323" class="idref" href="#alpha:323"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:323"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:323"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:322"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:323"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma3_0" class="idref" href="#KS_thm_2_4_lemma3_0"><span class="id" title="lemma">KS_thm_2_4_lemma3_0</span></a> : <span class="id" title="keyword">∀</span> <a id="i:324" class="idref" href="#i:324"><span class="id" title="binder">i</span></a> <a id="alpha:325" class="idref" href="#alpha:325"><span class="id" title="binder">alpha</span></a> <a id="n:326" class="idref" href="#n:326"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:325"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:324"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:325"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:326"><span class="id" title="variable">n</span></a>)  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:325"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:326"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma3" class="idref" href="#KS_thm_2_4_lemma3"><span class="id" title="lemma">KS_thm_2_4_lemma3</span></a> : <span class="id" title="keyword">∀</span> <a id="i:327" class="idref" href="#i:327"><span class="id" title="binder">i</span></a> <a id="n:328" class="idref" href="#n:328"><span class="id" title="binder">n</span></a> <a id="p:329" class="idref" href="#p:329"><span class="id" title="binder">p</span></a> <a id="alpha:330" class="idref" href="#alpha:330"><span class="id" title="binder">alpha</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:330"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:328"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:329"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:327"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:330"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:329"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:330"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:328"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma4" class="idref" href="#KS_thm_2_4_lemma4"><span class="id" title="lemma">KS_thm_2_4_lemma4</span></a> : <span class="id" title="keyword">∀</span> <a id="i:331" class="idref" href="#i:331"><span class="id" title="binder">i</span></a> <a id="alpha:332" class="idref" href="#alpha:332"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:332"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:331"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:332"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:332"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KS_thm_2_4_lemma5" class="idref" href="#KS_thm_2_4_lemma5"><span class="id" title="lemma">KS_thm_2_4_lemma5</span></a> : <span class="id" title="keyword">∀</span> <a id="i:333" class="idref" href="#i:333"><span class="id" title="binder">i</span></a> <a id="alpha:334" class="idref" href="#alpha:334"><span class="id" title="binder">alpha</span></a> <a id="beta:335" class="idref" href="#beta:335"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:333"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:334"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:335"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:334"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:334"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:333"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:334"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:335"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="KS_thm_2_4" class="idref" href="#KS_thm_2_4"><span class="id" title="lemma">KS_thm_2_4</span></a> (<a id="lambda:336" class="idref" href="#lambda:336"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:336"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:336"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:337" class="idref" href="#i:337"><span class="id" title="binder">i</span></a> <a id="j:338" class="idref" href="#j:338"><span class="id" title="binder">j</span></a>, (<a class="idref" href="hydras.Epsilon0.Paths.html#i:337"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:338"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:336"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:338"><span class="id" title="variable">j</span></a>)) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:336"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:337"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Corollary 12 of <span class="inlinecode"><span class="id" title="var">KS</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Cor12" class="idref" href="#Cor12"><span class="id" title="lemma">Cor12</span></a> (<a id="alpha:339" class="idref" href="#alpha:339"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:339"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:340" class="idref" href="#beta:340"><span class="id" title="binder">beta</span></a> <a id="i:341" class="idref" href="#i:341"><span class="id" title="binder">i</span></a> <a id="n:342" class="idref" href="#n:342"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#beta:340"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:339"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:341"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:342"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:341"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:339"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:340"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:342"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:339"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:340"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Cor12_1" class="idref" href="#Cor12_1"><span class="id" title="lemma">Cor12_1</span></a> (<a id="alpha:343" class="idref" href="#alpha:343"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:343"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:344" class="idref" href="#beta:344"><span class="id" title="binder">beta</span></a> <a id="i:345" class="idref" href="#i:345"><span class="id" title="binder">i</span></a> <a id="n:346" class="idref" href="#n:346"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#beta:344"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:343"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:345"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:346"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:345"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:343"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:344"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:346"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:343"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:344"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Cor12_2" class="idref" href="#Cor12_2"><span class="id" title="lemma">Cor12_2</span></a> (<a id="alpha:347" class="idref" href="#alpha:347"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:347"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:348" class="idref" href="#beta:348"><span class="id" title="binder">beta</span></a> <a id="i:349" class="idref" href="#i:349"><span class="id" title="binder">i</span></a> <a id="n:350" class="idref" href="#n:350"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:349"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:350"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:349"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:347"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:348"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:350"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:347"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:348"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Cor12_3" class="idref" href="#Cor12_3"><span class="id" title="lemma">Cor12_3</span></a> (<a id="alpha:351" class="idref" href="#alpha:351"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:351"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:352" class="idref" href="#beta:352"><span class="id" title="binder">beta</span></a> <a id="i:353" class="idref" href="#i:353"><span class="id" title="binder">i</span></a> <a id="n:354" class="idref" href="#n:354"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#beta:352"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:351"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:353"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:354"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:353"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:351"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:352"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:354"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:351"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:352"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_eps_zero" class="idref" href="#const_pathS_eps_zero"><span class="id" title="lemma">const_pathS_eps_zero</span></a> <a id="n:355" class="idref" href="#n:355"><span class="id" title="binder">n</span></a> <a id="alpha:356" class="idref" href="#alpha:356"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:355"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:356"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:356"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma2_6_1" class="idref" href="#Lemma2_6_1"><span class="id" title="lemma">Lemma2_6_1</span></a> (<a id="alpha:357" class="idref" href="#alpha:357"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:357"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:358" class="idref" href="#beta:358"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#beta:358"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:357"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:359" class="idref" href="#n:359"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:359"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:357"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:358"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="small_lemma" class="idref" href="#small_lemma"><span class="id" title="lemma">small_lemma</span></a> (<a id="i:360" class="idref" href="#i:360"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="beta:361" class="idref" href="#beta:361"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :  <span class="id" title="keyword">∀</span> <a id="alpha:362" class="idref" href="#alpha:362"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:360"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:362"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:361"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:362"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:361"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:362"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:360"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2_6_2" class="idref" href="#L2_6_2"><span class="id" title="lemma">L2_6_2</span></a> (<a id="p:363" class="idref" href="#p:363"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:364" class="idref" href="#alpha:364"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:364"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:365" class="idref" href="#beta:365"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:363"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:364"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:365"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:365"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:364"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:363"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:364"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:365"><span class="id" title="variable">beta</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Lemma_4_3_Proof" class="idref" href="#Lemma_4_3_Proof"><span class="id" title="section">Lemma_4_3_Proof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Lemma_4_3_Proof.alpha" class="idref" href="#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a id="Lemma_4_3_Proof.beta" class="idref" href="#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.H00" class="idref" href="#Lemma_4_3_Proof.H00"><span class="id" title="variable">H00</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.nf1" class="idref" href="#Lemma_4_3_Proof.nf1"><span class="id" title="variable">nf1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.nf2" class="idref" href="#Lemma_4_3_Proof.nf2"><span class="id" title="variable">nf2</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Lemma_4_3_Proof.n0" class="idref" href="#Lemma_4_3_Proof.n0"><span class="id" title="variable">n0</span></a> <a id="Lemma_4_3_Proof.n1" class="idref" href="#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a> <a id="Lemma_4_3_Proof.n2" class="idref" href="#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.H0" class="idref" href="#Lemma_4_3_Proof.H0"><span class="id" title="variable">H0</span></a>: (<a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.H1" class="idref" href="#Lemma_4_3_Proof.H1"><span class="id" title="variable">H1</span></a>: (<a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Lemma_4_3_Proof.H4" class="idref" href="#Lemma_4_3_Proof.H4"><span class="id" title="variable">H4</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4_3_1" class="idref" href="#R4_3_1"><span class="id" title="lemma">R4_3_1</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4_3_2" class="idref" href="#R4_3_2"><span class="id" title="lemma">R4_3_2</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a>)).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4_3_3" class="idref" href="#R4_3_3"><span class="id" title="lemma">R4_3_3</span></a>: <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4_3_4" class="idref" href="#R4_3_4"><span class="id" title="lemma">R4_3_4</span></a> : (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a>))%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_3_0" class="idref" href="#Lemma_4_3_0"><span class="id" title="lemma">Lemma_4_3_0</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2"><span class="id" title="variable">n2</span></a>)) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1"><span class="id" title="variable">n1</span></a>)).<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof"><span class="id" title="section">Lemma_4_3_Proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_3" class="idref" href="#Lemma_4_3"><span class="id" title="lemma">Lemma_4_3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:377" class="idref" href="#alpha:377"><span class="id" title="binder">alpha</span></a> <a id="beta:378" class="idref" href="#beta:378"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:377"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:378"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n0:379" class="idref" href="#n0:379"><span class="id" title="binder">n0</span></a> <a id="n1:380" class="idref" href="#n1:380"><span class="id" title="binder">n1</span></a> <a id="n2:381" class="idref" href="#n2:381"><span class="id" title="binder">n2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#n0:379"><span class="id" title="variable">n0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n2:381"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#n1:380"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n2:381"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n0:379"><span class="id" title="variable">n0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:377"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:378"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#n2:381"><span class="id" title="variable">n2</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:377"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n2:381"><span class="id" title="variable">n2</span></a>)) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:378"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n1:380"><span class="id" title="variable">n1</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="const_pathS_LT'" class="idref" href="#const_pathS_LT'"><span class="id" title="lemma">const_pathS_LT'</span></a> (<a id="i:382" class="idref" href="#i:382"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="alpha:383" class="idref" href="#alpha:383"><span class="id" title="binder">alpha</span></a> <a id="beta:384" class="idref" href="#beta:384"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="H:385" class="idref" href="#H:385"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:383"><span class="id" title="variable">alpha</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:382"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:383"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:384"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:383"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:384"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:383"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_4_1" class="idref" href="#Lemma_4_4_1"><span class="id" title="lemma">Lemma_4_4_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:386" class="idref" href="#s:386"><span class="id" title="binder">s</span></a> <a id="n:387" class="idref" href="#n:387"><span class="id" title="binder">n</span></a> <a id="alpha:388" class="idref" href="#alpha:388"><span class="id" title="binder">alpha</span></a> <a id="beta:389" class="idref" href="#beta:389"><span class="id" title="binder">beta</span></a> (<a id="Ha:390" class="idref" href="#Ha:390"><span class="id" title="binder">Ha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:388"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hb:391" class="idref" href="#Hb:391"><span class="id" title="binder">Hb</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:389"><span class="id" title="variable">beta</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:387"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:386"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:387"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:388"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:389"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:387"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:386"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:388"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:386"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Epsilon0.Paths.html#gnawS"><span class="id" title="definition">gnawS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:389"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#s:386"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KP_arrowS_zero" class="idref" href="#KP_arrowS_zero"><span class="id" title="lemma">KP_arrowS_zero</span></a> <a id="n:392" class="idref" href="#n:392"><span class="id" title="binder">n</span></a> <a id="beta:393" class="idref" href="#beta:393"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#KP_arrowS"><span class="id" title="definition">KP_arrowS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:392"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:393"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:393"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="KP_5_iii" class="idref" href="#KP_5_iii"><span class="id" title="lemma">KP_5_iii</span></a> <a id="n:394" class="idref" href="#n:394"><span class="id" title="binder">n</span></a> <a id="alpha:395" class="idref" href="#alpha:395"><span class="id" title="binder">alpha</span></a> <a id="beta:396" class="idref" href="#beta:396"><span class="id" title="binder">beta</span></a>:  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:395"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:396"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:395"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:396"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#KP_arrowS"><span class="id" title="definition">KP_arrowS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:395"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:396"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma_4_4_0" class="idref" href="#Lemma_4_4_0"><span class="id" title="lemma">Lemma_4_4_0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:397" class="idref" href="#n:397"><span class="id" title="binder">n</span></a> <a id="p:398" class="idref" href="#p:398"><span class="id" title="binder">p</span></a> <a id="alpha:399" class="idref" href="#alpha:399"><span class="id" title="binder">alpha</span></a> <a id="beta:400" class="idref" href="#beta:400"><span class="id" title="binder">beta</span></a> (<a id="Halpha:401" class="idref" href="#Halpha:401"><span class="id" title="binder">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:399"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:399"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:400"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#n:397"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:398"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:397"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:399"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:400"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS_eps"><span class="id" title="definition">const_pathS_eps</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:398"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:399"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:398"><span class="id" title="variable">p</span></a>)) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:400"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:398"><span class="id" title="variable">p</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_shift" class="idref" href="#standard_path_shift"><span class="id" title="lemma">standard_path_shift</span></a> <a id="i:402" class="idref" href="#i:402"><span class="id" title="binder">i</span></a> <a id="alpha:403" class="idref" href="#alpha:403"><span class="id" title="binder">alpha</span></a> <a id="j:404" class="idref" href="#j:404"><span class="id" title="binder">j</span></a> <a id="beta:405" class="idref" href="#beta:405"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:402"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:403"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:404"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:405"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:402"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:403"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:404"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:405"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_unshift_0" class="idref" href="#standard_path_unshift_0"><span class="id" title="lemma">standard_path_unshift_0</span></a> <a id="i:406" class="idref" href="#i:406"><span class="id" title="binder">i</span></a> <a id="alpha:407" class="idref" href="#alpha:407"><span class="id" title="binder">alpha</span></a> <a id="j:408" class="idref" href="#j:408"><span class="id" title="binder">j</span></a> <a id="beta:409" class="idref" href="#beta:409"><span class="id" title="binder">beta</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:406"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:407"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:408"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:409"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="k:410" class="idref" href="#k:410"><span class="id" title="binder">k</span></a> <a id="l:411" class="idref" href="#l:411"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:406"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:410"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:408"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:411"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#k:410"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:407"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:411"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:409"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_unshift" class="idref" href="#standard_path_unshift"><span class="id" title="lemma">standard_path_unshift</span></a> <a id="i:412" class="idref" href="#i:412"><span class="id" title="binder">i</span></a> <a id="alpha:413" class="idref" href="#alpha:413"><span class="id" title="binder">alpha</span></a> <a id="j:414" class="idref" href="#j:414"><span class="id" title="binder">j</span></a> <a id="beta:415" class="idref" href="#beta:415"><span class="id" title="binder">beta</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:412"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:413"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:414"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:415"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#i:412"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:413"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:414"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:415"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_toS_le_inv" class="idref" href="#standard_path_toS_le_inv"><span class="id" title="lemma">standard_path_toS_le_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="j:416" class="idref" href="#j:416"><span class="id" title="binder">j</span></a> <a id="beta:417" class="idref" href="#beta:417"><span class="id" title="binder">beta</span></a> <a id="i:418" class="idref" href="#i:418"><span class="id" title="binder">i</span></a> <a id="alpha:419" class="idref" href="#alpha:419"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:416"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:417"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:418"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:419"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:418"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:416"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_to_le_inv" class="idref" href="#standard_path_to_le_inv"><span class="id" title="lemma">standard_path_to_le_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="j:420" class="idref" href="#j:420"><span class="id" title="binder">j</span></a> <a id="beta:421" class="idref" href="#beta:421"><span class="id" title="binder">beta</span></a> <a id="i:422" class="idref" href="#i:422"><span class="id" title="binder">i</span></a> <a id="alpha:423" class="idref" href="#alpha:423"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:420"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:421"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:422"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:423"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:422"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:420"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_toS_zero" class="idref" href="#standard_path_toS_zero"><span class="id" title="lemma">standard_path_toS_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="j:424" class="idref" href="#j:424"><span class="id" title="binder">j</span></a> <a id="beta:425" class="idref" href="#beta:425"><span class="id" title="binder">beta</span></a> <a id="i:426" class="idref" href="#i:426"><span class="id" title="binder">i</span></a> <a id="alpha:427" class="idref" href="#alpha:427"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS"><span class="id" title="inductive">standard_path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:424"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:425"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:426"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:427"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:427"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_origin" class="idref" href="#standard_path_origin"><span class="id" title="lemma">standard_path_origin</span></a> : <span class="id" title="keyword">∀</span> <a id="j:428" class="idref" href="#j:428"><span class="id" title="binder">j</span></a> <a id="beta:429" class="idref" href="#beta:429"><span class="id" title="binder">beta</span></a> <a id="i:430" class="idref" href="#i:430"><span class="id" title="binder">i</span></a> <a id="alpha:431" class="idref" href="#alpha:431"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:428"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:429"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:430"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:431"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:431"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_pathS_path_toS" class="idref" href="#standard_pathS_path_toS"><span class="id" title="lemma">standard_pathS_path_toS</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:432" class="idref" href="#i:432"><span class="id" title="binder">i</span></a> <a id="alpha:433" class="idref" href="#alpha:433"><span class="id" title="binder">alpha</span></a> <a id="j:434" class="idref" href="#j:434"><span class="id" title="binder">j</span></a> <a id="beta:435" class="idref" href="#beta:435"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:432"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:433"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:434"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:435"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:432"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:434"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:435"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:432"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:434"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:433"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_path_to" class="idref" href="#standard_path_path_to"><span class="id" title="lemma">standard_path_path_to</span></a> <a id="i:436" class="idref" href="#i:436"><span class="id" title="binder">i</span></a> <a id="alpha:437" class="idref" href="#alpha:437"><span class="id" title="binder">alpha</span></a> <a id="j:438" class="idref" href="#j:438"><span class="id" title="binder">j</span></a> <a id="beta:439" class="idref" href="#beta:439"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:436"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:437"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:438"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:439"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:436"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:436"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:438"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:439"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:436"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:438"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:437"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_zero" class="idref" href="#standard_gnaw_zero"><span class="id" title="lemma">standard_gnaw_zero</span></a> <a id="l:440" class="idref" href="#l:440"><span class="id" title="binder">l</span></a> <a id="i:441" class="idref" href="#i:441"><span class="id" title="binder">i</span></a> :  <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:441"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:440"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_iota_from" class="idref" href="#standard_gnaw_iota_from"><span class="id" title="lemma">standard_gnaw_iota_from</span></a> <a id="i:442" class="idref" href="#i:442"><span class="id" title="binder">i</span></a> <a id="alpha:443" class="idref" href="#alpha:443"><span class="id" title="binder">alpha</span></a> <a id="l:444" class="idref" href="#l:444"><span class="id" title="binder">l</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:442"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:442"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:443"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:444"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gnaw"><span class="id" title="definition">gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:443"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:442"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:444"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_plus" class="idref" href="#standard_gnaw_plus"><span class="id" title="lemma">standard_gnaw_plus</span></a> : <span class="id" title="keyword">∀</span>  (<a id="t1:445" class="idref" href="#t1:445"><span class="id" title="binder">t1</span></a> <a id="t2:446" class="idref" href="#t2:446"><span class="id" title="binder">t2</span></a> <a id="i:447" class="idref" href="#i:447"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="alpha:448" class="idref" href="#alpha:448"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:447"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:448"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#t1:445"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t2:446"><span class="id" title="variable">t2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#t1:445"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#i:447"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> (<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:447"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:448"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t1:445"><span class="id" title="variable">t1</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#t2:446"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_LE" class="idref" href="#standard_path_LE"><span class="id" title="lemma">standard_path_LE</span></a> : <span class="id" title="keyword">∀</span>  <a id="j:449" class="idref" href="#j:449"><span class="id" title="binder">j</span></a> <a id="beta:450" class="idref" href="#beta:450"><span class="id" title="binder">beta</span></a> <a id="i:451" class="idref" href="#i:451"><span class="id" title="binder">i</span></a> <a id="alpha:452" class="idref" href="#alpha:452"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:451"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:452"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:449"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:450"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:451"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:449"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_to_nf" class="idref" href="#standard_path_to_nf"><span class="id" title="lemma">standard_path_to_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="i:453" class="idref" href="#i:453"><span class="id" title="binder">i</span></a> <a id="alpha:454" class="idref" href="#alpha:454"><span class="id" title="binder">alpha</span></a> <a id="j:455" class="idref" href="#j:455"><span class="id" title="binder">j</span></a> <a id="beta:456" class="idref" href="#beta:456"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#j:455"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:456"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:453"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:454"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:454"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:456"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_zero" class="idref" href="#standard_path_zero"><span class="id" title="lemma">standard_path_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="i:457" class="idref" href="#i:457"><span class="id" title="binder">i</span></a> <a id="j:458" class="idref" href="#j:458"><span class="id" title="binder">j</span></a> <a id="alpha:459" class="idref" href="#alpha:459"><span class="id" title="binder">alpha</span></a> <a id="beta:460" class="idref" href="#beta:460"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:457"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:459"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:458"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:460"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:459"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:460"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_lt2" class="idref" href="#standard_path_lt2"><span class="id" title="lemma">standard_path_lt2</span></a> : <span class="id" title="keyword">∀</span> <a id="i:461" class="idref" href="#i:461"><span class="id" title="binder">i</span></a> <a id="alpha:462" class="idref" href="#alpha:462"><span class="id" title="binder">alpha</span></a> <a id="j:463" class="idref" href="#j:463"><span class="id" title="binder">j</span></a> <a id="beta:464" class="idref" href="#beta:464"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:461"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:462"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_to"><span class="id" title="inductive">standard_path_to</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#j:463"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:464"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:461"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:462"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:464"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:462"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_compose" class="idref" href="#standard_path_compose"><span class="id" title="lemma">standard_path_compose</span></a>  <a id="j:465" class="idref" href="#j:465"><span class="id" title="binder">j</span></a> <a id="beta:466" class="idref" href="#beta:466"><span class="id" title="binder">beta</span></a> <a id="k:467" class="idref" href="#k:467"><span class="id" title="binder">k</span></a> <a id="gamma:468" class="idref" href="#gamma:468"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:465"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:466"><span class="id" title="variable">beta</span></a>   <a class="idref" href="hydras.Epsilon0.Paths.html#k:467"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:468"><span class="id" title="variable">gamma</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:469" class="idref" href="#i:469"><span class="id" title="binder">i</span></a> <a id="alpha:470" class="idref" href="#alpha:470"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:469"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:470"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:465"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:466"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:469"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:470"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:467"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:468"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_S_zero" class="idref" href="#standard_gnaw_S_zero"><span class="id" title="lemma">standard_gnaw_S_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="i:471" class="idref" href="#i:471"><span class="id" title="binder">i</span></a> <a id="n:472" class="idref" href="#n:472"><span class="id" title="binder">n</span></a> <a id="alpha:473" class="idref" href="#alpha:473"><span class="id" title="binder">alpha</span></a>  ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:472"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:473"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:471"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:472"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:473"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:471"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_to_path" class="idref" href="#standard_gnaw_to_path"><span class="id" title="lemma">standard_gnaw_to_path</span></a> : <span class="id" title="keyword">∀</span> <a id="t:474" class="idref" href="#t:474"><span class="id" title="binder">t</span></a> <a id="alpha:475" class="idref" href="#alpha:475"><span class="id" title="binder">alpha</span></a> <a id="i:476" class="idref" href="#i:476"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:475"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:476"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:476"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:475"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:474"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:476"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:475"><span class="id" title="variable">alpha</span></a>  (<a class="idref" href="hydras.Epsilon0.Paths.html#t:474"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:476"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:476"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:475"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:474"><span class="id" title="variable">t</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_nf" class="idref" href="#standard_gnaw_nf"><span class="id" title="lemma">standard_gnaw_nf</span></a> <a id="l:477" class="idref" href="#l:477"><span class="id" title="binder">l</span></a> : <span class="id" title="keyword">∀</span> <a id="i:478" class="idref" href="#i:478"><span class="id" title="binder">i</span></a> <a id="alpha:479" class="idref" href="#alpha:479"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:479"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:478"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:479"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:477"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_equiv_1" class="idref" href="#standard_path_equiv_1"><span class="id" title="lemma">standard_path_equiv_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:480" class="idref" href="#i:480"><span class="id" title="binder">i</span></a> <a id="j:481" class="idref" href="#j:481"><span class="id" title="binder">j</span></a> <a id="alpha:482" class="idref" href="#alpha:482"><span class="id" title="binder">alpha</span></a> <a id="beta:483" class="idref" href="#beta:483"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:480"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:482"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:481"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:483"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:483"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:480"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:482"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:481"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#i:480"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_equiv_2" class="idref" href="#standard_path_equiv_2"><span class="id" title="lemma">standard_path_equiv_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:484" class="idref" href="#l:484"><span class="id" title="binder">l</span></a> <a id="i:485" class="idref" href="#i:485"><span class="id" title="binder">i</span></a> <a id="alpha:486" class="idref" href="#alpha:486"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.Paths.html#i:485"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="beta:487" class="idref" href="#beta:487"><span class="id" title="binder">beta</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:485"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:486"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:484"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:487"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:485"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:486"><span class="id" title="variable">alpha</span></a>  (<a class="idref" href="hydras.Epsilon0.Paths.html#l:484"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:485"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:487"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_S_iota_from" class="idref" href="#path_to_S_iota_from"><span class="id" title="lemma">path_to_S_iota_from</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:488" class="idref" href="#l:488"><span class="id" title="binder">l</span></a> <a id="i:489" class="idref" href="#i:489"><span class="id" title="binder">i</span></a> <a id="alpha:490" class="idref" href="#alpha:490"><span class="id" title="binder">alpha</span></a> <a id="beta:491" class="idref" href="#beta:491"><span class="id" title="binder">beta</span></a>,  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:491"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:489"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:488"><span class="id" title="variable">l</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:490"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path_toS"><span class="id" title="inductive">standard_path_toS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#l:488"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:489"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:491"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:489"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:490"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_S_standard_pathS" class="idref" href="#path_to_S_standard_pathS"><span class="id" title="lemma">path_to_S_standard_pathS</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="i:492" class="idref" href="#i:492"><span class="id" title="binder">i</span></a> <a id="j:493" class="idref" href="#j:493"><span class="id" title="binder">j</span></a> <a id="alpha:494" class="idref" href="#alpha:494"><span class="id" title="binder">alpha</span></a> <a id="beta:495" class="idref" href="#beta:495"><span class="id" title="binder">beta</span></a>,  (<a class="idref" href="hydras.Epsilon0.Paths.html#i:492"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:493"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:495"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:492"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:493"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:494"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:492"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:494"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:493"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:495"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_standardS_equiv" class="idref" href="#path_toS_standardS_equiv"><span class="id" title="lemma">path_toS_standardS_equiv</span></a> <a id="i:496" class="idref" href="#i:496"><span class="id" title="binder">i</span></a> <a id="j:497" class="idref" href="#j:497"><span class="id" title="binder">j</span></a> <a id="alpha:498" class="idref" href="#alpha:498"><span class="id" title="binder">alpha</span></a> <a id="beta:499" class="idref" href="#beta:499"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:496"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:497"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:499"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:496"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:497"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:498"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:496"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:498"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:497"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:499"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_to_standard_equiv" class="idref" href="#path_to_standard_equiv"><span class="id" title="lemma">path_to_standard_equiv</span></a> <a id="i:500" class="idref" href="#i:500"><span class="id" title="binder">i</span></a> <a id="j:501" class="idref" href="#j:501"><span class="id" title="binder">j</span></a> <a id="alpha:502" class="idref" href="#alpha:502"><span class="id" title="binder">alpha</span></a> <a id="beta:503" class="idref" href="#beta:503"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#i:500"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:500"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:501"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:503"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:500"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:501"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:502"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:500"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:502"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:501"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:503"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_pathS_app" class="idref" href="#standard_pathS_app"><span class="id" title="lemma">standard_pathS_app</span></a> <a id="i:504" class="idref" href="#i:504"><span class="id" title="binder">i</span></a> <a id="alpha:505" class="idref" href="#alpha:505"><span class="id" title="binder">alpha</span></a> <a id="j:506" class="idref" href="#j:506"><span class="id" title="binder">j</span></a> <a id="beta:507" class="idref" href="#beta:507"><span class="id" title="binder">beta</span></a> <a id="k:508" class="idref" href="#k:508"><span class="id" title="binder">k</span></a> <a id="gamma:509" class="idref" href="#gamma:509"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:504"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:506"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#j:506"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:508"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:504"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:505"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:506"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:507"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:506"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:507"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:508"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:509"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:504"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:505"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:508"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:509"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cons_standard_pathS" class="idref" href="#cons_standard_pathS"><span class="id" title="lemma">cons_standard_pathS</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:510" class="idref" href="#alpha:510"><span class="id" title="binder">alpha</span></a> <a id="n:511" class="idref" href="#n:511"><span class="id" title="binder">n</span></a> <a id="beta:512" class="idref" href="#beta:512"><span class="id" title="binder">beta</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:510"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:512"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="gamma:513" class="idref" href="#gamma:513"><span class="id" title="binder">gamma</span></a> <a id="i:514" class="idref" href="#i:514"><span class="id" title="binder">i</span></a> <a id="j:515" class="idref" href="#j:515"><span class="id" title="binder">j</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#i:514"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:512"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:515"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:513"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:513"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_pathS"><span class="id" title="definition">standard_pathS</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#i:514"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:510"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:512"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#j:515"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:510"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:513"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cons_standard_path" class="idref" href="#cons_standard_path"><span class="id" title="lemma">cons_standard_path</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:516" class="idref" href="#alpha:516"><span class="id" title="binder">alpha</span></a> <a id="n:517" class="idref" href="#n:517"><span class="id" title="binder">n</span></a> <a id="beta:518" class="idref" href="#beta:518"><span class="id" title="binder">beta</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:516"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:517"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:518"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="gamma:519" class="idref" href="#gamma:519"><span class="id" title="binder">gamma</span></a> <a id="i:520" class="idref" href="#i:520"><span class="id" title="binder">i</span></a> <a id="j:521" class="idref" href="#j:521"><span class="id" title="binder">j</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:520"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:518"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:521"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:519"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:519"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:520"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:520"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:516"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:517"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:518"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#j:521"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:516"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:517"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:519"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="flatten" class="idref" href="#flatten"><span class="id" title="lemma">flatten</span></a> : <span class="id" title="keyword">∀</span> (<a id="i:522" class="idref" href="#i:522"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  <a id="alpha:523" class="idref" href="#alpha:523"><span class="id" title="binder">alpha</span></a> (<a id="j:524" class="idref" href="#j:524"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  <a id="beta:525" class="idref" href="#beta:525"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:522"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:523"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#j:524"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:525"><span class="id" title="variable">beta</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:523"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:523"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:522"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:524"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:523"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:525"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_to_zero" class="idref" href="#standard_gnaw_to_zero"><span class="id" title="lemma">standard_gnaw_to_zero</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:526" class="idref" href="#alpha:526"><span class="id" title="binder">alpha</span></a> ,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:526"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="i:527" class="idref" href="#i:527"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="p:528" class="idref" href="#p:528"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:527"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:526"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#p:528"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_path_to_zero" class="idref" href="#standard_path_to_zero"><span class="id" title="lemma">standard_path_to_zero</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="alpha:529" class="idref" href="#alpha:529"><span class="id" title="binder">alpha</span></a> <a id="i:530" class="idref" href="#i:530"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:529"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:529"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:531" class="idref" href="#j:531"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:530"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:529"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:531"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_gnaw_2_zero" class="idref" href="#standard_gnaw_2_zero"><span class="id" title="lemma">standard_gnaw_2_zero</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="alpha:532" class="idref" href="#alpha:532"><span class="id" title="binder">alpha</span></a> <a id="i:533" class="idref" href="#i:533"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:532"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="t:534" class="idref" href="#t:534"><span class="id" title="binder">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:533"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:532"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:534"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+This section is dedicated to the proof of Lemma p 300 of KS
+    We associate to any path  from alpha to beta with a constant index  
+    a path from alpha to beta with a sequence of indices n+1, n+2, n+3, ... 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Constant_to_standard_Proof" class="idref" href="#Constant_to_standard_Proof"><span class="id" title="section">Constant_to_standard_Proof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Constant_to_standard_Proof.alpha" class="idref" href="#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a id="Constant_to_standard_Proof.beta" class="idref" href="#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="Constant_to_standard_Proof.n" class="idref" href="#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypotheses</span> (<a id="Constant_to_standard_Proof.Halpha" class="idref" href="#Constant_to_standard_Proof.Halpha"><span class="id" title="variable">Halpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>) (<a id="Constant_to_standard_Proof.Hpos" class="idref" href="#Constant_to_standard_Proof.Hpos"><span class="id" title="variable">Hpos</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Constant_to_standard_Proof.Hpa" class="idref" href="#Constant_to_standard_Proof.Hpa"><span class="id" title="variable">Hpa</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem0" class="idref" href="#Rem0"><span class="id" title="lemma">Rem0</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem1" class="idref" href="#Rem1"><span class="id" title="lemma">Rem1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="k:541" class="idref" href="#k:541"><span class="id" title="binder">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="gamma:542" class="idref" href="#gamma:542"><span class="id" title="binder">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#k:541"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:542"><span class="id" title="variable">gamma</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#gamma:542"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem01" class="idref" href="#Rem01"><span class="id" title="lemma">Rem01</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem02" class="idref" href="#Rem02"><span class="id" title="lemma">Rem02</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem2" class="idref" href="#Rem2"><span class="id" title="lemma">Rem2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="t:543" class="idref" href="#t:543"><span class="id" title="binder">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#t:543"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.t" class="idref" href="#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Rem2"><span class="id" title="lemma">Rem2</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem03" class="idref" href="#Rem03"><span class="id" title="lemma">Rem03</span></a> :  (<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.P" class="idref" href="#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> (<a id="i:544" class="idref" href="#i:544"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:544"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem04" class="idref" href="#Rem04"><span class="id" title="lemma">Rem04</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> 0.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem05" class="idref" href="#Rem05"><span class="id" title="lemma">Rem05</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem06" class="idref" href="#Rem06"><span class="id" title="lemma">Rem06</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="P_dec" class="idref" href="#P_dec"><span class="id" title="instance">P_dec</span></a> <a id="i:545" class="idref" href="#i:545"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Decision"><span class="id" title="class">Decision</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:545"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.l_def" class="idref" href="#Constant_to_standard_Proof.l_def"><span class="id" title="variable">l_def</span></a> :=  <a class="idref" href="hydras.Prelude.First_toggle.html#first_toggle"><span class="id" title="definition">first_toggle</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#P_dec"><span class="id" title="instance">P_dec</span></a> 0 <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#Rem06"><span class="id" title="lemma">Rem06</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Rem04"><span class="id" title="lemma">Rem04</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Rem05"><span class="id" title="lemma">Rem05</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.l" class="idref" href="#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l_def"><span class="id" title="variable">l_def</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.gamma" class="idref" href="#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> := <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem08" class="idref" href="#Rem08"><span class="id" title="lemma">Rem08</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:546" class="idref" href="#i:546"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:546"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#i:546"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:546"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem09" class="idref" href="#Rem09"><span class="id" title="lemma">Rem09</span></a> : (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem10" class="idref" href="#Rem10"><span class="id" title="lemma">Rem10</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem11" class="idref" href="#Rem11"><span class="id" title="lemma">Rem11</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R12" class="idref" href="#R12"><span class="id" title="lemma">R12</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R13" class="idref" href="#R13"><span class="id" title="lemma">R13</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> ((<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R14" class="idref" href="#R14"><span class="id" title="lemma">R14</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R15" class="idref" href="#R15"><span class="id" title="lemma">R15</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.m" class="idref" href="#Constant_to_standard_Proof.m"><span class="id" title="variable">m</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_def" class="idref" href="#m_def"><span class="id" title="lemma">m_def</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R16" class="idref" href="#R16"><span class="id" title="lemma">R16</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R17" class="idref" href="#R17"><span class="id" title="lemma">R17</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R18" class="idref" href="#R18"><span class="id" title="lemma">R18</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R19" class="idref" href="#R19"><span class="id" title="lemma">R19</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R20" class="idref" href="#R20"><span class="id" title="lemma">R20</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R21" class="idref" href="#R21"><span class="id" title="lemma">R21</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Constant_to_standard_Proof.delta" class="idref" href="#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a> := <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R22" class="idref" href="#R22"><span class="id" title="lemma">R22</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R23" class="idref" href="#R23"><span class="id" title="lemma">R23</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R24" class="idref" href="#R24"><span class="id" title="lemma">R24</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R25" class="idref" href="#R25"><span class="id" title="lemma">R25</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R26" class="idref" href="#R26"><span class="id" title="lemma">R26</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R27" class="idref" href="#R27"><span class="id" title="lemma">R27</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R28" class="idref" href="#R28"><span class="id" title="lemma">R28</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta"><span class="id" title="variable">delta</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R29" class="idref" href="#R29"><span class="id" title="lemma">R29</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R30" class="idref" href="#R30"><span class="id" title="lemma">R30</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R31_0" class="idref" href="#R31_0"><span class="id" title="lemma">R31_0</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="gamma_positive" class="idref" href="#gamma_positive"><span class="id" title="lemma">gamma_positive</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R31" class="idref" href="#R31"><span class="id" title="lemma">R31</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="constant_to_standard_0" class="idref" href="#constant_to_standard_0"><span class="id" title="lemma">constant_to_standard_0</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="l:547" class="idref" href="#l:547"><span class="id" title="binder">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:547"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof"><span class="id" title="section">Constant_to_standard_Proof</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="constant_to_standard" class="idref" href="#constant_to_standard"><span class="id" title="lemma">constant_to_standard</span></a>  (<a id="alpha:548" class="idref" href="#alpha:548"><span class="id" title="binder">alpha</span></a> <a id="beta:549" class="idref" href="#beta:549"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n:550" class="idref" href="#n:550"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:548"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:550"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:548"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:549"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="l:551" class="idref" href="#l:551"><span class="id" title="binder">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_gnaw"><span class="id" title="definition">standard_gnaw</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:550"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:548"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#l:551"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:549"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="constant_to_standard_path" class="idref" href="#constant_to_standard_path"><span class="id" title="lemma">constant_to_standard_path</span></a> <br/>
+&nbsp;&nbsp;(<a id="alpha:552" class="idref" href="#alpha:552"><span class="id" title="binder">alpha</span></a> <a id="beta:553" class="idref" href="#beta:553"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="i:554" class="idref" href="#i:554"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:552"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#const_pathS"><span class="id" title="definition">const_pathS</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:554"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:552"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:553"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:552"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:555" class="idref" href="#j:555"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:554"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:552"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:555"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:553"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>. 
+<br/>
+<span class="id" title="keyword">Corollary</span>  <a id="LT_to_standard_path" class="idref" href="#LT_to_standard_path"><span class="id" title="lemma">LT_to_standard_path</span></a> (<a id="alpha:556" class="idref" href="#alpha:556"><span class="id" title="binder">alpha</span></a> <a id="beta:557" class="idref" href="#beta:557"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:557"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:556"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="n:558" class="idref" href="#n:558"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:559" class="idref" href="#j:559"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#standard_path"><span class="id" title="definition">standard_path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:558"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:556"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:559"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:557"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab86"></a><h1 class="section">Adaptation to <span class="inlinecode"><span class="id" title="var">E0</span></span></h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Canon_plus" class="idref" href="#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a id="i:560" class="idref" href="#i:560"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)( <a id="alpha:561" class="idref" href="#alpha:561"><span class="id" title="binder">alpha</span></a> <a id="beta:562" class="idref" href="#beta:562"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.Epsilon0.Paths.html#const_path"><span class="id" title="definition">const_path</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:560"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:561"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:562"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_plus_inv" class="idref" href="#Canon_plus_inv"><span class="id" title="lemma">Canon_plus_inv</span></a> <a id="n:563" class="idref" href="#n:563"><span class="id" title="binder">n</span></a> <a id="alpha:564" class="idref" href="#alpha:564"><span class="id" title="binder">alpha</span></a> <a id="beta:565" class="idref" href="#beta:565"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:563"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:564"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:565"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:565"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:564"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:563"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:563"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:564"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:563"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:565"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="KS_thm_2_4_E0" class="idref" href="#KS_thm_2_4_E0"><span class="id" title="lemma">KS_thm_2_4_E0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="lambda:566" class="idref" href="#lambda:566"><span class="id" title="binder">lambda</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:566"><span class="id" title="variable">lambda</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:567" class="idref" href="#i:567"><span class="id" title="binder">i</span></a> <a id="j:568" class="idref" href="#j:568"><span class="id" title="binder">j</span></a>, (<a class="idref" href="hydras.Epsilon0.Paths.html#i:567"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:568"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> 1 (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:566"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:568"><span class="id" title="variable">j</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#lambda:566"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:567"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Cor12_E0" class="idref" href="#Cor12_E0"><span class="id" title="lemma">Cor12_E0</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:569" class="idref" href="#alpha:569"><span class="id" title="binder">alpha</span></a> <a id="beta:570" class="idref" href="#beta:570"><span class="id" title="binder">beta</span></a> <a id="i:571" class="idref" href="#i:571"><span class="id" title="binder">i</span></a> <a id="n:572" class="idref" href="#n:572"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:570"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:569"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:571"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:572"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:571"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:569"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:570"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:572"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:569"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:570"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_mono1" class="idref" href="#Canon_mono1"><span class="id" title="lemma">Canon_mono1</span></a> <a id="alpha:573" class="idref" href="#alpha:573"><span class="id" title="binder">alpha</span></a> <a id="i:574" class="idref" href="#i:574"><span class="id" title="binder">i</span></a> <a id="j:575" class="idref" href="#j:575"><span class="id" title="binder">j</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:573"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.Paths.html#i:574"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:575"><span class="id" title="variable">j</span></a>)% <span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:573"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:574"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:573"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#j:575"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">e0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonS_plus_1" class="idref" href="#CanonS_plus_1"><span class="id" title="lemma">CanonS_plus_1</span></a> <a id="alpha:576" class="idref" href="#alpha:576"><span class="id" title="binder">alpha</span></a> <a id="beta:577" class="idref" href="#beta:577"><span class="id" title="binder">beta</span></a> <a id="k:578" class="idref" href="#k:578"><span class="id" title="binder">k</span></a> <a id="i:579" class="idref" href="#i:579"><span class="id" title="binder">i</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:577"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:576"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#beta:577"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:576"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:576"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:579"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:577"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">e0</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:578"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:576"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.Paths.html#i:579"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:577"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#k:578"><span class="id" title="variable">k</span></a>)<a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">e0</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="CanonS_Phi0_Succ_eqn" class="idref" href="#CanonS_Phi0_Succ_eqn"><span class="id" title="lemma">CanonS_Phi0_Succ_eqn</span></a> <a id="i:580" class="idref" href="#i:580"><span class="id" title="binder">i</span></a> <a id="gamma:581" class="idref" href="#gamma:581"><span class="id" title="binder">gamma</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:581"><span class="id" title="variable">gamma</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:580"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#Omega_term"><span class="id" title="instance">Omega_term</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#gamma:581"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:580"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Lemma2_6_1_E0" class="idref" href="#Lemma2_6_1_E0"><span class="id" title="lemma">Lemma2_6_1_E0</span></a> (<a id="alpha:582" class="idref" href="#alpha:582"><span class="id" title="binder">alpha</span></a> <a id="beta:583" class="idref" href="#beta:583"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.Paths.html#beta:583"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:582"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">e0</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:584" class="idref" href="#n:584"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#n:584"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:582"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:583"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Lemmas used by <span class="inlinecode"><span class="id" title="var">F_alpha</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_plus_first_step" class="idref" href="#Canon_plus_first_step"><span class="id" title="lemma">Canon_plus_first_step</span></a>: <span class="id" title="keyword">∀</span> <a id="i:585" class="idref" href="#i:585"><span class="id" title="binder">i</span></a> <a id="alpha:586" class="idref" href="#alpha:586"><span class="id" title="binder">alpha</span></a> <a id="beta:587" class="idref" href="#beta:587"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:585"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:586"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:587"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#alpha:586"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:587"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:585"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:586"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:587"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Canon_plus_first_step_lim" class="idref" href="#Canon_plus_first_step_lim"><span class="id" title="lemma">Canon_plus_first_step_lim</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:588" class="idref" href="#i:588"><span class="id" title="binder">i</span></a> <a id="alpha:589" class="idref" href="#alpha:589"><span class="id" title="binder">alpha</span></a> <a id="beta:590" class="idref" href="#beta:590"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:589"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:588"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:589"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#beta:590"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#beta:590"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:589"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:588"><span class="id" title="variable">i</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#Canon_plus"><span class="id" title="definition">Canon_plus</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:588"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#alpha:589"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#i:588"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Epsilon0.Paths.html#beta:590"><span class="id" title="variable">beta</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Epsilon0.T1.html b/theories/html/hydras.Epsilon0.T1.html
new file mode 100644
index 00000000..f7b82b2a
--- /dev/null
+++ b/theories/html/hydras.Epsilon0.T1.html
@@ -0,0 +1,2392 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Epsilon0.T1</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Epsilon0.T1</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  A type for ordinals terms in Cantor Normal Form
+
+<div class="paragraph"> </div>
+
+After Manolios and Vroon's work on ACL2 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Max.html#"><span class="id" title="library">Max</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Operators_Properties.html#"><span class="id" title="library">Operators_Properties</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.PArith.PArith.html#"><span class="id" title="library">PArith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a>  <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Restriction</span></a>   <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.OrdNotations.html#"><span class="id" title="library">OrdNotations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.Comparable.html#"><span class="id" title="library">Prelude.Comparable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#"><span class="id" title="library">STDPP_compat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Create HintDb</span> <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">t1</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">Sortclass</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab39"></a><h1 class="section">Definitions</h1>
+
+
+<div class="paragraph"> </div>
+
+ 
+<div class="paragraph"> </div>
+
+<a id="lab40"></a><h2 class="section">A type of terms (not necessarily in normal form)</h2>
+
+
+<div class="paragraph"> </div>
+
+<span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span>  is intended to represent
+  the ordinal  <span class="inlinecode"><span class="id" title="var">omega</span>^<span class="id" title="var">a</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span>  <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">b</span></span>
+
+<div class="paragraph"> </div>
+
+Note that <span class="inlinecode"><span class="id" title="var">T1</span></span> contains terms which are not in Cantor normal  form.
+This issue is solved later which the help of the predicate <span class="inlinecode"><span class="id" title="var">nf</span></span>
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="T1" class="idref" href="#T1"><span class="id" title="inductive">T1</span></a> : <span class="id" title="keyword">Set</span>  :=<br/>
+| <a id="zero" class="idref" href="#zero"><span class="id" title="constructor">zero</span></a> <br/>
+| <a id="cons" class="idref" href="#cons"><span class="id" title="constructor">cons</span></a> (<a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1:1"><span class="id" title="inductive">T1</span></a>) (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="beta:5" class="idref" href="#beta:5"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1:1"><span class="id" title="inductive">T1</span></a>) .<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Basic functions and predicates on <span class="inlinecode"><span class="id" title="var">T1</span></span> 
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span>-th ordinal 
+</div>
+<div class="code">
+<span class="id" title="keyword">Notation</span> <a id="FS" class="idref" href="#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="one" class="idref" href="#one"><span class="id" title="abbreviation">one</span></a> := (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> 0).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+the <span class="inlinecode"><span class="id" title="var">n</span></span>-th (finite) ordinal  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1nat" class="idref" href="#T1nat"><span class="id" title="definition">T1nat</span></a> (<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#n:6"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="d9b48d1d60bd5708755ece6f90c63f3a" class="idref" href="#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">&quot;</span></a>\F n" := (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <span class="id" title="var">n</span>)  (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 29): <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ten" class="idref" href="#ten"><span class="id" title="definition">ten</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> := 10.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1nat" )]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="fin" class="idref" href="#fin"><span class="id" title="abbreviation">fin</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="T1omega" class="idref" href="#T1omega"><span class="id" title="abbreviation">T1omega</span></a> := (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1omega")]<br/>
+&nbsp;<span class="id" title="keyword">Notation</span> <a id="omega" class="idref" href="#omega"><span class="id" title="abbreviation">omega</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Successor and limits (syntactic definitions) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T1is_succ" class="idref" href="#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:9"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">_alpha</span> <span class="id" title="var">_n</span> <span class="id" title="keyword">beta</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ:10"><span class="id" title="definition">T1is_succ</span></a> <span class="id" title="keyword">beta</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T1limit" class="idref" href="#T1limit"><span class="id" title="definition">T1limit</span></a> <a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:12"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="keyword">beta</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#T1limit:13"><span class="id" title="definition">T1limit</span></a> <span class="id" title="keyword">beta</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1is_succ")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="succb" class="idref" href="#succb"><span class="id" title="abbreviation">succb</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1limit")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="limitb" class="idref" href="#limitb"><span class="id" title="abbreviation">limitb</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Exponential of base <span class="inlinecode"><span class="id" title="var">omega</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="phi0" class="idref" href="#phi0"><span class="id" title="abbreviation">phi0</span></a> <span class="id" title="var">alpha</span> := (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+multiples of <span class="inlinecode"><span class="id" title="var">phi0</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="omega_term" class="idref" href="#omega_term"><span class="id" title="definition">omega_term</span></a> (<a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)(<a id="k:16" class="idref" href="#k:16"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#k:16"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ omega towers 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="omega_tower" class="idref" href="#omega_tower"><span class="id" title="definition">omega_tower</span></a> (<a id="height:17" class="idref" href="#height:17"><span class="id" title="binder">height</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#height:17"><span class="id" title="variable">height</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| 0 ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">h</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#omega_tower:18"><span class="id" title="definition">omega_tower</span></a> <span class="id" title="var">h</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Additive principal ordinals
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="ap" class="idref" href="#ap"><span class="id" title="inductive">ap</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="ap_intro" class="idref" href="#ap_intro"><span class="id" title="constructor">ap_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#ap:20"><span class="id" title="inductive">ap</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:22"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab41"></a><h2 class="section">A linear  strict order on <span class="inlinecode"><span class="id" title="var">T1</span></span></h2>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="compare_T1" class="idref" href="#compare_T1"><span class="id" title="instance">compare_T1</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">cmp</span> (<a id="alpha:23" class="idref" href="#alpha:23"><span class="id" title="binder">alpha</span></a> <a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:23"><span class="id" title="variable">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#beta:24"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span>   , <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;| (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>),(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#cmp:25"><span class="id" title="variable">cmp</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ (<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#4a243c6f4b9487508e7110341eeedb0b"><span class="id" title="notation">?=</span></a> <span class="id" title="var">n'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cmp:25"><span class="id" title="variable">cmp</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">comp</span> ⇒ <span class="id" title="var">comp</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> (<a id="alpha:28" class="idref" href="#alpha:28"><span class="id" title="binder">alpha</span></a> <a id="beta:29" class="idref" href="#beta:29"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:28"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:29"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="le" class="idref" href="#le"><span class="id" title="abbreviation">le</span></a>  := (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="E1" class="idref" href="#E1"><span class="id" title="definition">E1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> 56 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> 3).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="E2" class="idref" href="#E2"><span class="id" title="definition">E2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> 3) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> 5 (<a class="idref" href="hydras.Epsilon0.T1.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> 3))%<span class="id" title="var">t1</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab42"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">compare</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_cons" class="idref" href="#compare_cons"><span class="id" title="lemma">compare_cons</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> <a id="b:32" class="idref" href="#b:32"><span class="id" title="binder">b</span></a> <a id="a':33" class="idref" href="#a':33"><span class="id" title="binder">a'</span></a> <a id="n':34" class="idref" href="#n':34"><span class="id" title="binder">n'</span></a> <a id="b':35" class="idref" href="#b':35"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:32"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':33"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':34"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':35"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':33"><span class="id" title="variable">a'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ (<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#4a243c6f4b9487508e7110341eeedb0b"><span class="id" title="notation">?=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':34"><span class="id" title="variable">n'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:32"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':35"><span class="id" title="variable">b'</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">comp</span> ⇒ <span class="id" title="var">comp</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_rev" class="idref" href="#compare_rev"><span class="id" title="lemma">compare_rev</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> <a id="beta:37" class="idref" href="#beta:37"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompOpp"><span class="id" title="definition">CompOpp</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:37"><span class="id" title="variable">beta</span></a>).  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:38" class="idref" href="#alpha:38"><span class="id" title="binder">alpha</span></a> <a id="beta:39" class="idref" href="#beta:39"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:38"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:39"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:38"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:39"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#alpha:38"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:39"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:39"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:38"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> (<a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a> <a id="beta:41" class="idref" href="#beta:41"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:40"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:41"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:40"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:41"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab43"></a><h2 class="section">Properties of Eq</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_refl" class="idref" href="#compare_refl"><span class="id" title="lemma">compare_refl</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_eq_iff" class="idref" href="#compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a> (<a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:44"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab44"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">lt</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="not_lt_zero" class="idref" href="#not_lt_zero"><span class="id" title="lemma">not_lt_zero</span></a> <a id="alpha:45" class="idref" href="#alpha:45"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:45"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">not_lt_zero</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_lt_impl" class="idref" href="#compare_lt_impl"><span class="id" title="lemma">compare_lt_impl</span></a> <a id="a:46" class="idref" href="#a:46"><span class="id" title="binder">a</span></a> <a id="b:47" class="idref" href="#b:47"><span class="id" title="binder">b</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:47"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_lt_iff" class="idref" href="#compare_lt_iff"><span class="id" title="lemma">compare_lt_iff</span></a> <a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="b:49" class="idref" href="#b:49"><span class="id" title="binder">b</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:49"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:49"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab45"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">lt</span></span> inv</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Inductive</span> <a id="lt_cases" class="idref" href="#lt_cases"><span class="id" title="inductive">lt_cases</span></a> (<a id="a:50" class="idref" href="#a:50"><span class="id" title="binder">a</span></a>  <a id="b:51" class="idref" href="#b:51"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a':53" class="idref" href="#a':53"><span class="id" title="binder">a'</span></a> <a id="b':54" class="idref" href="#b':54"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n':55" class="idref" href="#n':55"><span class="id" title="binder">n'</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;&nbsp;| <a id="lt_left" class="idref" href="#lt_left"><span class="id" title="constructor">lt_left</span></a> (<a id="H:58" class="idref" href="#H:58"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':53"><span class="id" title="variable">a'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="lt_middle" class="idref" href="#lt_middle"><span class="id" title="constructor">lt_middle</span></a> (<a id="H:59" class="idref" href="#H:59"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':53"><span class="id" title="variable">a'</span></a>)(<a id="H1:60" class="idref" href="#H1:60"><span class="id" title="binder">H1</span></a> : (<a class="idref" href="hydras.Epsilon0.T1.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':55"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span>)<br/>
+&nbsp;&nbsp;| <a id="lt_right" class="idref" href="#lt_right"><span class="id" title="constructor">lt_right</span></a> (<a id="H:61" class="idref" href="#H:61"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':53"><span class="id" title="variable">a'</span></a>)(<a id="H1:62" class="idref" href="#H1:62"><span class="id" title="binder">H1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':55"><span class="id" title="variable">n'</span></a>)(<a id="H2:63" class="idref" href="#H2:63"><span class="id" title="binder">H2</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:51"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':54"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inv_strong" class="idref" href="#lt_inv_strong"><span class="id" title="lemma">lt_inv_strong</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a> <a id="n:65" class="idref" href="#n:65"><span class="id" title="binder">n</span></a> <a id="b:66" class="idref" href="#b:66"><span class="id" title="binder">b</span></a> <a id="a':67" class="idref" href="#a':67"><span class="id" title="binder">a'</span></a> <a id="n':68" class="idref" href="#n':68"><span class="id" title="binder">n'</span></a> <a id="b':69" class="idref" href="#b':69"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:66"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':67"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':68"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':69"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt_cases"><span class="id" title="inductive">lt_cases</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:66"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':67"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':69"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':68"><span class="id" title="variable">n'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_irrefl" class="idref" href="#lt_irrefl"><span class="id" title="lemma">lt_irrefl</span></a> (<a id="alpha:70" class="idref" href="#alpha:70"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:70"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_inv" class="idref" href="#lt_inv"><span class="id" title="lemma">lt_inv</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:71" class="idref" href="#a:71"><span class="id" title="binder">a</span></a> <a id="n:72" class="idref" href="#n:72"><span class="id" title="binder">n</span></a> <a id="b:73" class="idref" href="#b:73"><span class="id" title="binder">b</span></a> <a id="a':74" class="idref" href="#a':74"><span class="id" title="binder">a'</span></a> <a id="n':75" class="idref" href="#n':75"><span class="id" title="binder">n'</span></a> <a id="b':76" class="idref" href="#b':76"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:72"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:73"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':74"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':75"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':76"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:71"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a':74"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':74"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:72"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':75"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':74"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:72"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':75"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:73"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b':76"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inv_coeff" class="idref" href="#lt_inv_coeff"><span class="id" title="lemma">lt_inv_coeff</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:77" class="idref" href="#a:77"><span class="id" title="binder">a</span></a> <a id="n:78" class="idref" href="#n:78"><span class="id" title="binder">n</span></a> <a id="n':79" class="idref" href="#n':79"><span class="id" title="binder">n'</span></a> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a> <a id="b':81" class="idref" href="#b':81"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:78"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:80"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':79"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':81"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:78"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':79"><span class="id" title="variable">n'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inv_coeff_dec" class="idref" href="#lt_inv_coeff_dec"><span class="id" title="lemma">lt_inv_coeff_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:82" class="idref" href="#a:82"><span class="id" title="binder">a</span></a> <a id="n:83" class="idref" href="#n:83"><span class="id" title="binder">n</span></a> <a id="n':84" class="idref" href="#n':84"><span class="id" title="binder">n'</span></a> <a id="b:85" class="idref" href="#b:85"><span class="id" title="binder">b</span></a> <a id="b':86" class="idref" href="#b':86"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:82"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:83"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:85"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:82"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':84"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':86"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a>(<a class="idref" href="hydras.Epsilon0.T1.html#n:83"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':84"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:83"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':84"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:85"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':86"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inv_tail" class="idref" href="#lt_inv_tail"><span class="id" title="lemma">lt_inv_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:87" class="idref" href="#a:87"><span class="id" title="binder">a</span></a> <a id="n:88" class="idref" href="#n:88"><span class="id" title="binder">n</span></a> <a id="b:89" class="idref" href="#b:89"><span class="id" title="binder">b</span></a> <a id="b':90" class="idref" href="#b':90"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:87"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:88"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:89"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:87"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:88"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':90"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':90"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="head_lt" class="idref" href="#head_lt"><span class="id" title="lemma">head_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:91" class="idref" href="#alpha:91"><span class="id" title="binder">alpha</span></a> <a id="alpha':92" class="idref" href="#alpha':92"><span class="id" title="binder">alpha'</span></a> <a id="n:93" class="idref" href="#n:93"><span class="id" title="binder">n</span></a> <a id="n':94" class="idref" href="#n':94"><span class="id" title="binder">n'</span></a> <a id="beta:95" class="idref" href="#beta:95"><span class="id" title="binder">beta</span></a> <a id="beta':96" class="idref" href="#beta':96"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:91"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha':92"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:91"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:93"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:95"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':92"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':94"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':96"><span class="id" title="variable">beta'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="coeff_lt" class="idref" href="#coeff_lt"><span class="id" title="lemma">coeff_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:97" class="idref" href="#alpha:97"><span class="id" title="binder">alpha</span></a> <a id="n:98" class="idref" href="#n:98"><span class="id" title="binder">n</span></a> <a id="n':99" class="idref" href="#n':99"><span class="id" title="binder">n'</span></a> <a id="beta:100" class="idref" href="#beta:100"><span class="id" title="binder">beta</span></a> <a id="beta':101" class="idref" href="#beta':101"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#n:98"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':99"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:97"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:98"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:100"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:97"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':99"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':101"><span class="id" title="variable">beta'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="tail_lt" class="idref" href="#tail_lt"><span class="id" title="lemma">tail_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:102" class="idref" href="#alpha:102"><span class="id" title="binder">alpha</span></a> <a id="n:103" class="idref" href="#n:103"><span class="id" title="binder">n</span></a> <a id="beta:104" class="idref" href="#beta:104"><span class="id" title="binder">beta</span></a> <a id="beta':105" class="idref" href="#beta':105"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:104"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':105"><span class="id" title="variable">beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:102"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:103"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:104"><span class="id" title="variable">beta</span></a>)  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:102"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:103"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':105"><span class="id" title="variable">beta'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_fin_rw" class="idref" href="#compare_fin_rw"><span class="id" title="lemma">compare_fin_rw</span></a> (<a id="n:106" class="idref" href="#n:106"><span class="id" title="binder">n</span></a> <a id="n1:107" class="idref" href="#n1:107"><span class="id" title="binder">n1</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:106"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n1:107"><span class="id" title="variable">n1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#n:106"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#4a243c6f4b9487508e7110341eeedb0b"><span class="id" title="notation">?=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n1:107"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_fin_iff" class="idref" href="#lt_fin_iff"><span class="id" title="lemma">lt_fin_iff</span></a> (<a id="i:108" class="idref" href="#i:108"><span class="id" title="binder">i</span></a> <a id="j:109" class="idref" href="#j:109"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:108"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#j:109"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt"><span class="id" title="definition">Nat.lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:108"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#j:109"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_trans" class="idref" href="#lt_trans"><span class="id" title="lemma">lt_trans</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:110" class="idref" href="#alpha:110"><span class="id" title="binder">alpha</span></a> <a id="beta:111" class="idref" href="#beta:111"><span class="id" title="binder">beta</span></a> <a id="gamma:112" class="idref" href="#gamma:112"><span class="id" title="binder">gamma</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:112"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:112"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="t1_strorder" class="idref" href="#t1_strorder"><span class="id" title="instance">t1_strorder</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>. 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inv_head" class="idref" href="#lt_inv_head"><span class="id" title="lemma">lt_inv_head</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:113" class="idref" href="#a:113"><span class="id" title="binder">a</span></a> <a id="n:114" class="idref" href="#n:114"><span class="id" title="binder">n</span></a> <a id="b:115" class="idref" href="#b:115"><span class="id" title="binder">b</span></a> <a id="a':116" class="idref" href="#a':116"><span class="id" title="binder">a'</span></a> <a id="n':117" class="idref" href="#n':117"><span class="id" title="binder">n'</span></a> <a id="b':118" class="idref" href="#b':118"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:113"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:114"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:115"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':116"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':117"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':118"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:113"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':116"><span class="id" title="variable">a'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab46"></a><h2 class="section">The predicate "to be in normal form"</h2>
+
+
+<div class="paragraph"> </div>
+
+Cantor normal form needs the exponents of omega to be
+   in strict decreasing order 
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="lt_dec" class="idref" href="#lt_dec"><span class="id" title="instance">lt_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> :=<br/>
+<span class="id" title="keyword">fun</span> <a id="alpha:119" class="idref" href="#alpha:119"><span class="id" title="binder">alpha</span></a> <a id="beta:120" class="idref" href="#beta:120"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide"><span class="id" title="definition">decide</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:119"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:120"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nf_b" class="idref" href="#nf_b"><span class="id" title="definition">nf_b</span></a> (<a id="alpha:121" class="idref" href="#alpha:121"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:121"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#nf_b:122"><span class="id" title="definition">nf_b</span></a> <span class="id" title="var">a</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> ((<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">b</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#nf_b:122"><span class="id" title="definition">nf_b</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf_b:122"><span class="id" title="definition">nf_b</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">a</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">bool</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="nf" class="idref" href="#nf"><span class="id" title="definition">nf</span></a> <a id="alpha:124" class="idref" href="#alpha:124"><span class="id" title="binder">alpha</span></a> :<span class="id" title="keyword">Prop</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_b"><span class="id" title="definition">nf_b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:124"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="bad_term" class="idref" href="#bad_term"><span class="id" title="definition">bad_term</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> 1 1 (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> 2 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+epsilon0 as a set 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="epsilon_0" class="idref" href="#epsilon_0"><span class="id" title="definition">epsilon_0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab47"></a><h2 class="section">Arithmetic functions</h2>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab48"></a><h3 class="section">Successor</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="succ" class="idref" href="#succ"><span class="id" title="definition">succ</span></a> (<a id="a:125" class="idref" href="#a:125"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#a:125"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.T1.html#succ:126"><span class="id" title="definition">succ</span></a> <span class="id" title="var">c</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab49"></a><h3 class="section">Predecessor (partial function</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="pred" class="idref" href="#pred"><span class="id" title="definition">pred</span></a> (<a id="c:128" class="idref" href="#c:128"><span class="id" title="binder">c</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#c:128"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Epsilon0.T1.html#pred:129"><span class="id" title="definition">pred</span></a> <span class="id" title="var">b</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">c</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#c:128"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab50"></a><h3 class="section">Addition</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T1add" class="idref" href="#T1add"><span class="id" title="definition">T1add</span></a> (<a id="a:131" class="idref" href="#a:131"><span class="id" title="binder">a</span></a> <a id="b:132" class="idref" href="#b:132"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#a:131"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#b:132"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">y</span>  ⇒ <span class="id" title="var">y</span><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">x</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  ⇒ <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:131"><span class="id" title="variable">a</span></a> <span class="id" title="var">a'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:131"><span class="id" title="variable">a</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1add:133"><span class="id" title="definition">T1add</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:132"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>  ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:131"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><span class="id" title="var">n'</span>)) <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="f1825676a507995dc4cc5d57158dc2ac" class="idref" href="#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">&quot;</span></a>alpha + beta" := (<a class="idref" href="hydras.Epsilon0.T1.html#T1add:133"><span class="id" title="definition">T1add</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>) : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1add")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="plus" class="idref" href="#plus"><span class="id" title="abbreviation">plus</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1add</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab51"></a><h3 class="section">multiplication</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T1mul" class="idref" href="#T1mul"><span class="id" title="definition">T1mul</span></a> (<a id="a:136" class="idref" href="#a:136"><span class="id" title="binder">a</span></a> <a id="b:137" class="idref" href="#b:137"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#a:136"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#b:137"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">_</span>  ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)) <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:136"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)) <a class="idref" href="hydras.Epsilon0.T1.html#b:137"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:136"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <span class="id" title="var">a'</span>) <span class="id" title="var">n'</span> (<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:136"><span class="id" title="variable">a</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#b:137"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="ec5016939656baf2f20020d4a10c3866" class="idref" href="#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">&quot;</span></a>a * b" := (<a class="idref" href="hydras.Epsilon0.T1.html#T1mul:138"><span class="id" title="definition">T1mul</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use T1mul")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="mult" class="idref" href="#mult"><span class="id" title="abbreviation">mult</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1mul</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab52"></a><h3 class="section">Substraction  (used as a helper for exponentiation)</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="minus" class="idref" href="#minus"><span class="id" title="definition">minus</span></a> (<a id="alpha:141" class="idref" href="#alpha:141"><span class="id" title="binder">alpha</span></a> <a id="beta:142" class="idref" href="#beta:142"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:141"><span class="id" title="variable">alpha</span></a>,<a class="idref" href="hydras.Epsilon0.T1.html#beta:142"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">y</span>  ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> <span class="id" title="var">n</span> <span class="id" title="var">n'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span>  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a> (<span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a><span class="id" title="var">n'</span>)) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span><br/>
+&nbsp;|  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ (<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a> <span class="id" title="var">n</span> <span class="id" title="var">n'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a> (<span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">n'</span>)) <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <span class="id" title="var">b</span> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;<span class="id" title="keyword">where</span>  <a id="::t1_scope:x_'-'_x" class="idref" href="#::t1_scope:x_'-'_x"><span class="id" title="notation">&quot;</span></a>alpha - beta" := (<a class="idref" href="hydras.Epsilon0.T1.html#minus:143"><span class="id" title="definition">minus</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>):<span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab53"></a><h3 class="section">exponentiation functions</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="exp_F" class="idref" href="#exp_F"><span class="id" title="definition">exp_F</span></a> (<a id="alpha:146" class="idref" href="#alpha:146"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)(<a id="n:147" class="idref" href="#n:147"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#n:147"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;| 0 ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> 0<br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#alpha:146"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#exp_F:148"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:146"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">p</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="exp" class="idref" href="#exp"><span class="id" title="definition">exp</span></a>  (<a id="alpha:150" class="idref" href="#alpha:150"><span class="id" title="binder">alpha</span></a> <a id="beta:151" class="idref" href="#beta:151"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:150"><span class="id" title="variable">alpha</span></a> ,<a class="idref" href="hydras.Epsilon0.T1.html#beta:151"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;|  <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <span class="id" title="var">_</span> , <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">_</span>         ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;| <span class="id" title="var">x</span> , <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">_</span> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#exp_F"><span class="id" title="definition">exp_F</span></a> <span class="id" title="var">x</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>)<br/>
+&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <span class="id" title="var">n'</span>  <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n'</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a>  <span class="id" title="var">b'</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a> (<span class="id" title="var">a'</span> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 1) <span class="id" title="var">n'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <span class="id" title="var">b'</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a><br/>
+&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span>  <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a>   (<span class="id" title="var">a</span> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <span class="id" title="var">b'</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>)<br/>
+<span class="id" title="keyword">end</span> <br/>
+<span class="id" title="keyword">where</span> <a id="6d507108061d062f40e7ec3dfffa8239" class="idref" href="#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">&quot;</span></a>alpha ^ beta " := (<a class="idref" href="hydras.Epsilon0.T1.html#exp:152"><span class="id" title="definition">exp</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>) : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab54"></a><h1 class="section">Lemmas</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_of_phi0" class="idref" href="#compare_of_phi0"><span class="id" title="lemma">compare_of_phi0</span></a> <a id="alpha:155" class="idref" href="#alpha:155"><span class="id" title="binder">alpha</span></a> <a id="beta:156" class="idref" href="#beta:156"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:155"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:156"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:155"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:156"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_lt" class="idref" href="#zero_lt"><span class="id" title="lemma">zero_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:157" class="idref" href="#alpha:157"><span class="id" title="binder">alpha</span></a> <a id="n:158" class="idref" href="#n:158"><span class="id" title="binder">n</span></a> <a id="beta:159" class="idref" href="#beta:159"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:158"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:159"><span class="id" title="variable">beta</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_lt</span> <span class="id" title="var">head_lt</span> <span class="id" title="var">coeff_lt</span> <span class="id" title="var">tail_lt</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_nf" class="idref" href="#zero_nf"><span class="id" title="lemma">zero_nf</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="single_nf" class="idref" href="#single_nf"><span class="id" title="lemma">single_nf</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:160" class="idref" href="#a:160"><span class="id" title="binder">a</span></a> <a id="n:161" class="idref" href="#n:161"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:160"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:160"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:161"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cons_nf" class="idref" href="#cons_nf"><span class="id" title="lemma">cons_nf</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:162" class="idref" href="#a:162"><span class="id" title="binder">a</span></a> <a id="n:163" class="idref" href="#n:163"><span class="id" title="binder">n</span></a> <a id="a':164" class="idref" href="#a':164"><span class="id" title="binder">a'</span></a> <a id="n':165" class="idref" href="#n':165"><span class="id" title="binder">n'</span></a> <a id="b:166" class="idref" href="#b:166"><span class="id" title="binder">b</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':164"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':164"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':165"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:166"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:163"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':164"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':165"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:166"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_nf</span> <span class="id" title="var">single_nf</span> <span class="id" title="var">cons_nf</span>: <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_inv1" class="idref" href="#nf_inv1"><span class="id" title="lemma">nf_inv1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:167" class="idref" href="#a:167"><span class="id" title="binder">a</span></a> <a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a> <a id="b:169" class="idref" href="#b:169"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:167"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:168"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:169"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:167"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_inv2" class="idref" href="#nf_inv2"><span class="id" title="lemma">nf_inv2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:170" class="idref" href="#a:170"><span class="id" title="binder">a</span></a> <a id="n:171" class="idref" href="#n:171"><span class="id" title="binder">n</span></a> <a id="b:172" class="idref" href="#b:172"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:170"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:171"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:172"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:172"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_inv3" class="idref" href="#nf_inv3"><span class="id" title="lemma">nf_inv3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:173" class="idref" href="#a:173"><span class="id" title="binder">a</span></a> <a id="n:174" class="idref" href="#n:174"><span class="id" title="binder">n</span></a>  <a id="a':175" class="idref" href="#a':175"><span class="id" title="binder">a'</span></a> <a id="n':176" class="idref" href="#n':176"><span class="id" title="binder">n'</span></a> <a id="b':177" class="idref" href="#b':177"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:173"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:174"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':175"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':176"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':177"><span class="id" title="variable">b'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':175"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:173"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_cons_inv" class="idref" href="#nf_cons_inv"><span class="id" title="lemma">nf_cons_inv</span></a> <a id="a:178" class="idref" href="#a:178"><span class="id" title="binder">a</span></a> <a id="n:179" class="idref" href="#n:179"><span class="id" title="binder">n</span></a> <a id="b:180" class="idref" href="#b:180"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:178"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:179"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:180"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:178"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:180"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:180"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:178"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_cons_iff" class="idref" href="#nf_cons_iff"><span class="id" title="lemma">nf_cons_iff</span></a> <a id="a:181" class="idref" href="#a:181"><span class="id" title="binder">a</span></a> <a id="n:182" class="idref" href="#n:182"><span class="id" title="binder">n</span></a> <a id="b:183" class="idref" href="#b:183"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:181"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:182"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:183"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:181"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:183"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:183"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:181"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+already in Stdlib ? 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bool_eq_iff" class="idref" href="#bool_eq_iff"><span class="id" title="lemma">bool_eq_iff</span></a> (<a id="b:184" class="idref" href="#b:184"><span class="id" title="binder">b</span></a> <a id="b':185" class="idref" href="#b':185"><span class="id" title="binder">b'</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#b:184"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':185"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#b:184"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':185"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_b_cons_eq" class="idref" href="#nf_b_cons_eq"><span class="id" title="lemma">nf_b_cons_eq</span></a> <a id="a:186" class="idref" href="#a:186"><span class="id" title="binder">a</span></a> <a id="n:187" class="idref" href="#n:187"><span class="id" title="binder">n</span></a> <a id="b:188" class="idref" href="#b:188"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf_b"><span class="id" title="definition">nf_b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:186"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:187"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:188"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf_b"><span class="id" title="definition">nf_b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:186"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf_b"><span class="id" title="definition">nf_b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:188"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:188"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:186"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">nf_decomp</span> <span class="id" title="var">H</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">nf0</span> := <span class="id" title="tactic">fresh</span> "nf"<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">nf1</span> := <span class="id" title="tactic">fresh</span> "nf"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">Hlp</span> := <span class="id" title="tactic">fresh</span> "lt"<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">H</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">t</span> ?<span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) ⇒ <span class="id" title="tactic">assert</span> (<span class="id" title="var">nf0</span>:= <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv1"><span class="id" title="lemma">nf_inv1</span></a> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">n</span> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">t2</span> ?<span class="id" title="var">p</span> ?<span class="id" title="var">t3</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⇒ <span class="id" title="tactic">assert</span> (<span class="id" title="var">nf0</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv1"><span class="id" title="lemma">nf_inv1</span></a> <span class="id" title="var">H</span>); <span class="id" title="tactic">assert</span>(<span class="id" title="var">nf1</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv2"><span class="id" title="lemma">nf_inv2</span></a> <span class="id" title="var">H</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">lt</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv3"><span class="id" title="lemma">nf_inv3</span></a> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">t2</span>) ⇒ <span class="id" title="tactic">assert</span> (<span class="id" title="var">nf0</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv1"><span class="id" title="lemma">nf_inv1</span></a> <span class="id" title="var">H</span>); <span class="id" title="tactic">assert</span>(<span class="id" title="var">nf1</span> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv2"><span class="id" title="lemma">nf_inv2</span></a> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ lt alpha (phi0 beta)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Inductive</span> <a id="lt_a_phi0_b" class="idref" href="#lt_a_phi0_b"><span class="id" title="inductive">lt_a_phi0_b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="lt_a_phi0_b_z" class="idref" href="#lt_a_phi0_b_z"><span class="id" title="constructor">lt_a_phi0_b_z</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:191" class="idref" href="#alpha:191"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt_a_phi0_b:189"><span class="id" title="inductive">lt_a_phi0_b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:191"><span class="id" title="variable">alpha</span></a><br/>
+| <a id="lt_a_phi0_b_c" class="idref" href="#lt_a_phi0_b_c"><span class="id" title="constructor">lt_a_phi0_b_c</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:192" class="idref" href="#alpha:192"><span class="id" title="binder">alpha</span></a> <a id="alpha':193" class="idref" href="#alpha':193"><span class="id" title="binder">alpha'</span></a> <a id="n':194" class="idref" href="#n':194"><span class="id" title="binder">n'</span></a> <a id="beta':195" class="idref" href="#beta':195"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':193"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:192"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt_a_phi0_b:189"><span class="id" title="inductive">lt_a_phi0_b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':193"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':194"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':195"><span class="id" title="variable">beta'</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#alpha:192"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt_a_phi0_b"><span class="id" title="inductive">lt_a_phi0_b</span></a> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x '&lt;_phi0' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'&lt;_phi0'_x" class="idref" href="#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&quot;</span></a>&lt;_phi0" := <a class="idref" href="hydras.Epsilon0.T1.html#lt_a_phi0_b"><span class="id" title="inductive">lt_a_phi0_b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="get_decomposition" class="idref" href="#get_decomposition"><span class="id" title="definition">get_decomposition</span></a> : <span class="id" title="keyword">∀</span> <a id="c:196" class="idref" href="#c:196"><span class="id" title="binder">c</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:196"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="a:197" class="idref" href="#a:197"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="n:198" class="idref" href="#n:198"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b:199" class="idref" href="#b:199"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:196"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:197"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:198"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:199"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">decomp_exhib</span> <span class="id" title="var">H</span> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="var">e</span>:= <br/>
+&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">Ha</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">Hn</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">Hb</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">H</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ?<span class="id" title="var">c</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Epsilon0.T1.html#get_decomposition"><span class="id" title="definition">get_decomposition</span></a>  <span class="id" title="var">H</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span> <span class="id" title="var">Ha</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">Ha</span>;<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span> <span class="id" title="var">Hn</span>; <span class="id" title="tactic">case</span> <span class="id" title="var">Hn</span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">b</span> <span class="id" title="var">e</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">Ha</span> <span class="id" title="var">Hn</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_FS" class="idref" href="#nf_FS"><span class="id" title="lemma">nf_FS</span></a> : <span class="id" title="keyword">∀</span> <a id="n:200" class="idref" href="#n:200"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:200"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_fin" class="idref" href="#nf_fin"><span class="id" title="lemma">nf_fin</span></a> : <span class="id" title="keyword">∀</span> <a id="n:201" class="idref" href="#n:201"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:201"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab55"></a><h2 class="section">Successors, limits and zero</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_not_zero" class="idref" href="#succ_not_zero"><span class="id" title="lemma">succ_not_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:202" class="idref" href="#alpha:202"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:202"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_is_succ" class="idref" href="#succ_is_succ"><span class="id" title="lemma">succ_is_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:203" class="idref" href="#alpha:203"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:203"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab56"></a><h2 class="section">Second part on <span class="inlinecode"><span class="id" title="var">lt</span></span> and <span class="inlinecode"><span class="id" title="var">le</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_lt" class="idref" href="#finite_lt"><span class="id" title="lemma">finite_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:204" class="idref" href="#n:204"><span class="id" title="binder">n</span></a> <a id="p:205" class="idref" href="#p:205"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="hydras.Epsilon0.T1.html#n:204"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:205"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:204"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:205"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_ltR" class="idref" href="#finite_ltR"><span class="id" title="lemma">finite_ltR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:206" class="idref" href="#n:206"><span class="id" title="binder">n</span></a> <a id="p:207" class="idref" href="#p:207"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:206"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:207"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:206"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:207"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_eq_lt_dec" class="idref" href="#le_eq_lt_dec"><span class="id" title="lemma">le_eq_lt_dec</span></a> <a id="alpha:208" class="idref" href="#alpha:208"><span class="id" title="binder">alpha</span></a> <a id="beta:209" class="idref" href="#beta:209"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:208"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:209"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:208"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:209"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:208"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:209"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ" class="idref" href="#lt_succ"><span class="id" title="lemma">lt_succ</span></a> (<a id="alpha:210" class="idref" href="#alpha:210"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:210"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:210"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_a" class="idref" href="#lt_a_phi0_a"><span class="id" title="lemma">lt_a_phi0_a</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:211" class="idref" href="#a:211"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:211"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:211"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_lt" class="idref" href="#phi0_lt"><span class="id" title="lemma">phi0_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:212" class="idref" href="#a:212"><span class="id" title="binder">a</span></a> <a id="b:213" class="idref" href="#b:213"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:212"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:213"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:212"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:213"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_ltR" class="idref" href="#phi0_ltR"><span class="id" title="lemma">phi0_ltR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:214" class="idref" href="#a:214"><span class="id" title="binder">a</span></a> <a id="b:215" class="idref" href="#b:215"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:214"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:215"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:214"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:215"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_of_finite" class="idref" href="#nf_of_finite"><span class="id" title="lemma">nf_of_finite</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:216" class="idref" href="#n:216"><span class="id" title="binder">n</span></a> <a id="b:217" class="idref" href="#b:217"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:216"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:217"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:217"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="zero_le" class="idref" href="#zero_le"><span class="id" title="lemma">zero_le</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:218" class="idref" href="#a:218"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:218"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_inv" class="idref" href="#le_inv"><span class="id" title="lemma">le_inv</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:219" class="idref" href="#a:219"><span class="id" title="binder">a</span></a> <a id="n:220" class="idref" href="#n:220"><span class="id" title="binder">n</span></a> <a id="b:221" class="idref" href="#b:221"><span class="id" title="binder">b</span></a> <a id="a':222" class="idref" href="#a':222"><span class="id" title="binder">a'</span></a> <a id="n':223" class="idref" href="#n':223"><span class="id" title="binder">n'</span></a> <a id="b':224" class="idref" href="#b':224"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:219"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:220"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:221"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':222"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':223"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':224"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:219"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':222"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:219"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':222"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:220"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':223"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:219"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':222"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:220"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':223"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:221"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b':224"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Epsilon0.T1.html#le_inv"><span class="id" title="lemma">le_inv</span></a> [<span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>] <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_inv" class="idref" href="#lt_a_phi0_b_inv"><span class="id" title="lemma">lt_a_phi0_b_inv</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:225" class="idref" href="#a:225"><span class="id" title="binder">a</span></a> <a id="n:226" class="idref" href="#n:226"><span class="id" title="binder">n</span></a> <a id="b:227" class="idref" href="#b:227"><span class="id" title="binder">b</span></a> <a id="a':228" class="idref" href="#a':228"><span class="id" title="binder">a'</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:225"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:226"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:227"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':228"><span class="id" title="variable">a'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:225"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':228"><span class="id" title="variable">a'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_zero_inv" class="idref" href="#le_zero_inv"><span class="id" title="lemma">le_zero_inv</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:229" class="idref" href="#a:229"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:229"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:229"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_tail" class="idref" href="#le_tail"><span class="id" title="lemma">le_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:230" class="idref" href="#a:230"><span class="id" title="binder">a</span></a> <a id="n:231" class="idref" href="#n:231"><span class="id" title="binder">n</span></a> <a id="b:232" class="idref" href="#b:232"><span class="id" title="binder">b</span></a> <a id="b':233" class="idref" href="#b':233"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:232"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':233"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:230"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:231"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:232"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:230"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:231"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':233"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_le</span> <span class="id" title="var">le_tail</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_phi0" class="idref" href="#le_phi0"><span class="id" title="lemma">le_phi0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:234" class="idref" href="#a:234"><span class="id" title="binder">a</span></a> <a id="n:235" class="idref" href="#n:235"><span class="id" title="binder">n</span></a> <a id="b:236" class="idref" href="#b:236"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:234"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:234"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:235"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:236"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="head_lt_cons" class="idref" href="#head_lt_cons"><span class="id" title="lemma">head_lt_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:237" class="idref" href="#a:237"><span class="id" title="binder">a</span></a> <a id="n:238" class="idref" href="#n:238"><span class="id" title="binder">n</span></a> <a id="b:239" class="idref" href="#b:239"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:237"><span class="id" title="variable">a</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:237"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:238"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:239"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">head_lt_cons</span>: <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1_eq_dec" class="idref" href="#T1_eq_dec"><span class="id" title="definition">T1_eq_dec</span></a>  (<a id="alpha:240" class="idref" href="#alpha:240"><span class="id" title="binder">alpha</span></a> <a id="beta:241" class="idref" href="#beta:241"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:240"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:241"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:240"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:241"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:242" class="idref" href="#alpha:242"><span class="id" title="binder">alpha</span></a> <a id="beta:243" class="idref" href="#beta:243"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:242"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:243"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:242"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:243"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:243"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:242"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt_le_dec" class="idref" href="#lt_le_dec"><span class="id" title="definition">lt_le_dec</span></a> (<a id="alpha:244" class="idref" href="#alpha:244"><span class="id" title="binder">alpha</span></a> <a id="beta:245" class="idref" href="#beta:245"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:244"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:245"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:245"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:244"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="epsilon0_pre_order" class="idref" href="#epsilon0_pre_order"><span class="id" title="instance">epsilon0_pre_order</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="epsilon0_dec" class="idref" href="#epsilon0_dec"><span class="id" title="instance">epsilon0_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">auto_nf</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> ?<span class="id" title="var">alpha</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;( <span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#cons_nf"><span class="id" title="lemma">cons_nf</span></a> || <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#single_nf"><span class="id" title="lemma">single_nf</span></a> || <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero_nf"><span class="id" title="lemma">zero_nf</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|| (<span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv1"><span class="id" title="lemma">nf_inv1</span></a> || <span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Epsilon0.T1.html#nf_inv2"><span class="id" title="lemma">nf_inv2</span></a>); <span class="id" title="tactic">eauto</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_tail_lt_nf" class="idref" href="#nf_tail_lt_nf"><span class="id" title="lemma">nf_tail_lt_nf</span></a> <a id="b:246" class="idref" href="#b:246"><span class="id" title="binder">b</span></a> <a id="b':247" class="idref" href="#b':247"><span class="id" title="binder">b'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':247"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:246"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':247"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:248" class="idref" href="#a:248"><span class="id" title="binder">a</span></a> <a id="n:249" class="idref" href="#n:249"><span class="id" title="binder">n</span></a>,   <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:248"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:249"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:246"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:248"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:249"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':247"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="tail_lt_cons" class="idref" href="#tail_lt_cons"><span class="id" title="lemma">tail_lt_cons</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:250" class="idref" href="#b:250"><span class="id" title="binder">b</span></a> <a id="n:251" class="idref" href="#n:251"><span class="id" title="binder">n</span></a> <a id="a:252" class="idref" href="#a:252"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:252"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:251"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:250"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:250"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:252"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:251"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:250"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_inv1" class="idref" href="#lt_a_phi0_b_inv1"><span class="id" title="lemma">lt_a_phi0_b_inv1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:253" class="idref" href="#a:253"><span class="id" title="binder">a</span></a> <a id="n:254" class="idref" href="#n:254"><span class="id" title="binder">n</span></a> <a id="b:255" class="idref" href="#b:255"><span class="id" title="binder">b</span></a> <a id="a':256" class="idref" href="#a':256"><span class="id" title="binder">a'</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:253"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:254"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:255"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':256"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:253"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':256"><span class="id" title="variable">a'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_intro" class="idref" href="#nf_intro"><span class="id" title="lemma">nf_intro</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:257" class="idref" href="#a:257"><span class="id" title="binder">a</span></a> <a id="n:258" class="idref" href="#n:258"><span class="id" title="binder">n</span></a> <a id="b:259" class="idref" href="#b:259"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:257"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:259"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:259"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:257"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:257"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:258"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:259"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_intro'" class="idref" href="#nf_intro'"><span class="id" title="lemma">nf_intro'</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:260" class="idref" href="#a:260"><span class="id" title="binder">a</span></a> <a id="n:261" class="idref" href="#n:261"><span class="id" title="binder">n</span></a> <a id="b:262" class="idref" href="#b:262"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:260"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:262"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:262"><span class="id" title="variable">b</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:260"><span class="id" title="variable">a</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:260"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:261"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:262"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_intro" class="idref" href="#lt_a_phi0_b_intro"><span class="id" title="lemma">lt_a_phi0_b_intro</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:263" class="idref" href="#a:263"><span class="id" title="binder">a</span></a> <a id="n:264" class="idref" href="#n:264"><span class="id" title="binder">n</span></a> <a id="b:265" class="idref" href="#b:265"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:263"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:264"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:265"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:265"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:263"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_coeff_irrelevance" class="idref" href="#nf_coeff_irrelevance"><span class="id" title="lemma">nf_coeff_irrelevance</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:266" class="idref" href="#a:266"><span class="id" title="binder">a</span></a> <a id="b:267" class="idref" href="#b:267"><span class="id" title="binder">b</span></a> <a id="n:268" class="idref" href="#n:268"><span class="id" title="binder">n</span></a> <a id="p:269" class="idref" href="#p:269"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:266"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:268"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:267"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:266"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:269"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:267"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_phi0" class="idref" href="#lt_a_phi0_b_phi0"><span class="id" title="lemma">lt_a_phi0_b_phi0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:270" class="idref" href="#a:270"><span class="id" title="binder">a</span></a> <a id="b:271" class="idref" href="#b:271"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#b:271"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:270"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:271"><span class="id" title="variable">b</span></a> ( <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:270"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_phi0R" class="idref" href="#lt_a_phi0_b_phi0R"><span class="id" title="lemma">lt_a_phi0_b_phi0R</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:272" class="idref" href="#a:272"><span class="id" title="binder">a</span></a> <a id="b:273" class="idref" href="#b:273"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:273"><span class="id" title="variable">b</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:272"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:273"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:272"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_def" class="idref" href="#lt_a_phi0_b_def"><span class="id" title="lemma">lt_a_phi0_b_def</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:274" class="idref" href="#a:274"><span class="id" title="binder">a</span></a> <a id="b:275" class="idref" href="#b:275"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#b:275"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:274"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:275"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:274"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi0_b_iff" class="idref" href="#lt_a_phi0_b_iff"><span class="id" title="lemma">lt_a_phi0_b_iff</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:276" class="idref" href="#a:276"><span class="id" title="binder">a</span></a> <a id="b:277" class="idref" href="#b:277"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:276"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:277"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#b:277"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x"><span class="id" title="notation">_phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:276"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <span class="id" title="keyword">∀</span> <a id="n:278" class="idref" href="#n:278"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:276"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:278"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:277"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_omega_tower" class="idref" href="#nf_omega_tower"><span class="id" title="lemma">nf_omega_tower</span></a> : <span class="id" title="keyword">∀</span> <a id="n:279" class="idref" href="#n:279"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:279"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+A strong induction scheme for nf 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="nf_rect" class="idref" href="#nf_rect"><span class="id" title="definition">nf_rect</span></a> : <span class="id" title="keyword">∀</span> <a id="P:280" class="idref" href="#P:280"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:281" class="idref" href="#n:281"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:281"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:282" class="idref" href="#a:282"><span class="id" title="binder">a</span></a> <a id="n:283" class="idref" href="#n:283"><span class="id" title="binder">n</span></a> <a id="b:284" class="idref" href="#b:284"><span class="id" title="binder">b</span></a> <a id="n':285" class="idref" href="#n':285"><span class="id" title="binder">n'</span></a> <a id="b':286" class="idref" href="#b':286"><span class="id" title="binder">b'</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:282"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:283"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:284"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:282"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:283"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:284"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':286"><span class="id" title="variable">b'</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:282"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:283"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:284"><span class="id" title="variable">b</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':286"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':286"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:282"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:283"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:284"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#n':285"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':286"><span class="id" title="variable">b'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:287" class="idref" href="#a:287"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:287"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#P:280"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:287"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab57"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">compare</span></span>: second part</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="compare_reflectR" class="idref" href="#compare_reflectR"><span class="id" title="lemma">compare_reflectR</span></a> ( <a id="alpha:288" class="idref" href="#alpha:288"><span class="id" title="binder">alpha</span></a> <a id="beta:289" class="idref" href="#beta:289"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:288"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:289"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inleft"><span class="id" title="constructor">inleft</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inleft"><span class="id" title="constructor">inleft</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inright"><span class="id" title="constructor">inright</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:288"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:289"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab58"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">max</span></span></h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="max_nf" class="idref" href="#max_nf"><span class="id" title="lemma">max_nf</span></a> (<a id="alpha:290" class="idref" href="#alpha:290"><span class="id" title="binder">alpha</span></a> <a id="beta:291" class="idref" href="#beta:291"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:290"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:291"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:290"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:291"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab59"></a><h2 class="section">Restriction of lt and le to terms in normal form</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&gt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&gt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;=' y 't1&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;=' y 't1&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;' y 't1&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't1&lt;' y 't1&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LT" class="idref" href="#LT"><span class="id" title="definition">LT</span></a> := <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="::t1_scope:x_'t1&lt;'_x" class="idref" href="#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&quot;</span></a>t1&lt;" := <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="LE" class="idref" href="#LE"><span class="id" title="definition">LE</span></a> := <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>).<br/>
+<span class="id" title="keyword">Infix</span> <a id="f097afe6f86551e3dd19af72feb179da" class="idref" href="#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">&quot;</span></a>t1&lt;=" := <a class="idref" href="hydras.Epsilon0.T1.html#LE"><span class="id" title="definition">LE</span></a> : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x" class="idref" href="#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&quot;</span></a>alpha t1&lt; beta t1&lt; gamma" :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">gamma</span>) : <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Elements" class="idref" href="#Elements"><span class="id" title="definition">Elements</span></a> <a id="alpha:292" class="idref" href="#alpha:292"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="beta:293" class="idref" href="#beta:293"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#beta:293"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:292"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Epsilon0.T1.html#Elements"><span class="id" title="definition">Elements</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Elements"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Elements"><span class="id" title="definition">T1</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Elements"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Elements"><span class="id" title="definition">Ensemble</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_nf_l" class="idref" href="#LT_nf_l"><span class="id" title="lemma">LT_nf_l</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:294" class="idref" href="#alpha:294"><span class="id" title="binder">alpha</span></a> <a id="beta:295" class="idref" href="#beta:295"><span class="id" title="binder">beta</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#alpha:294"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:295"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:294"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_nf_r" class="idref" href="#LT_nf_r"><span class="id" title="lemma">LT_nf_r</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:296" class="idref" href="#alpha:296"><span class="id" title="binder">alpha</span></a> <a id="beta:297" class="idref" href="#beta:297"><span class="id" title="binder">beta</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#alpha:296"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:297"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:297"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_lt" class="idref" href="#LT_lt"><span class="id" title="lemma">LT_lt</span></a> <a id="alpha:298" class="idref" href="#alpha:298"><span class="id" title="binder">alpha</span></a> <a id="beta:299" class="idref" href="#beta:299"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#alpha:298"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:299"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:298"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:299"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_nf_l" class="idref" href="#LE_nf_l"><span class="id" title="lemma">LE_nf_l</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:300" class="idref" href="#alpha:300"><span class="id" title="binder">alpha</span></a> <a id="beta:301" class="idref" href="#beta:301"><span class="id" title="binder">beta</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#alpha:300"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:301"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:300"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_nf_r" class="idref" href="#LE_nf_r"><span class="id" title="lemma">LE_nf_r</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:302" class="idref" href="#alpha:302"><span class="id" title="binder">alpha</span></a> <a id="beta:303" class="idref" href="#beta:303"><span class="id" title="binder">beta</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#alpha:302"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:303"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:303"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_le" class="idref" href="#LE_le"><span class="id" title="lemma">LE_le</span></a> <a id="alpha:304" class="idref" href="#alpha:304"><span class="id" title="binder">alpha</span></a> <a id="beta:305" class="idref" href="#beta:305"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#alpha:304"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:305"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:304"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:305"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">LT_nf_r</span> <span class="id" title="var">LT_nf_l</span> <span class="id" title="var">LT_lt</span> <span class="id" title="var">LE_nf_r</span> <span class="id" title="var">LE_nf_l</span> <span class="id" title="var">LE_le</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_zero_gt_0" class="idref" href="#not_zero_gt_0"><span class="id" title="lemma">not_zero_gt_0</span></a> (<a id="alpha:306" class="idref" href="#alpha:306"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#alpha:306"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:306"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_zero_lt" class="idref" href="#not_zero_lt"><span class="id" title="lemma">not_zero_lt</span></a> (<a id="alpha:307" class="idref" href="#alpha:307"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:307"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:307"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_zero" class="idref" href="#LE_zero"><span class="id" title="lemma">LE_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:308" class="idref" href="#alpha:308"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:308"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:308"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_refl" class="idref" href="#LE_refl"><span class="id" title="lemma">LE_refl</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:309" class="idref" href="#alpha:309"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:309"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:309"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:309"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_trans" class="idref" href="#LT_trans"><span class="id" title="lemma">LT_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:310" class="idref" href="#a:310"><span class="id" title="binder">a</span></a> <a id="b:311" class="idref" href="#b:311"><span class="id" title="binder">b</span></a> <a id="c:312" class="idref" href="#c:312"><span class="id" title="binder">c</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#a:310"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:311"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:311"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:312"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:310"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:312"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="LE_trans" class="idref" href="#LE_trans"><span class="id" title="lemma">LE_trans</span></a> (<a id="alpha:313" class="idref" href="#alpha:313"><span class="id" title="binder">alpha</span></a> <a id="beta:314" class="idref" href="#beta:314"><span class="id" title="binder">beta</span></a> <a id="gamma:315" class="idref" href="#gamma:315"><span class="id" title="binder">gamma</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:313"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:314"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:314"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#gamma:315"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:313"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:315"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_antisym" class="idref" href="#LE_antisym"><span class="id" title="lemma">LE_antisym</span></a> (<a id="alpha:316" class="idref" href="#alpha:316"><span class="id" title="binder">alpha</span></a> <a id="beta:317" class="idref" href="#beta:317"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:316"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:317"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:317"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:316"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:316"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:317"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT1" class="idref" href="#LT1"><span class="id" title="lemma">LT1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:318" class="idref" href="#alpha:318"><span class="id" title="binder">alpha</span></a> <a id="n:319" class="idref" href="#n:319"><span class="id" title="binder">n</span></a> <a id="beta:320" class="idref" href="#beta:320"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:318"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:319"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:320"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:318"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:319"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:320"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT2" class="idref" href="#LT2"><span class="id" title="lemma">LT2</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:321" class="idref" href="#alpha:321"><span class="id" title="binder">alpha</span></a> <a id="alpha':322" class="idref" href="#alpha':322"><span class="id" title="binder">alpha'</span></a> <a id="n:323" class="idref" href="#n:323"><span class="id" title="binder">n</span></a> <a id="n':324" class="idref" href="#n':324"><span class="id" title="binder">n'</span></a> <a id="beta:325" class="idref" href="#beta:325"><span class="id" title="binder">beta</span></a> <a id="beta':326" class="idref" href="#beta':326"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:321"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:323"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:325"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':322"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':324"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':326"><span class="id" title="variable">beta'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:321"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':322"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:321"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:323"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:325"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha':322"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':324"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':326"><span class="id" title="variable">beta'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT3" class="idref" href="#LT3"><span class="id" title="lemma">LT3</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:327" class="idref" href="#alpha:327"><span class="id" title="binder">alpha</span></a>  <a id="n:328" class="idref" href="#n:328"><span class="id" title="binder">n</span></a> <a id="n':329" class="idref" href="#n':329"><span class="id" title="binder">n'</span></a> <a id="beta:330" class="idref" href="#beta:330"><span class="id" title="binder">beta</span></a> <a id="beta':331" class="idref" href="#beta':331"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:327"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:328"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:330"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:327"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':329"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':331"><span class="id" title="variable">beta'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#n:328"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':329"><span class="id" title="variable">n'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:327"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:328"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:330"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:327"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':329"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':331"><span class="id" title="variable">beta'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT4" class="idref" href="#LT4"><span class="id" title="lemma">LT4</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:332" class="idref" href="#alpha:332"><span class="id" title="binder">alpha</span></a>  <a id="n:333" class="idref" href="#n:333"><span class="id" title="binder">n</span></a>  <a id="beta:334" class="idref" href="#beta:334"><span class="id" title="binder">beta</span></a> <a id="beta':335" class="idref" href="#beta':335"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:332"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:333"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:334"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:332"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:333"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':335"><span class="id" title="variable">beta'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:334"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':335"><span class="id" title="variable">beta'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:332"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:333"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:334"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:332"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:333"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':335"><span class="id" title="variable">beta'</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">LT1</span> <span class="id" title="var">LT2</span> <span class="id" title="var">LT3</span> <span class="id" title="var">LT4</span>: <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_irrefl" class="idref" href="#LT_irrefl"><span class="id" title="lemma">LT_irrefl</span></a> (<a id="alpha:336" class="idref" href="#alpha:336"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:336"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:336"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_LT_trans" class="idref" href="#LE_LT_trans"><span class="id" title="lemma">LE_LT_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:337" class="idref" href="#alpha:337"><span class="id" title="binder">alpha</span></a> <a id="beta:338" class="idref" href="#beta:338"><span class="id" title="binder">beta</span></a> <a id="gamma:339" class="idref" href="#gamma:339"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:337"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:338"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:338"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:339"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:337"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:339"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_LE_trans" class="idref" href="#LT_LE_trans"><span class="id" title="lemma">LT_LE_trans</span></a> (<a id="alpha:340" class="idref" href="#alpha:340"><span class="id" title="binder">alpha</span></a> <a id="beta:341" class="idref" href="#beta:341"><span class="id" title="binder">beta</span></a> <a id="gamma:342" class="idref" href="#gamma:342"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#alpha:340"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:341"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:341"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:342"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:340"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:342"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_LT_zero" class="idref" href="#not_LT_zero"><span class="id" title="lemma">not_LT_zero</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:343" class="idref" href="#alpha:343"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:343"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="LT_St" class="idref" href="#LT_St"><span class="id" title="instance">LT_St</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_cons_LT" class="idref" href="#nf_cons_LT"><span class="id" title="lemma">nf_cons_LT</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:344" class="idref" href="#a:344"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n:345" class="idref" href="#n:345"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a':346" class="idref" href="#a':346"><span class="id" title="binder">a'</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n':347" class="idref" href="#n':347"><span class="id" title="binder">n'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="b:348" class="idref" href="#b:348"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a':346"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:344"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:344"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':346"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':347"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:348"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:344"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:345"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':346"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':347"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:348"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_cons_LT</span>: <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_inv1</span> <span class="id" title="var">nf_inv2</span> <span class="id" title="var">nf_inv3</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="head_LT_cons" class="idref" href="#head_LT_cons"><span class="id" title="lemma">head_LT_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:349" class="idref" href="#alpha:349"><span class="id" title="binder">alpha</span></a> <a id="n:350" class="idref" href="#n:350"><span class="id" title="binder">n</span></a> <a id="beta:351" class="idref" href="#beta:351"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:349"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:350"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:351"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:349"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:349"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:350"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:351"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="tail_LT_cons" class="idref" href="#tail_LT_cons"><span class="id" title="lemma">tail_LT_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:352" class="idref" href="#alpha:352"><span class="id" title="binder">alpha</span></a> <a id="n:353" class="idref" href="#n:353"><span class="id" title="binder">n</span></a> <a id="beta:354" class="idref" href="#beta:354"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:352"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:353"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:354"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:354"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:352"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:353"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:354"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="LT_inv" class="idref" href="#LT_inv"><span class="id" title="lemma">LT_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="a:355" class="idref" href="#a:355"><span class="id" title="binder">a</span></a> <a id="n:356" class="idref" href="#n:356"><span class="id" title="binder">n</span></a> <a id="b:357" class="idref" href="#b:357"><span class="id" title="binder">b</span></a> <a id="a':358" class="idref" href="#a':358"><span class="id" title="binder">a'</span></a> <a id="n':359" class="idref" href="#n':359"><span class="id" title="binder">n'</span></a> <a id="b':360" class="idref" href="#b':360"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:356"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:357"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':358"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':359"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':360"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':358"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:355"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':358"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#n:356"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':359"><span class="id" title="variable">n'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:356"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':359"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>   <a class="idref" href="hydras.Epsilon0.T1.html#b:357"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':360"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="LT_cases" class="idref" href="#LT_cases"><span class="id" title="inductive">LT_cases</span></a> (<a id="a:361" class="idref" href="#a:361"><span class="id" title="binder">a</span></a>  <a id="b:362" class="idref" href="#b:362"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n:363" class="idref" href="#n:363"><span class="id" title="binder">n</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a':364" class="idref" href="#a':364"><span class="id" title="binder">a'</span></a> <a id="b':365" class="idref" href="#b':365"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="n':366" class="idref" href="#n':366"><span class="id" title="binder">n'</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Type</span> :=<br/>
+| <a id="LT_left" class="idref" href="#LT_left"><span class="id" title="constructor">LT_left</span></a> (<a id="H:369" class="idref" href="#H:369"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:361"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':364"><span class="id" title="variable">a'</span></a>)<br/>
+| <a id="LT_middle" class="idref" href="#LT_middle"><span class="id" title="constructor">LT_middle</span></a> (<a id="H:370" class="idref" href="#H:370"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:361"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':364"><span class="id" title="variable">a'</span></a>)(<a id="H1:371" class="idref" href="#H1:371"><span class="id" title="binder">H1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#n:363"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':366"><span class="id" title="variable">n'</span></a>)<br/>
+| <a id="LT_right" class="idref" href="#LT_right"><span class="id" title="constructor">LT_right</span></a> (<a id="H:372" class="idref" href="#H:372"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:361"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':364"><span class="id" title="variable">a'</span></a>)(<a id="H1:373" class="idref" href="#H1:373"><span class="id" title="binder">H1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#n:363"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':366"><span class="id" title="variable">n'</span></a>)(<a id="H2:374" class="idref" href="#H2:374"><span class="id" title="binder">H2</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#b:362"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':365"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="LT_inv_strong" class="idref" href="#LT_inv_strong"><span class="id" title="lemma">LT_inv_strong</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:375" class="idref" href="#a:375"><span class="id" title="binder">a</span></a>  <a id="b:376" class="idref" href="#b:376"><span class="id" title="binder">b</span></a> <a id="n:377" class="idref" href="#n:377"><span class="id" title="binder">n</span></a> <a id="a':378" class="idref" href="#a':378"><span class="id" title="binder">a'</span></a>  <a id="b':379" class="idref" href="#b':379"><span class="id" title="binder">b'</span></a> <a id="n':380" class="idref" href="#n':380"><span class="id" title="binder">n'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:375"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:377"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:376"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':378"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':380"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':379"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT_cases"><span class="id" title="inductive">LT_cases</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:375"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:376"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:377"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':378"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':379"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':380"><span class="id" title="variable">n'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="remove_first_sumand" class="idref" href="#remove_first_sumand"><span class="id" title="lemma">remove_first_sumand</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:381" class="idref" href="#a:381"><span class="id" title="binder">a</span></a> <a id="n:382" class="idref" href="#n:382"><span class="id" title="binder">n</span></a> <a id="b:383" class="idref" href="#b:383"><span class="id" title="binder">b</span></a>  <a id="b':384" class="idref" href="#b':384"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:381"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:382"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:383"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:381"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:382"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':384"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:383"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':384"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_cons_0" class="idref" href="#LT_cons_0"><span class="id" title="lemma">LT_cons_0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:385" class="idref" href="#a:385"><span class="id" title="binder">a</span></a> <a id="n:386" class="idref" href="#n:386"><span class="id" title="binder">n</span></a> <a id="b:387" class="idref" href="#b:387"><span class="id" title="binder">b</span></a> <a id="a':388" class="idref" href="#a':388"><span class="id" title="binder">a'</span></a> <a id="b':389" class="idref" href="#b':389"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:385"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:386"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:387"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':388"><span class="id" title="variable">a'</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#b':389"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:385"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':388"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:386"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:385"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':388"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:387"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':389"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_phi0" class="idref" href="#LE_phi0"><span class="id" title="lemma">LE_phi0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:390" class="idref" href="#a:390"><span class="id" title="binder">a</span></a> <a id="n:391" class="idref" href="#n:391"><span class="id" title="binder">n</span></a> <a id="b:392" class="idref" href="#b:392"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:390"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:391"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:392"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:390"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:390"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:391"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:392"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_tail_lt" class="idref" href="#nf_tail_lt"><span class="id" title="lemma">nf_tail_lt</span></a> <a id="alpha:393" class="idref" href="#alpha:393"><span class="id" title="binder">alpha</span></a> <a id="n:394" class="idref" href="#n:394"><span class="id" title="binder">n</span></a> <a id="beta:395" class="idref" href="#beta:395"><span class="id" title="binder">beta</span></a> <a id="gamma:396" class="idref" href="#gamma:396"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:393"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:395"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#gamma:396"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:395"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:393"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:394"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:396"><span class="id" title="variable">gamma</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab60"></a><h2 class="section">Well foundedness of <span class="inlinecode"><span class="id" title="var">LT</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Direct_proof" class="idref" href="#Direct_proof"><span class="id" title="module">Direct_proof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Direct_proof.well_foundedness_proof" class="idref" href="#Direct_proof.well_foundedness_proof"><span class="id" title="section">well_foundedness_proof</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a>: <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Direct_proof.Acc_zero" class="idref" href="#Direct_proof.Acc_zero"><span class="id" title="lemma">Acc_zero</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Direct_proof.well_foundedness_proof.First_attempt" class="idref" href="#Direct_proof.well_foundedness_proof.First_attempt"><span class="id" title="section">First_attempt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Direct_proof.wf_LT" class="idref" href="#Direct_proof.wf_LT"><span class="id" title="lemma">wf_LT</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:397" class="idref" href="#alpha:397"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:397"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:397"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.First_attempt"><span class="id" title="section">First_attempt</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab61"></a><h3 class="section">Strong accessibility (inspired by Tait's proof)</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Direct_proof.well_foundedness_proof.Acc_strong" class="idref" href="#Direct_proof.well_foundedness_proof.Acc_strong"><span class="id" title="variable">Acc_strong</span></a> (<a id="alpha:398" class="idref" href="#alpha:398"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:399" class="idref" href="#n:399"><span class="id" title="binder">n</span></a> <a id="beta:400" class="idref" href="#beta:400"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:398"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:399"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:400"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:398"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#n:399"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:400"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Direct_proof.acc_impl" class="idref" href="#Direct_proof.acc_impl"><span class="id" title="lemma">acc_impl</span></a> {<a id="A:401" class="idref" href="#A:401"><span class="id" title="binder">A</span></a>} {<a id="R:402" class="idref" href="#R:402"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#A:401"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#A:401"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>} (<a id="a:403" class="idref" href="#a:403"><span class="id" title="binder">a</span></a> <a id="b:404" class="idref" href="#b:404"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#A:401"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#R:402"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:403"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:404"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#R:402"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:404"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#R:402"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:403"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Direct_proof.Acc_strong_stronger" class="idref" href="#Direct_proof.Acc_strong_stronger"><span class="id" title="lemma">Acc_strong_stronger</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:405" class="idref" href="#alpha:405"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:405"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.Acc_strong"><span class="id" title="variable">Acc_strong</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:405"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:405"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Direct_proof.Acc_implies_Acc_strong" class="idref" href="#Direct_proof.Acc_implies_Acc_strong"><span class="id" title="lemma">Acc_implies_Acc_strong</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:406" class="idref" href="#alpha:406"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:406"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.Acc_strong"><span class="id" title="variable">Acc_strong</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:406"><span class="id" title="variable">alpha</span></a>.   
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab62"></a><h3 class="section">A (last) structural induction</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Direct_proof.nf_Acc" class="idref" href="#Direct_proof.nf_Acc"><span class="id" title="lemma">nf_Acc</span></a> (<a id="alpha:409" class="idref" href="#alpha:409"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:409"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:409"><span class="id" title="variable">alpha</span></a>.  
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof"><span class="id" title="section">well_foundedness_proof</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof"><span class="id" title="module">Direct_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="nf_Acc" class="idref" href="#nf_Acc"><span class="id" title="definition">nf_Acc</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#nf_Acc"><span class="id" title="lemma">Direct_proof.nf_Acc</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="nf_Wf" class="idref" href="#nf_Wf"><span class="id" title="lemma">nf_Wf</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="T1_wf" class="idref" href="#T1_wf"><span class="id" title="lemma">T1_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a>. 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transfinite_recursor_lt" class="idref" href="#transfinite_recursor_lt"><span class="id" title="definition">transfinite_recursor_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:410" class="idref" href="#P:410"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:411" class="idref" href="#x:411"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:412" class="idref" href="#y:412"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#x:411"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#y:412"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#y:412"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#x:411"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#P:410"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#y:412"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#P:410"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#x:411"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:413" class="idref" href="#alpha:413"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#P:410"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:413"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transfinite_recursor" class="idref" href="#transfinite_recursor"><span class="id" title="definition">transfinite_recursor</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded_induction_type"><span class="id" title="lemma">well_founded_induction_type</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Epsilon0.T1.html#transfinite_recursor"><span class="id" title="definition">transfinite_recursor</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#Direct_proof"><span class="id" title="module">Direct_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">transfinite_induction_lt</span> <span class="id" title="var">alpha</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">pattern</span> <span class="id" title="var">alpha</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#transfinite_recursor_lt"><span class="id" title="definition">transfinite_recursor_lt</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">transfinite_induction</span> <span class="id" title="var">alpha</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">pattern</span> <span class="id" title="var">alpha</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#transfinite_recursor"><span class="id" title="definition">transfinite_recursor</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab63"></a><h2 class="section">Properties of successor</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_nf" class="idref" href="#succ_nf"><span class="id" title="lemma">succ_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:418" class="idref" href="#alpha:418"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:418"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:418"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab64"></a><h2 class="section">properties of addition</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_zero" class="idref" href="#plus_zero"><span class="id" title="lemma">plus_zero</span></a> <a id="alpha:419" class="idref" href="#alpha:419"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:419"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:419"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_zero_r" class="idref" href="#plus_zero_r"><span class="id" title="lemma">plus_zero_r</span></a> <a id="alpha:420" class="idref" href="#alpha:420"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#alpha:420"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:420"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_is_plus_one" class="idref" href="#succ_is_plus_one"><span class="id" title="lemma">succ_is_plus_one</span></a> (<a id="a:421" class="idref" href="#a:421"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :  <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:421"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:421"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_of_plus_finite" class="idref" href="#succ_of_plus_finite"><span class="id" title="lemma">succ_of_plus_finite</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:422" class="idref" href="#a:422"><span class="id" title="binder">a</span></a> (<a id="H:423" class="idref" href="#H:423"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:422"><span class="id" title="variable">a</span></a>) (<a id="i:424" class="idref" href="#i:424"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) , <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:422"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:424"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:422"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:424"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab65"></a><h2 class="section"><span class="inlinecode"><span class="id" title="var">plus</span></span> and <span class="inlinecode"><span class="id" title="var">LT</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_cons_cons_rw1" class="idref" href="#plus_cons_cons_rw1"><span class="id" title="lemma">plus_cons_cons_rw1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:425" class="idref" href="#a:425"><span class="id" title="binder">a</span></a> <a id="n:426" class="idref" href="#n:426"><span class="id" title="binder">n</span></a> <a id="b:427" class="idref" href="#b:427"><span class="id" title="binder">b</span></a> <a id="a':428" class="idref" href="#a':428"><span class="id" title="binder">a'</span></a> <a id="n':429" class="idref" href="#n':429"><span class="id" title="binder">n'</span></a> <a id="b':430" class="idref" href="#b':430"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:425"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a':428"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:425"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:426"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:427"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':428"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':429"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':430"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':428"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':429"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':430"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_cons_cons_rw2" class="idref" href="#plus_cons_cons_rw2"><span class="id" title="lemma">plus_cons_cons_rw2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:431" class="idref" href="#a:431"><span class="id" title="binder">a</span></a> <a id="n:432" class="idref" href="#n:432"><span class="id" title="binder">n</span></a> <a id="b:433" class="idref" href="#b:433"><span class="id" title="binder">b</span></a> <a id="n':434" class="idref" href="#n':434"><span class="id" title="binder">n'</span></a> <a id="b':435" class="idref" href="#b':435"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:431"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:432"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:433"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:431"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':434"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':435"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:431"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:432"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:433"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:431"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':434"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':435"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:431"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:432"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':434"><span class="id" title="variable">n'</span></a>)) <a class="idref" href="hydras.Epsilon0.T1.html#b':435"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_cons_cons_rw3" class="idref" href="#plus_cons_cons_rw3"><span class="id" title="lemma">plus_cons_cons_rw3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:436" class="idref" href="#a:436"><span class="id" title="binder">a</span></a> <a id="n:437" class="idref" href="#n:437"><span class="id" title="binder">n</span></a> <a id="b:438" class="idref" href="#b:438"><span class="id" title="binder">b</span></a> <a id="a':439" class="idref" href="#a':439"><span class="id" title="binder">a'</span></a> <a id="n':440" class="idref" href="#n':440"><span class="id" title="binder">n'</span></a> <a id="b':441" class="idref" href="#b':441"><span class="id" title="binder">b'</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':439"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:436"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:436"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:437"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:438"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':439"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':440"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':441"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:436"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:437"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:438"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':439"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':440"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':441"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:436"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:437"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#b:438"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':439"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':440"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':441"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab66"></a><h2 class="section">On additive principal ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ap_plus" class="idref" href="#ap_plus"><span class="id" title="lemma">ap_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="a:442" class="idref" href="#a:442"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ap"><span class="id" title="inductive">ap</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:442"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:443" class="idref" href="#b:443"><span class="id" title="binder">b</span></a> <a id="c:444" class="idref" href="#c:444"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:443"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:444"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:443"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:442"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:444"><span class="id" title="variable">c</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:442"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#b:443"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:444"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#a:442"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ap_plusR" class="idref" href="#ap_plusR"><span class="id" title="lemma">ap_plusR</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="c:445" class="idref" href="#c:445"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:446" class="idref" href="#a:446"><span class="id" title="binder">a</span></a> <a id="b:447" class="idref" href="#b:447"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:446"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:447"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:446"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:447"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:446"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:447"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ap"><span class="id" title="inductive">ap</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:445"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Technical lemma for proving <span class="inlinecode"><span class="id" title="var">plus_nf</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_nf0" class="idref" href="#plus_nf0"><span class="id" title="lemma">plus_nf0</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:448" class="idref" href="#a:448"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:448"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:449" class="idref" href="#b:449"><span class="id" title="binder">b</span></a> <a id="c:450" class="idref" href="#c:450"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:449"><span class="id" title="variable">b</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:448"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:450"><span class="id" title="variable">c</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:448"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:449"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:450"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#b:449"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:450"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_nf" class="idref" href="#plus_nf"><span class="id" title="lemma">plus_nf</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:451" class="idref" href="#a:451"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:451"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:452" class="idref" href="#b:452"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:452"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:451"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:452"><span class="id" title="variable">b</span></a>). 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_term_plus_rw" class="idref" href="#omega_term_plus_rw"><span class="id" title="lemma">omega_term_plus_rw</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:453" class="idref" href="#a:453"><span class="id" title="binder">a</span></a> <a id="n:454" class="idref" href="#n:454"><span class="id" title="binder">n</span></a> <a id="b:455" class="idref" href="#b:455"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:453"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:454"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:455"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:453"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:454"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:455"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:453"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:454"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:455"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_is_zero" class="idref" href="#plus_is_zero"><span class="id" title="lemma">plus_is_zero</span></a> <a id="alpha:456" class="idref" href="#alpha:456"><span class="id" title="binder">alpha</span></a> <a id="beta:457" class="idref" href="#beta:457"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:456"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:457"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:456"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:457"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:456"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:457"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1add_not_monotonous_l" class="idref" href="#T1add_not_monotonous_l"><span class="id" title="lemma">T1add_not_monotonous_l</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:458" class="idref" href="#a:458"><span class="id" title="binder">a</span></a> <a id="b:459" class="idref" href="#b:459"><span class="id" title="binder">b</span></a> <a id="c:460" class="idref" href="#c:460"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:458"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:459"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:458"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:460"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:459"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:460"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1mul_not_monotonous" class="idref" href="#T1mul_not_monotonous"><span class="id" title="lemma">T1mul_not_monotonous</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:461" class="idref" href="#a:461"><span class="id" title="binder">a</span></a> <a id="b:462" class="idref" href="#b:462"><span class="id" title="binder">b</span></a> <a id="c:463" class="idref" href="#c:463"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:463"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:461"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:462"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:461"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:463"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:462"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:463"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab67"></a><h2 class="section">monotonicity of <span class="inlinecode"><span class="id" title="var">succ</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_strict_mono" class="idref" href="#succ_strict_mono"><span class="id" title="lemma">succ_strict_mono</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:464" class="idref" href="#a:464"><span class="id" title="binder">a</span></a> <a id="b:465" class="idref" href="#b:465"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:464"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:465"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:464"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:465"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:464"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:465"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_strict_mono_LT" class="idref" href="#succ_strict_mono_LT"><span class="id" title="lemma">succ_strict_mono_LT</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:466" class="idref" href="#alpha:466"><span class="id" title="binder">alpha</span></a> <a id="beta:467" class="idref" href="#beta:467"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:466"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:467"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:466"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:467"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_mono" class="idref" href="#succ_mono"><span class="id" title="lemma">succ_mono</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:468" class="idref" href="#a:468"><span class="id" title="binder">a</span></a> <a id="b:469" class="idref" href="#b:469"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:468"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:469"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:468"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:469"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:468"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:469"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le_R" class="idref" href="#lt_succ_le_R"><span class="id" title="lemma">lt_succ_le_R</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:470" class="idref" href="#a:470"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:470"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:471" class="idref" href="#b:471"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:471"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:470"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#b:471"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:470"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:471"><span class="id" title="variable">b</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_LT" class="idref" href="#le_lt_LT"><span class="id" title="lemma">le_lt_LT</span></a> <a id="alpha:472" class="idref" href="#alpha:472"><span class="id" title="binder">alpha</span></a> <a id="beta:473" class="idref" href="#beta:473"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:472"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:473"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:472"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:473"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:472"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:473"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_succ_LE_R" class="idref" href="#LT_succ_LE_R"><span class="id" title="lemma">LT_succ_LE_R</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:474" class="idref" href="#alpha:474"><span class="id" title="binder">alpha</span></a> <a id="beta:475" class="idref" href="#beta:475"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:474"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:474"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:475"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:474"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:475"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le_2" class="idref" href="#lt_succ_le_2"><span class="id" title="lemma">lt_succ_le_2</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:476" class="idref" href="#a:476"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:476"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:477" class="idref" href="#b:477"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:477"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:476"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:477"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:476"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:477"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+TODO: bulletize this proof ! 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le" class="idref" href="#lt_succ_le"><span class="id" title="lemma">lt_succ_le</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:478" class="idref" href="#a:478"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:478"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:479" class="idref" href="#b:479"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:479"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:478"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:479"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:478"><span class="id" title="variable">a</span></a>)  <a class="idref" href="hydras.Epsilon0.T1.html#b:479"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_succ_LE" class="idref" href="#LT_succ_LE"><span class="id" title="lemma">LT_succ_LE</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:480" class="idref" href="#alpha:480"><span class="id" title="binder">alpha</span></a> <a id="beta:481" class="idref" href="#beta:481"><span class="id" title="binder">beta</span></a> ,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:480"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:481"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:480"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:481"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_succ_LE_2" class="idref" href="#LT_succ_LE_2"><span class="id" title="lemma">LT_succ_LE_2</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:482" class="idref" href="#alpha:482"><span class="id" title="binder">alpha</span></a> <a id="beta:483" class="idref" href="#beta:483"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:483"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:482"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:483"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:482"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:483"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_strict_monoR" class="idref" href="#succ_strict_monoR"><span class="id" title="lemma">succ_strict_monoR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:484" class="idref" href="#alpha:484"><span class="id" title="binder">alpha</span></a> <a id="beta:485" class="idref" href="#beta:485"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:484"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:485"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:484"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:485"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:484"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:485"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_monomorphism" class="idref" href="#succ_monomorphism"><span class="id" title="lemma">succ_monomorphism</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:486" class="idref" href="#alpha:486"><span class="id" title="binder">alpha</span></a> (<a id="H:487" class="idref" href="#H:487"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:486"><span class="id" title="variable">alpha</span></a>) <a id="beta:488" class="idref" href="#beta:488"><span class="id" title="binder">beta</span></a> (<a id="H':489" class="idref" href="#H':489"><span class="id" title="binder">H'</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:488"><span class="id" title="variable">beta</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:486"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:488"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:486"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:488"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_injective" class="idref" href="#succ_injective"><span class="id" title="lemma">succ_injective</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:490" class="idref" href="#alpha:490"><span class="id" title="binder">alpha</span></a> (<a id="H:491" class="idref" href="#H:491"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:490"><span class="id" title="variable">alpha</span></a>) <a id="beta:492" class="idref" href="#beta:492"><span class="id" title="binder">beta</span></a> (<a id="H':493" class="idref" href="#H':493"><span class="id" title="binder">H'</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:492"><span class="id" title="variable">beta</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:490"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:492"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:490"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:492"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_compatS" class="idref" href="#succ_compatS"><span class="id" title="lemma">succ_compatS</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:494" class="idref" href="#n:494"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:494"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:494"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_compat" class="idref" href="#succ_compat"><span class="id" title="lemma">succ_compat</span></a> (<a id="n:495" class="idref" href="#n:495"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:495"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:495"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab68"></a><h2 class="section">monotonicity of <span class="inlinecode"><span class="id" title="var">phi0</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono_strict" class="idref" href="#phi0_mono_strict"><span class="id" title="lemma">phi0_mono_strict</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:496" class="idref" href="#a:496"><span class="id" title="binder">a</span></a> <a id="b:497" class="idref" href="#b:497"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:496"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:497"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:496"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:497"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono_strict_LT" class="idref" href="#phi0_mono_strict_LT"><span class="id" title="lemma">phi0_mono_strict_LT</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:498" class="idref" href="#alpha:498"><span class="id" title="binder">alpha</span></a> <a id="beta:499" class="idref" href="#beta:499"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:498"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:499"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:498"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:499"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono" class="idref" href="#phi0_mono"><span class="id" title="lemma">phi0_mono</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:500" class="idref" href="#a:500"><span class="id" title="binder">a</span></a> <a id="b:501" class="idref" href="#b:501"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:500"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:501"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:500"><span class="id" title="variable">a</span></a>) ( <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:501"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_left_absorb" class="idref" href="#plus_left_absorb"><span class="id" title="lemma">plus_left_absorb</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:502" class="idref" href="#a:502"><span class="id" title="binder">a</span></a> <a id="n:503" class="idref" href="#n:503"><span class="id" title="binder">n</span></a> <a id="b:504" class="idref" href="#b:504"><span class="id" title="binder">b</span></a> <a id="c:505" class="idref" href="#c:505"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:505"><span class="id" title="variable">c</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:502"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:505"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:502"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:503"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:504"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:502"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:503"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:504"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_compat" class="idref" href="#plus_compat"><span class="id" title="lemma">plus_compat</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:506" class="idref" href="#n:506"><span class="id" title="binder">n</span></a> <a id="p:507" class="idref" href="#p:507"><span class="id" title="binder">p</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:506"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:507"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:506"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:507"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab69"></a><h2 class="section">Multiplication</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_fin_omega" class="idref" href="#mult_fin_omega"><span class="id" title="lemma">mult_fin_omega</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:508" class="idref" href="#n:508"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:508"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_plus_mult" class="idref" href="#phi0_plus_mult"><span class="id" title="lemma">phi0_plus_mult</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:509" class="idref" href="#a:509"><span class="id" title="binder">a</span></a> <a id="b:510" class="idref" href="#b:510"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:509"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:510"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:509"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:510"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:509"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:510"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_compat" class="idref" href="#mult_compat"><span class="id" title="lemma">mult_compat</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:511" class="idref" href="#n:511"><span class="id" title="binder">n</span></a> <a id="p:512" class="idref" href="#p:512"><span class="id" title="binder">p</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:512"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:512"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:511"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:512"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_a_0" class="idref" href="#mult_a_0"><span class="id" title="lemma">mult_a_0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:513" class="idref" href="#a:513"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#a:513"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_1_a" class="idref" href="#mult_1_a"><span class="id" title="lemma">mult_1_a</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:514" class="idref" href="#a:514"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:514"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  1 <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:514"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:514"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_a_1" class="idref" href="#mult_a_1"><span class="id" title="lemma">mult_a_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:515" class="idref" href="#a:515"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:515"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:515"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 1  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:515"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_nf_fin" class="idref" href="#mult_nf_fin"><span class="id" title="lemma">mult_nf_fin</span></a> <a id="alpha:516" class="idref" href="#alpha:516"><span class="id" title="binder">alpha</span></a> <a id="n:517" class="idref" href="#n:517"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:516"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#alpha:516"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:517"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab70"></a><h2 class="section">About minus</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minus_lt" class="idref" href="#minus_lt"><span class="id" title="lemma">minus_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:518" class="idref" href="#a:518"><span class="id" title="binder">a</span></a> <a id="b:519" class="idref" href="#b:519"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:518"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#b:519"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:518"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:519"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minus_a_a" class="idref" href="#minus_a_a"><span class="id" title="lemma">minus_a_a</span></a> : <span class="id" title="keyword">∀</span> <a id="a:520" class="idref" href="#a:520"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#a:520"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:520"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minus_le" class="idref" href="#minus_le"><span class="id" title="lemma">minus_le</span></a> : <span class="id" title="keyword">∀</span> <a id="a:521" class="idref" href="#a:521"><span class="id" title="binder">a</span></a> <a id="b:522" class="idref" href="#b:522"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:521"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:522"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:521"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:522"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab71"></a><h2 class="section">Exponential</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp_fin_omega" class="idref" href="#exp_fin_omega"><span class="id" title="lemma">exp_fin_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="n:523" class="idref" href="#n:523"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:523"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab72"></a><h2 class="section">Relations between <span class="inlinecode"><span class="id" title="var">cons</span></span>, <span class="inlinecode"><span class="id" title="var">phi0</span></span> and <span class="inlinecode">+</span></h2>
+
+
+<div class="paragraph"> </div>
+
+ The next three lemmas express the consistency between
+     the intuitive meaning given to the constructor cons and
+     its derivates : phi0, omega-term, and the arithmetic 
+     operations on ordinals which belong to epsilon0 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_eq_bad" class="idref" href="#phi0_eq_bad"><span class="id" title="lemma">phi0_eq_bad</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:524" class="idref" href="#alpha:524"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:524"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:524"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_eq" class="idref" href="#phi0_eq"><span class="id" title="lemma">phi0_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:525" class="idref" href="#alpha:525"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:525"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:525"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:525"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_term_def" class="idref" href="#omega_term_def"><span class="id" title="lemma">omega_term_def</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:526" class="idref" href="#a:526"><span class="id" title="binder">a</span></a> <a id="n:527" class="idref" href="#n:527"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:526"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#omega_term"><span class="id" title="definition">omega_term</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:526"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:527"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:526"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:527"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cons_def" class="idref" href="#cons_def"><span class="id" title="lemma">cons_def</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:528" class="idref" href="#a:528"><span class="id" title="binder">a</span></a> <a id="n:529" class="idref" href="#n:529"><span class="id" title="binder">n</span></a> <a id="b:530" class="idref" href="#b:530"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:528"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:529"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:530"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:528"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:529"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:530"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:528"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:529"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:530"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="unique_decomposition" class="idref" href="#unique_decomposition"><span class="id" title="lemma">unique_decomposition</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:531" class="idref" href="#a:531"><span class="id" title="binder">a</span></a> <a id="n:532" class="idref" href="#n:532"><span class="id" title="binder">n</span></a> <a id="b:533" class="idref" href="#b:533"><span class="id" title="binder">b</span></a> <a id="a':534" class="idref" href="#a':534"><span class="id" title="binder">a'</span></a> <a id="n':535" class="idref" href="#n':535"><span class="id" title="binder">n'</span></a> <a id="b':536" class="idref" href="#b':536"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:531"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:532"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:533"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':534"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':535"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':536"><span class="id" title="variable">b'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:531"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:532"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:533"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':534"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':535"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':536"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:531"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':534"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:532"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':535"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:533"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':536"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Cantor_normal_form" class="idref" href="#Cantor_normal_form"><span class="id" title="lemma">Cantor_normal_form</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="o:537" class="idref" href="#o:537"><span class="id" title="binder">o</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#o:537"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#o:537"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="a:538" class="idref" href="#a:538"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="n:539" class="idref" href="#n:539"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b:540" class="idref" href="#b:540"><span class="id" title="binder">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#o:537"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:538"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:539"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:540"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:538"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:539"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:540"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_one" class="idref" href="#LT_one"><span class="id" title="lemma">LT_one</span></a> <a id="alpha:541" class="idref" href="#alpha:541"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:541"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:541"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_inv" class="idref" href="#lt_omega_inv"><span class="id" title="lemma">lt_omega_inv</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:542" class="idref" href="#alpha:542"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:542"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:542"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:543" class="idref" href="#n:543"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:542"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:543"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">T1_inversion</span> <span class="id" title="var">H</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">H</span> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.Epsilon0.T1.html#not_lt_zero"><span class="id" title="lemma">not_lt_zero</span></a> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt"><span class="id" title="definition">Nat.lt</span></a> <span class="id" title="var">_</span> 0 ⇒ <span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.nlt_0_r"><span class="id" title="lemma">Nat.nlt_0_r</span></a> <span class="id" title="var">_</span> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt"><span class="id" title="definition">Nat.lt</span></a> ?<span class="id" title="var">x</span> ?<span class="id" title="var">x</span> ⇒ <span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt_irrefl"><span class="id" title="lemma">Nat.lt_irrefl</span></a> <span class="id" title="var">_</span> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> ?<span class="id" title="var">x</span> ?<span class="id" title="var">x</span> ⇒ <span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.Epsilon0.T1.html#lt_irrefl"><span class="id" title="lemma">lt_irrefl</span></a>  <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>) (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.Epsilon0.T1.html#lt_inv"><span class="id" title="lemma">lt_inv</span></a> <span class="id" title="var">H</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">y</span>) ⇒ <span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <span class="id" title="tactic">fresh</span> "e" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="hydras.Epsilon0.T1.html#nf_of_finite"><span class="id" title="lemma">nf_of_finite</span></a> <span class="id" title="var">H</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span>  <span class="id" title="var">e</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_of_finite" class="idref" href="#LT_of_finite"><span class="id" title="lemma">LT_of_finite</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:544" class="idref" href="#alpha:544"><span class="id" title="binder">alpha</span></a> <a id="n:545" class="idref" href="#n:545"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#alpha:544"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:545"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:544"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p:546" class="idref" href="#p:546"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:546"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:545"><span class="id" title="variable">n</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:544"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:546"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_omega" class="idref" href="#nf_omega"><span class="id" title="lemma">nf_omega</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="nf_phi0" class="idref" href="#nf_phi0"><span class="id" title="lemma">nf_phi0</span></a> <a id="alpha:547" class="idref" href="#alpha:547"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:547"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:547"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_phi0</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="omega_omega" class="idref" href="#omega_omega"><span class="id" title="definition">omega_omega</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_omega_omega" class="idref" href="#nf_omega_omega"><span class="id" title="lemma">nf_omega_omega</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#omega_omega"><span class="id" title="definition">omega_omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_0_a" class="idref" href="#mult_0_a"><span class="id" title="lemma">mult_0_a</span></a> : <span class="id" title="keyword">∀</span> <a id="a:548" class="idref" href="#a:548"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:548"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_Sn_add" class="idref" href="#mult_Sn_add"><span class="id" title="lemma">mult_Sn_add</span></a> (<a id="alpha:549" class="idref" href="#alpha:549"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) <a id="n:550" class="idref" href="#n:550"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:549"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:549"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:550"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:549"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#n:550"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:549"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cases_for_mult" class="idref" href="#cases_for_mult"><span class="id" title="lemma">cases_for_mult</span></a> (<a id="alpha:551" class="idref" href="#alpha:551"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:551"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:551"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:552" class="idref" href="#n:552"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:551"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:552"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:553" class="idref" href="#a:553"><span class="id" title="binder">a</span></a> <a id="n:554" class="idref" href="#n:554"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:553"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:551"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:553"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:554"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:555" class="idref" href="#a:555"><span class="id" title="binder">a</span></a> <a id="n:556" class="idref" href="#n:556"><span class="id" title="binder">n</span></a> <a id="b:557" class="idref" href="#b:557"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#a:555"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:557"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:551"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:555"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:556"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:557"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L03" class="idref" href="#L03"><span class="id" title="lemma">L03</span></a> <a id="alpha:558" class="idref" href="#alpha:558"><span class="id" title="binder">alpha</span></a> <a id="n:559" class="idref" href="#n:559"><span class="id" title="binder">n</span></a> <a id="beta:560" class="idref" href="#beta:560"><span class="id" title="binder">beta</span></a> <a id="p:561" class="idref" href="#p:561"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:558"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:558"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:559"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:560"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:561"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:558"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#p:561"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:559"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:561"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#beta:560"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L05" class="idref" href="#L05"><span class="id" title="lemma">L05</span></a> <a id="a:562" class="idref" href="#a:562"><span class="id" title="binder">a</span></a> <a id="n:563" class="idref" href="#n:563"><span class="id" title="binder">n</span></a> <a id="b:564" class="idref" href="#b:564"><span class="id" title="binder">b</span></a> <a id="c:565" class="idref" href="#c:565"><span class="id" title="binder">c</span></a> <a id="p:566" class="idref" href="#p:566"><span class="id" title="binder">p</span></a> <a id="d:567" class="idref" href="#d:567"><span class="id" title="binder">d</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:562"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:565"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:562"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:563"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:564"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:565"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:566"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:567"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:562"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:565"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#p:566"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:562"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:563"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:564"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:567"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_LT_iff" class="idref" href="#nf_LT_iff"><span class="id" title="lemma">nf_LT_iff</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:568" class="idref" href="#alpha:568"><span class="id" title="binder">alpha</span></a> <a id="n:569" class="idref" href="#n:569"><span class="id" title="binder">n</span></a> <a id="beta:570" class="idref" href="#beta:570"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:568"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:569"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:570"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:568"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:570"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:570"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:568"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_plus_l" class="idref" href="#lt_plus_l"><span class="id" title="lemma">lt_plus_l</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> {<a id="a:571" class="idref" href="#a:571"><span class="id" title="binder">a</span></a> <a id="b:572" class="idref" href="#b:572"><span class="id" title="binder">b</span></a> <a id="c:573" class="idref" href="#c:573"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>} {<a id="n:574" class="idref" href="#n:574"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:571"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:571"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:572"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:574"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:573"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_plus_r" class="idref" href="#lt_plus_r"><span class="id" title="lemma">lt_plus_r</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> {<a id="a:575" class="idref" href="#a:575"><span class="id" title="binder">a</span></a> <a id="b:576" class="idref" href="#b:576"><span class="id" title="binder">b</span></a> <a id="c:577" class="idref" href="#c:577"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>} {<a id="n:578" class="idref" href="#n:578"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:575"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:576"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:578"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:577"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#a:575"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduce_lt_plus" class="idref" href="#reduce_lt_plus"><span class="id" title="lemma">reduce_lt_plus</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:579" class="idref" href="#a:579"><span class="id" title="binder">a</span></a> <a id="b:580" class="idref" href="#b:580"><span class="id" title="binder">b</span></a> <a id="c:581" class="idref" href="#c:581"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:579"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:580"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.Epsilon0.T1.html#a:579"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:581"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:580"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:581"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_smono_LT_r" class="idref" href="#plus_smono_LT_r"><span class="id" title="lemma">plus_smono_LT_r</span></a> (<a id="alpha:582" class="idref" href="#alpha:582"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:583" class="idref" href="#beta:583"><span class="id" title="binder">beta</span></a> <a id="gamma:584" class="idref" href="#gamma:584"><span class="id" title="binder">gamma</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:582"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:583"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:584"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:582"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:583"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:582"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:584"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_add" class="idref" href="#LT_add"><span class="id" title="lemma">LT_add</span></a> (<a id="alpha:585" class="idref" href="#alpha:585"><span class="id" title="binder">alpha</span></a> <a id="beta:586" class="idref" href="#beta:586"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>): <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:585"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:586"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:586"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:585"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:585"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:586"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_mult_nf" class="idref" href="#Proof_of_mult_nf"><span class="id" title="section">Proof_of_mult_nf</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_mult_nf.alpha" class="idref" href="#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_mult_nf.Halpha" class="idref" href="#Proof_of_mult_nf.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_mult_nf.P" class="idref" href="#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> (<a id="beta:589" class="idref" href="#beta:589"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:589"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:589"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="gamma:590" class="idref" href="#gamma:590"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#gamma:590"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:589"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:590"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:589"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_mult_nf.Induction" class="idref" href="#Proof_of_mult_nf.Induction"><span class="id" title="section">Induction</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_mult_nf.Induction.beta" class="idref" href="#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_mult_nf.Induction.Hbeta" class="idref" href="#Proof_of_mult_nf.Induction.Hbeta"><span class="id" title="variable">Hbeta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_mult_nf.Induction.IHbeta" class="idref" href="#Proof_of_mult_nf.Induction.IHbeta"><span class="id" title="variable">IHbeta</span></a> : <span class="id" title="keyword">∀</span> <a id="delta:593" class="idref" href="#delta:593"><span class="id" title="binder">delta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#delta:593"><span class="id" title="variable">delta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#delta:593"><span class="id" title="variable">delta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L3" class="idref" href="#L3"><span class="id" title="lemma">L3</span></a> <a id="n:595" class="idref" href="#n:595"><span class="id" title="binder">n</span></a> <a id="p:596" class="idref" href="#p:596"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:595"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:596"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L4" class="idref" href="#L4"><span class="id" title="lemma">L4</span></a> : <span class="id" title="keyword">∀</span> <a id="a:597" class="idref" href="#a:597"><span class="id" title="binder">a</span></a> <a id="n:598" class="idref" href="#n:598"><span class="id" title="binder">n</span></a> <a id="b:599" class="idref" href="#b:599"><span class="id" title="binder">b</span></a> <a id="p:600" class="idref" href="#p:600"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#a:597"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:597"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:598"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:599"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:600"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L5" class="idref" href="#L5"><span class="id" title="lemma">L5</span></a> <a id="a:601" class="idref" href="#a:601"><span class="id" title="binder">a</span></a> <a id="n:602" class="idref" href="#n:602"><span class="id" title="binder">n</span></a> <a id="b:603" class="idref" href="#b:603"><span class="id" title="binder">b</span></a> <a id="c:604" class="idref" href="#c:604"><span class="id" title="binder">c</span></a> <a id="p:605" class="idref" href="#p:605"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#a:601"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:604"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:601"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:602"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:603"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:604"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:605"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#a:601"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:604"><span class="id" title="variable">c</span></a>)  <a class="idref" href="hydras.Epsilon0.T1.html#p:605"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L6" class="idref" href="#L6"><span class="id" title="lemma">L6</span></a> <a id="n:606" class="idref" href="#n:606"><span class="id" title="binder">n</span></a> <a id="c:607" class="idref" href="#c:607"><span class="id" title="binder">c</span></a> <a id="p:608" class="idref" href="#p:608"><span class="id" title="binder">p</span></a> <a id="d:609" class="idref" href="#d:609"><span class="id" title="binder">d</span></a> :  <a class="idref" href="hydras.Epsilon0.T1.html#c:607"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:606"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:607"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:608"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:609"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:607"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:608"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:606"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:609"><span class="id" title="variable">d</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L7" class="idref" href="#L7"><span class="id" title="lemma">L7</span></a> <a id="n:610" class="idref" href="#n:610"><span class="id" title="binder">n</span></a> <a id="c:611" class="idref" href="#c:611"><span class="id" title="binder">c</span></a> <a id="p:612" class="idref" href="#p:612"><span class="id" title="binder">p</span></a> :  <a class="idref" href="hydras.Epsilon0.T1.html#c:611"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:610"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:611"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:612"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:611"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:612"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L8" class="idref" href="#L8"><span class="id" title="lemma">L8</span></a> <a id="n:613" class="idref" href="#n:613"><span class="id" title="binder">n</span></a>  <a id="c:614" class="idref" href="#c:614"><span class="id" title="binder">c</span></a> <a id="p:615" class="idref" href="#p:615"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:613"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:614"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:615"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:614"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L9" class="idref" href="#L9"><span class="id" title="lemma">L9</span></a> : <span class="id" title="keyword">∀</span> <a id="n:616" class="idref" href="#n:616"><span class="id" title="binder">n</span></a> <a id="c:617" class="idref" href="#c:617"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:617"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:617"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:616"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:617"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L10" class="idref" href="#L10"><span class="id" title="lemma">L10</span></a> : <span class="id" title="keyword">∀</span> <a id="a:618" class="idref" href="#a:618"><span class="id" title="binder">a</span></a> <a id="n:619" class="idref" href="#n:619"><span class="id" title="binder">n</span></a> <a id="b:620" class="idref" href="#b:620"><span class="id" title="binder">b</span></a> <a id="c:621" class="idref" href="#c:621"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:621"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:618"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:619"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:620"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:618"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:621"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:618"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:619"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:620"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:621"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L11" class="idref" href="#L11"><span class="id" title="lemma">L11</span></a> <a id="n:622" class="idref" href="#n:622"><span class="id" title="binder">n</span></a>  <a id="c:623" class="idref" href="#c:623"><span class="id" title="binder">c</span></a> <a id="p:624" class="idref" href="#p:624"><span class="id" title="binder">p</span></a> <a id="d:625" class="idref" href="#d:625"><span class="id" title="binder">d</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:622"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:623"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:624"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:625"><span class="id" title="variable">d</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:623"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#d:625"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L12" class="idref" href="#L12"><span class="id" title="lemma">L12</span></a> : <span class="id" title="keyword">∀</span> <a id="a:626" class="idref" href="#a:626"><span class="id" title="binder">a</span></a> <a id="n:627" class="idref" href="#n:627"><span class="id" title="binder">n</span></a> <a id="b:628" class="idref" href="#b:628"><span class="id" title="binder">b</span></a> <a id="c:629" class="idref" href="#c:629"><span class="id" title="binder">c</span></a> <a id="p:630" class="idref" href="#p:630"><span class="id" title="binder">p</span></a> <a id="d:631" class="idref" href="#d:631"><span class="id" title="binder">d</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#a:626"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:629"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:626"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:627"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:628"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:629"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:630"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d:631"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L13" class="idref" href="#L13"><span class="id" title="lemma">L13</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction"><span class="id" title="section">Induction</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L14" class="idref" href="#L14"><span class="id" title="lemma">L14</span></a>  <a id="beta:632" class="idref" href="#beta:632"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:632"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:632"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_mult_nf"><span class="id" title="section">Proof_of_mult_nf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="mult_nf" class="idref" href="#mult_nf"><span class="id" title="lemma">mult_nf</span></a> <a id="alpha:633" class="idref" href="#alpha:633"><span class="id" title="binder">alpha</span></a> <a id="beta:634" class="idref" href="#beta:634"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:633"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:634"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#alpha:633"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:634"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="mult_mono" class="idref" href="#mult_mono"><span class="id" title="lemma">mult_mono</span></a> <a id="alpha:635" class="idref" href="#alpha:635"><span class="id" title="binder">alpha</span></a> <a id="beta:636" class="idref" href="#beta:636"><span class="id" title="binder">beta</span></a> <a id="gamma:637" class="idref" href="#gamma:637"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:635"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:635"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:636"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:637"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:635"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:636"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:635"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:637"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_exp_F" class="idref" href="#nf_exp_F"><span class="id" title="lemma">nf_exp_F</span></a> <a id="alpha:638" class="idref" href="#alpha:638"><span class="id" title="binder">alpha</span></a> <a id="n:639" class="idref" href="#n:639"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:638"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:638"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:639"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp_F_eq" class="idref" href="#exp_F_eq"><span class="id" title="lemma">exp_F_eq</span></a> <a id="alpha:640" class="idref" href="#alpha:640"><span class="id" title="binder">alpha</span></a> <a id="n:641" class="idref" href="#n:641"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:640"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#exp_F"><span class="id" title="definition">exp_F</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:640"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:641"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:640"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:641"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_cases" class="idref" href="#T1limit_cases"><span class="id" title="lemma">T1limit_cases</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:642" class="idref" href="#alpha:642"><span class="id" title="binder">alpha</span></a> <a id="n:643" class="idref" href="#n:643"><span class="id" title="binder">n</span></a> <a id="beta:644" class="idref" href="#beta:644"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:642"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:643"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:644"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:642"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:643"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:644"><span class="id" title="variable">beta</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:642"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:644"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:642"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:644"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_succ" class="idref" href="#pred_of_succ"><span class="id" title="lemma">pred_of_succ</span></a> : <span class="id" title="keyword">∀</span>  <a id="beta:645" class="idref" href="#beta:645"><span class="id" title="binder">beta</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:645"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:645"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:645"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_limit" class="idref" href="#pred_of_limit"><span class="id" title="lemma">pred_of_limit</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:646" class="idref" href="#alpha:646"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="zero_limit_succ_dec" class="idref" href="#zero_limit_succ_dec"><span class="id" title="definition">zero_limit_succ_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:647" class="idref" href="#alpha:647"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:647"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:647"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:647"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:648" class="idref" href="#beta:648"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:648"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:647"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:648"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_limitR" class="idref" href="#pred_of_limitR"><span class="id" title="lemma">pred_of_limitR</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:649" class="idref" href="#alpha:649"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:649"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:649"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:649"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:649"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_LT" class="idref" href="#pred_LT"><span class="id" title="lemma">pred_LT</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:650" class="idref" href="#alpha:650"><span class="id" title="binder">alpha</span></a> <a id="beta:651" class="idref" href="#beta:651"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:650"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:650"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:651"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:651"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:650"><span class="id" title="variable">alpha</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_nf" class="idref" href="#pred_nf"><span class="id" title="lemma">pred_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:652" class="idref" href="#alpha:652"><span class="id" title="binder">alpha</span></a> <a id="beta:653" class="idref" href="#beta:653"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:652"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:652"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:653"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:653"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_succ" class="idref" href="#T1limit_succ"><span class="id" title="lemma">T1limit_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:654" class="idref" href="#alpha:654"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:654"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:654"><span class="id" title="variable">alpha</span></a>) .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_succ" class="idref" href="#LT_succ"><span class="id" title="lemma">LT_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:655" class="idref" href="#alpha:655"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:655"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:655"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:655"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_not_zero" class="idref" href="#T1limit_not_zero"><span class="id" title="lemma">T1limit_not_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:656" class="idref" href="#alpha:656"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:656"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:656"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:656"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T1limit_not_zero</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1limit_succ_tail" class="idref" href="#T1limit_succ_tail"><span class="id" title="lemma">T1limit_succ_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:657" class="idref" href="#alpha:657"><span class="id" title="binder">alpha</span></a> <a id="n:658" class="idref" href="#n:658"><span class="id" title="binder">n</span></a> <a id="beta:659" class="idref" href="#beta:659"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:659"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:657"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:658"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:659"><span class="id" title="variable">beta</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_not_limit" class="idref" href="#succ_not_limit"><span class="id" title="lemma">succ_not_limit</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:660" class="idref" href="#alpha:660"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:660"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:660"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1is_succ_def" class="idref" href="#T1is_succ_def"><span class="id" title="lemma">T1is_succ_def</span></a> <a id="alpha:661" class="idref" href="#alpha:661"><span class="id" title="binder">alpha</span></a> (<a id="Halpha:662" class="idref" href="#Halpha:662"><span class="id" title="binder">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:661"><span class="id" title="variable">alpha</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:661"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><a id="beta:663" class="idref" href="#beta:663"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:663"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:661"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:663"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1is_succ_iff" class="idref" href="#T1is_succ_iff"><span class="id" title="lemma">T1is_succ_iff</span></a> <a id="alpha:664" class="idref" href="#alpha:664"><span class="id" title="binder">alpha</span></a> (<a id="Halpha:665" class="idref" href="#Halpha:665"><span class="id" title="binder">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:664"><span class="id" title="variable">alpha</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:664"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:666" class="idref" href="#beta:666"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:666"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:664"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:666"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_r" class="idref" href="#LE_r"><span class="id" title="lemma">LE_r</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:667" class="idref" href="#alpha:667"><span class="id" title="binder">alpha</span></a> <a id="beta:668" class="idref" href="#beta:668"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#alpha:667"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:668"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:667"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:668"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LE_LT_eq_dec" class="idref" href="#LE_LT_eq_dec"><span class="id" title="lemma">LE_LT_eq_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:669" class="idref" href="#alpha:669"><span class="id" title="binder">alpha</span></a> <a id="beta:670" class="idref" href="#beta:670"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#alpha:669"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:670"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:669"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:670"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:669"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:670"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_eq_LT_dec" class="idref" href="#LT_eq_LT_dec"><span class="id" title="lemma">LT_eq_LT_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:671" class="idref" href="#alpha:671"><span class="id" title="binder">alpha</span></a> <a id="beta:672" class="idref" href="#beta:672"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:671"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:672"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:671"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:672"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:671"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:672"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#beta:672"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:671"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_cons_phi0_inv" class="idref" href="#lt_cons_phi0_inv"><span class="id" title="lemma">lt_cons_phi0_inv</span></a> <a id="alpha:673" class="idref" href="#alpha:673"><span class="id" title="binder">alpha</span></a> <a id="n:674" class="idref" href="#n:674"><span class="id" title="binder">n</span></a> <a id="beta:675" class="idref" href="#beta:675"><span class="id" title="binder">beta</span></a> <a id="gamma:676" class="idref" href="#gamma:676"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:673"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:674"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:675"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#gamma:676"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:675"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:673"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:673"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#gamma:676"><span class="id" title="variable">gamma</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_LT_right" class="idref" href="#nf_LT_right"><span class="id" title="lemma">nf_LT_right</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:677" class="idref" href="#alpha:677"><span class="id" title="binder">alpha</span></a> <a id="n:678" class="idref" href="#n:678"><span class="id" title="binder">n</span></a> <a id="beta:679" class="idref" href="#beta:679"><span class="id" title="binder">beta</span></a> <a id="beta':680" class="idref" href="#beta':680"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:677"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:678"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:679"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta':680"><span class="id" title="variable">beta'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:679"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:677"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:678"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta':680"><span class="id" title="variable">beta'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eq_succ_LT" class="idref" href="#eq_succ_LT"><span class="id" title="lemma">eq_succ_LT</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:681" class="idref" href="#alpha:681"><span class="id" title="binder">alpha</span></a> <a id="beta:682" class="idref" href="#beta:682"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:682"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:681"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:682"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#beta:682"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:681"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eq_succ_lt" class="idref" href="#eq_succ_lt"><span class="id" title="lemma">eq_succ_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:683" class="idref" href="#alpha:683"><span class="id" title="binder">alpha</span></a> <a id="beta:684" class="idref" href="#beta:684"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:684"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:683"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:684"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#beta:684"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:683"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strict_lub" class="idref" href="#strict_lub"><span class="id" title="definition">strict_lub</span></a> (<a id="s:685" class="idref" href="#s:685"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) (<a id="lambda:686" class="idref" href="#lambda:686"><span class="id" title="binder">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:687" class="idref" href="#i:687"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#s:685"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:687"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lambda:686"><span class="id" title="variable">lambda</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="alpha:688" class="idref" href="#alpha:688"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:689" class="idref" href="#i:689"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#s:685"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:689"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:688"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lambda:686"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:688"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strict_lub_lub" class="idref" href="#strict_lub_lub"><span class="id" title="definition">strict_lub_lub</span></a> : <span class="id" title="keyword">∀</span> <a id="s:690" class="idref" href="#s:690"><span class="id" title="binder">s</span></a> <a id="l:691" class="idref" href="#l:691"><span class="id" title="binder">l</span></a> <a id="alpha:692" class="idref" href="#alpha:692"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#s:690"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l:691"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:693" class="idref" href="#i:693"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#s:690"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:693"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#alpha:692"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#l:691"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:692"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strict_lub_maj" class="idref" href="#strict_lub_maj"><span class="id" title="definition">strict_lub_maj</span></a> : <span class="id" title="keyword">∀</span> <a id="s:694" class="idref" href="#s:694"><span class="id" title="binder">s</span></a> <a id="l:695" class="idref" href="#l:695"><span class="id" title="binder">l</span></a> ,  <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#s:694"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l:695"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:696" class="idref" href="#i:696"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#s:694"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:696"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l:695"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="strict_lub_unique" class="idref" href="#strict_lub_unique"><span class="id" title="lemma">strict_lub_unique</span></a> : <span class="id" title="keyword">∀</span> <a id="s:697" class="idref" href="#s:697"><span class="id" title="binder">s</span></a> <a id="l:698" class="idref" href="#l:698"><span class="id" title="binder">l</span></a> <a id="l':699" class="idref" href="#l':699"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#s:697"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l:698"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#s:697"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l':699"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#l:698"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#l':699"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="strict_lub_T1limit" class="idref" href="#strict_lub_T1limit"><span class="id" title="lemma">strict_lub_T1limit</span></a> : <span class="id" title="keyword">∀</span> (<a id="alpha:700" class="idref" href="#alpha:700"><span class="id" title="binder">alpha</span></a> :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)(<a id="s:701" class="idref" href="#s:701"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:700"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#s:701"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:700"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:700"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_one" class="idref" href="#lt_one"><span class="id" title="lemma">lt_one</span></a> (<a id="alpha:704" class="idref" href="#alpha:704"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:704"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:704"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="omega_limit" class="idref" href="#omega_limit"><span class="id" title="lemma">omega_limit</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#strict_lub"><span class="id" title="definition">strict_lub</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1nat"><span class="id" title="definition">T1nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LT_succ_LT_eq_dec" class="idref" href="#LT_succ_LT_eq_dec"><span class="id" title="lemma">LT_succ_LT_eq_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:705" class="idref" href="#alpha:705"><span class="id" title="binder">alpha</span></a> <a id="beta:706" class="idref" href="#beta:706"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:705"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:706"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#alpha:705"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:706"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:705"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:706"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Epsilon0.T1.html#alpha:705"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:706"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le_2'" class="idref" href="#lt_succ_le_2'"><span class="id" title="lemma">lt_succ_le_2'</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:707" class="idref" href="#a:707"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:707"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:708" class="idref" href="#b:708"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:708"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:707"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:708"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:707"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:708"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:707"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:708"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_lt_limit" class="idref" href="#succ_lt_limit"><span class="id" title="lemma">succ_lt_limit</span></a> <a id="alpha:709" class="idref" href="#alpha:709"><span class="id" title="binder">alpha</span></a> (<a id="Halpha:710" class="idref" href="#Halpha:710"><span class="id" title="binder">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:709"><span class="id" title="variable">alpha</span></a>)(<a id="H:711" class="idref" href="#H:711"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:709"><span class="id" title="variable">alpha</span></a> ):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:712" class="idref" href="#beta:712"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#beta:712"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:709"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:712"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:709"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_cons" class="idref" href="#succ_cons"><span class="id" title="lemma">succ_cons</span></a> <a id="alpha:713" class="idref" href="#alpha:713"><span class="id" title="binder">alpha</span></a> <a id="n:714" class="idref" href="#n:714"><span class="id" title="binder">n</span></a> <a id="beta:715" class="idref" href="#beta:715"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#beta:715"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:713"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:714"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:715"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:713"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:714"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:715"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:713"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:714"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:715"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_cons'" class="idref" href="#succ_cons'"><span class="id" title="lemma">succ_cons'</span></a> <a id="alpha:716" class="idref" href="#alpha:716"><span class="id" title="binder">alpha</span></a> <a id="i:717" class="idref" href="#i:717"><span class="id" title="binder">i</span></a> <a id="beta:718" class="idref" href="#beta:718"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#alpha:716"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:716"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:717"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:718"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:716"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:717"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:718"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:716"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#i:717"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:718"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_rw1" class="idref" href="#succ_rw1"><span class="id" title="lemma">succ_rw1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:719" class="idref" href="#alpha:719"><span class="id" title="binder">alpha</span></a> <a id="n:720" class="idref" href="#n:720"><span class="id" title="binder">n</span></a> <a id="beta:721" class="idref" href="#beta:721"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#alpha:719"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:719"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:720"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:721"><span class="id" title="variable">beta</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:719"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:720"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#beta:721"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_cons_cons_eqn" class="idref" href="#plus_cons_cons_eqn"><span class="id" title="lemma">plus_cons_cons_eqn</span></a> <a id="a:722" class="idref" href="#a:722"><span class="id" title="binder">a</span></a> <a id="n:723" class="idref" href="#n:723"><span class="id" title="binder">n</span></a> <a id="b:724" class="idref" href="#b:724"><span class="id" title="binder">b</span></a> <a id="a':725" class="idref" href="#a':725"><span class="id" title="binder">a'</span></a> <a id="n':726" class="idref" href="#n':726"><span class="id" title="binder">n'</span></a> <a id="b':727" class="idref" href="#b':727"><span class="id" title="binder">b'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:722"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:723"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:724"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':725"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':726"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':727"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:722"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':725"><span class="id" title="variable">a'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:722"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#n:723"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':726"><span class="id" title="variable">n'</span></a>)) <a class="idref" href="hydras.Epsilon0.T1.html#b':727"><span class="id" title="variable">b'</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':725"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':726"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':727"><span class="id" title="variable">b'</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:722"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:723"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1add</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:724"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a':725"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n':726"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b':727"><span class="id" title="variable">b'</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1addA" class="idref" href="#T1addA"><span class="id" title="lemma">T1addA</span></a> (<a id="x:728" class="idref" href="#x:728"><span class="id" title="binder">x</span></a> <a id="y:729" class="idref" href="#y:729"><span class="id" title="binder">y</span></a> <a id="z:730" class="idref" href="#z:730"><span class="id" title="binder">z</span></a> :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#x:728"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#y:729"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#z:730"><span class="id" title="variable">z</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#x:728"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#y:729"><span class="id" title="variable">y</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#z:730"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="T1addAssoc" class="idref" href="#T1addAssoc"><span class="id" title="instance">T1addAssoc</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1add</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_dist" class="idref" href="#Proof_of_dist"><span class="id" title="section">Proof_of_dist</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_dist.P" class="idref" href="#Proof_of_dist.P"><span class="id" title="variable">P</span></a> (<a id="b:731" class="idref" href="#b:731"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:732" class="idref" href="#a:732"><span class="id" title="binder">a</span></a> <a id="c:733" class="idref" href="#c:733"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:732"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:731"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:733"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:732"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#b:731"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:733"><span class="id" title="variable">c</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:732"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:731"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:732"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:733"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Ltac</span> <span class="id" title="var">rewrite_ind</span> <span class="id" title="var">Hind</span> <span class="id" title="var">b</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">pose</span> <span class="id" title="var">proof</span> (<span class="id" title="var">Hind</span> <span class="id" title="var">b</span>) <span class="id" title="keyword">as</span> -&gt;; [ | <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Epsilon0.T1.html#tail_LT_cons"><span class="id" title="lemma">tail_LT_cons</span></a>| | | ];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eauto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L0" class="idref" href="#L0"><span class="id" title="lemma">L0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:734" class="idref" href="#p:734"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_dist.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#p:734"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="mult_plus_distr_l" class="idref" href="#mult_plus_distr_l"><span class="id" title="lemma">mult_plus_distr_l</span></a> (<a id="a:735" class="idref" href="#a:735"><span class="id" title="binder">a</span></a> <a id="b:736" class="idref" href="#b:736"><span class="id" title="binder">b</span></a> <a id="c:737" class="idref" href="#c:737"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:735"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:736"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:737"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#a:735"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#b:736"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:737"><span class="id" title="variable">c</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:735"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#b:736"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#a:735"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#c:737"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#Proof_of_dist"><span class="id" title="section">Proof_of_dist</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab73"></a><h2 class="section">An abstract syntax for ordinals in Cantor normal form</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">ppT1_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ppT1_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">pT1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="ppT1" class="idref" href="#ppT1"><span class="id" title="inductive">ppT1</span></a> :=<br/>
+| <a id="PP_fin" class="idref" href="#PP_fin"><span class="id" title="constructor">PP_fin</span></a> (<span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+| <a id="PP_add" class="idref" href="#PP_add"><span class="id" title="constructor">PP_add</span></a> (<span class="id" title="var">_</span> <span class="id" title="var">_</span> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1:738"><span class="id" title="inductive">ppT1</span></a>)<br/>
+| <a id="PP_mult" class="idref" href="#PP_mult"><span class="id" title="constructor">PP_mult</span></a> (<span class="id" title="var">_</span> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1:738"><span class="id" title="inductive">ppT1</span></a>) (<span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+| <a id="PP_exp" class="idref" href="#PP_exp"><span class="id" title="constructor">PP_exp</span></a> (<span class="id" title="var">_</span> <span class="id" title="var">_</span> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1:738"><span class="id" title="inductive">ppT1</span></a>)<br/>
+| <a id="PP_omega" class="idref" href="#PP_omega"><span class="id" title="constructor">PP_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">PP_fin</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">:</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">nat</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">&gt;-&gt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">ppT1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="72f214ae79555f1ca58c18aadc6d29d1" class="idref" href="#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">&quot;</span></a>alpha + beta" := (<a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>) : <span class="id" title="var">ppT1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="d1152fe3a7d79cfdcdc4303f763052d6" class="idref" href="#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">&quot;</span></a>alpha * n" := (<a class="idref" href="hydras.Epsilon0.T1.html#PP_mult"><span class="id" title="constructor">PP_mult</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">n</span>) : <span class="id" title="var">ppT1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="3f1bc0399e791f84036bad45d79e1159" class="idref" href="#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">&quot;</span></a>alpha ^ beta" := (<a class="idref" href="hydras.Epsilon0.T1.html#PP_exp"><span class="id" title="constructor">PP_exp</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>) : <span class="id" title="var">ppT1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="45bf03a575f6e81359314e906fb2bff3" class="idref" href="#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#PP_omega"><span class="id" title="constructor">PP_omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> 2 <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">pT1</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="pp0" class="idref" href="#pp0"><span class="id" title="definition">pp0</span></a> (<a id="alpha:740" class="idref" href="#alpha:740"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:740"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">PP_fin</span></a> 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">PP_fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> 0 <span class="id" title="keyword">beta</span> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> <span class="id" title="var">n</span> <span class="id" title="keyword">beta</span> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:740"><span class="id" title="variable">alpha</span></a>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> 0 <span class="id" title="keyword">beta</span> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:740"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:740"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">alpha</span> <span class="id" title="var">n</span> <span class="id" title="keyword">beta</span> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#alpha:740"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#pp0:741"><span class="id" title="definition">pp0</span></a> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="eval_pp" class="idref" href="#eval_pp"><span class="id" title="definition">eval_pp</span></a> (<a id="e:743" class="idref" href="#e:743"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#e:743"><span class="id" title="variable">e</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">PP_fin</span></a> 0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_fin"><span class="id" title="constructor">PP_fin</span></a> <span class="id" title="var">n</span> ⇒ <span class="id" title="var">n</span>       <br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> ⇒ ( <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#eval_pp:744"><span class="id" title="definition">eval_pp</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#e:743"><span class="id" title="variable">e</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#eval_pp:744"><span class="id" title="definition">eval_pp</span></a> <span class="id" title="var">f</span><a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">t1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_mult"><span class="id" title="constructor">PP_mult</span></a> <span class="id" title="var">e</span> <span class="id" title="var">n</span> ⇒ ( <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#eval_pp:744"><span class="id" title="definition">eval_pp</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#e:743"><span class="id" title="variable">e</span></a><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">t1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_exp"><span class="id" title="constructor">PP_exp</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> ⇒ (<a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#eval_pp:744"><span class="id" title="definition">eval_pp</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#e:743"><span class="id" title="variable">e</span></a><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#eval_pp:744"><span class="id" title="definition">eval_pp</span></a> <span class="id" title="var">f</span><a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">t1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3"><span class="id" title="abbreviation">ω</span></a>   ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="reassoc" class="idref" href="#reassoc"><span class="id" title="definition">reassoc</span></a> (<a id="exp:746" class="idref" href="#exp:746"><span class="id" title="binder">exp</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a>) (<a id="fuel:747" class="idref" href="#fuel:747"><span class="id" title="binder">fuel</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#exp:746"><span class="id" title="variable">exp</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#fuel:747"><span class="id" title="variable">fuel</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#exp"><span class="id" title="definition">exp</span></a>, 0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#exp:746"><span class="id" title="variable">exp</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">e</span> (<a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">f</span> <span class="id" title="var">g</span>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a>  <span class="id" title="var">e</span> <span class="id" title="var">n</span>) (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">g</span> <span class="id" title="var">n</span>)) <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒  <a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">e</span> <span class="id" title="var">n</span>) (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_mult"><span class="id" title="constructor">PP_mult</span></a> <span class="id" title="var">e</span> <span class="id" title="var">i</span> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>⇒ <a class="idref" href="hydras.Epsilon0.T1.html#PP_mult"><span class="id" title="constructor">PP_mult</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">e</span> <span class="id" title="var">n</span>) <span class="id" title="var">i</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_exp"><span class="id" title="constructor">PP_exp</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#PP_exp"><span class="id" title="constructor">PP_exp</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">e</span> <span class="id" title="var">n</span>) (<a class="idref" href="hydras.Epsilon0.T1.html#reassoc:748"><span class="id" title="definition">reassoc</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span>, <span class="id" title="var">_</span>⇒ <span class="id" title="var">e</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="pp_size" class="idref" href="#pp_size"><span class="id" title="definition">pp_size</span></a> (<a id="exp:751" class="idref" href="#exp:751"><span class="id" title="binder">exp</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Epsilon0.T1.html#exp:751"><span class="id" title="variable">exp</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#PP_add"><span class="id" title="constructor">PP_add</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> |  <a class="idref" href="hydras.Epsilon0.T1.html#PP_exp"><span class="id" title="constructor">PP_exp</span></a> <span class="id" title="var">e</span> <span class="id" title="var">f</span> ⇒ (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#pp_size:752"><span class="id" title="definition">pp_size</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#pp_size:752"><span class="id" title="definition">pp_size</span></a> <span class="id" title="var">f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a>))%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#PP_mult"><span class="id" title="constructor">PP_mult</span></a> <span class="id" title="var">e</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#pp_size:752"><span class="id" title="definition">pp_size</span></a> <span class="id" title="var">e</span>)<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ 1%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="pp" class="idref" href="#pp"><span class="id" title="definition">pp</span></a> (<a id="e:754" class="idref" href="#e:754"><span class="id" title="binder">e</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#ppT1"><span class="id" title="inductive">ppT1</span></a>  := <span class="id" title="keyword">let</span> <a id="t:755" class="idref" href="#t:755"><span class="id" title="binder">t</span></a> := <a class="idref" href="hydras.Epsilon0.T1.html#pp0"><span class="id" title="definition">pp0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#e:754"><span class="id" title="variable">e</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.Epsilon0.T1.html#reassoc"><span class="id" title="definition">reassoc</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#t:755"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#pp_size"><span class="id" title="definition">pp_size</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#t:755"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span>  <span class="id" title="keyword">fun</span> <a id="n:756" class="idref" href="#n:756"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#pp"><span class="id" title="definition">pp</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:756"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#n:756"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> 1<a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a>))%<span class="id" title="var">t1</span> .<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">is_closed</span> <span class="id" title="var">alpha</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">alpha</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">idtac</span><br/>
+&nbsp;&nbsp;| 0 ⇒ <span class="id" title="tactic">idtac</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ?<span class="id" title="var">n</span> ⇒ <span class="id" title="var">is_closed</span> <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;|<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">a</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">b</span> ⇒ <span class="id" title="var">is_closed</span> <span class="id" title="var">a</span> ; <span class="id" title="var">is_closed</span> <span class="id" title="var">n</span> ; <span class="id" title="var">is_closed</span> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;| ?<span class="id" title="var">other</span> ⇒ <span class="id" title="tactic">fail</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">pp0tac</span> <span class="id" title="var">alpha</span>   :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">alpha</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">exact</span> 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ?<span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">exact</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">exact</span> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> 0 ?<span class="id" title="keyword">beta</span> ⇒ <span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <span class="id" title="keyword">ltac</span> :(<span class="id" title="var">pp0tac</span> <span class="id" title="keyword">beta</span>))%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> ?<span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#one"><span class="id" title="abbreviation">one</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="keyword">beta</span> ⇒ <span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <span class="id" title="keyword">ltac</span>: (<span class="id" title="var">pp0tac</span> <span class="id" title="keyword">beta</span>))%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">alpha</span> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <span class="id" title="keyword">ltac</span>: (<span class="id" title="var">pp0tac</span> <span class="id" title="var">alpha</span>))%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">alpha</span> 0 ?<span class="id" title="keyword">beta</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <span class="id" title="keyword">ltac</span> :(<span class="id" title="var">pp0tac</span> <span class="id" title="var">alpha</span>) <a class="idref" href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1"><span class="id" title="notation">+</span></a> <span class="id" title="keyword">ltac</span>: (<span class="id" title="var">pp0tac</span> <span class="id" title="keyword">beta</span>))%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">alpha</span> ?<span class="id" title="var">n</span> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159"><span class="id" title="notation">^</span></a> <span class="id" title="keyword">ltac</span>: (<span class="id" title="var">pp0tac</span> <span class="id" title="var">alpha</span>) <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">pT1</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> ?<span class="id" title="var">alpha</span> ?<span class="id" title="var">n</span> ?<span class="id" title="keyword">beta</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <span class="id" title="keyword">ltac</span>: (<span class="id" title="var">pp0tac</span> <span class="id" title="var">alpha</span>) <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">ltac</span> : (<span class="id" title="var">pp0tac</span> <span class="id" title="keyword">beta</span>)%<span class="id" title="var">pT1</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">pptac</span> <span class="id" title="var">term</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">t</span> := <span class="id" title="tactic">eval</span> <span class="id" title="var">cbn</span> <span class="id" title="tactic">in</span> <span class="id" title="var">term</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="var">tryif</span> <span class="id" title="var">is_closed</span> <span class="id" title="var">t</span> <span class="id" title="keyword">then</span> <span class="id" title="tactic">exact</span> (<a class="idref" href="hydras.Epsilon0.T1.html#pp"><span class="id" title="definition">pp</span></a> <span class="id" title="var">t</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="tactic">exact</span> <span class="id" title="var">term</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="essai" class="idref" href="#essai"><span class="id" title="section">essai</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="essai.n" class="idref" href="#essai.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Epsilon0.T1.html#essai"><span class="id" title="section">essai</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#FS"><span class="id" title="abbreviation">FS</span></a> 6))).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab74"></a><h1 class="section">Examples</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> :  42 <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> 42.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex3" class="idref" href="#Ex3"><span class="id" title="definition">Ex3</span></a> : 5 <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex4" class="idref" href="#Ex4"><span class="id" title="definition">Ex4</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 5.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex5" class="idref" href="#Ex5"><span class="id" title="definition">Ex5</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> 5<a class="idref" href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239"><span class="id" title="notation">)</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="alpha_0" class="idref" href="#alpha_0"><span class="id" title="definition">alpha_0</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 2 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;4<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> 1 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="alpha_0_eq" class="idref" href="#alpha_0_eq"><span class="id" title="definition">alpha_0_eq</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#alpha_0"><span class="id" title="definition">alpha_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> 3 <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a> 5 <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> 2.<br/>
+ </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Gamma0.Gamma0</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Gamma0.Gamma0</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab112"></a><h1 class="section">A notation system for ordinals less than  Gamma0</h1>
+
+<div class="paragraph"> </div>
+
+   Pierre Casteran 
+    LaBRI, Université Bordeaux 1, and LaBRI  (CNRS, UMR 5800)
+     with a contribution by Evelyne Contejean 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Restriction</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.term.html#"><span class="id" title="library">rpo.term</span></a>  <a class="idref" href="hydras.rpo.rpo.html#"><span class="id" title="library">rpo.rpo</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Gamma0.T2.html#"><span class="id" title="library">T2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#"><span class="id" title="module">Datatypes</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_a" class="idref" href="#nf_a"><span class="id" title="lemma">nf_a</span></a> : <span class="id" title="keyword">∀</span> <a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="b:2" class="idref" href="#b:2"><span class="id" title="binder">b</span></a> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> <a id="c:4" class="idref" href="#c:4"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:2"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:4"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:1"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_b" class="idref" href="#nf_b"><span class="id" title="lemma">nf_b</span></a> : <span class="id" title="keyword">∀</span> <a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> <a id="c:8" class="idref" href="#c:8"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:6"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:7"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:8"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:6"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_c" class="idref" href="#nf_c"><span class="id" title="lemma">nf_c</span></a> : <span class="id" title="keyword">∀</span> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a> <a id="c:12" class="idref" href="#c:12"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:11"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:12"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:12"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_a</span> <span class="id" title="var">nf_b</span> <span class="id" title="var">nf_c</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">nf_inv</span> := ((<span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#nf_a"><span class="id" title="lemma">nf_a</span></a>; <span class="id" title="var">eassumption</span>)|| <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#nf_b"><span class="id" title="lemma">nf_b</span></a>; <span class="id" title="var">eassumption</span>)|| <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#nf_c"><span class="id" title="lemma">nf_c</span></a>; <span class="id" title="var">eassumption</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_lt_succ" class="idref" href="#zero_lt_succ"><span class="id" title="lemma">zero_lt_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:13"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_lt_zero" class="idref" href="#not_lt_zero"><span class="id" title="lemma">not_lt_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_irr" class="idref" href="#lt_irr"><span class="id" title="lemma">lt_irr</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:15"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">lt_clean</span> := <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">try</span> (<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="var">ineq</span> : <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> ?<span class="id" title="var">a</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> |- <span class="id" title="var">_</span> ] ⇒ <span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Gamma0.Gamma0.html#not_lt_zero"><span class="id" title="lemma">not_lt_zero</span></a> <span class="id" title="var">ineq</span>);<span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|[<span class="id" title="var">ineq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> ?<span class="id" title="var">a</span> 0 |- <span class="id" title="var">_</span> ] ⇒ <span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.nlt_0_r"><span class="id" title="lemma">Nat.nlt_0_r</span></a> <span class="id" title="var">a</span>);<span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|[<span class="id" title="var">ref</span> : <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> ?<span class="id" title="var">a</span> ?<span class="id" title="var">a</span> |- <span class="id" title="var">_</span>] ⇒ <span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Gamma0.Gamma0.html#lt_irr"><span class="id" title="lemma">lt_irr</span></a> <span class="id" title="var">ref</span>);<span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|[<span class="id" title="var">ref</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> ?<span class="id" title="var">a</span> ?<span class="id" title="var">a</span> |- <span class="id" title="var">_</span>] ⇒ <span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Gamma0.Gamma0.html#lt_irr"><span class="id" title="lemma">lt_irr</span></a> <span class="id" title="var">ref</span>);<span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_zero_alpha" class="idref" href="#le_zero_alpha"><span class="id" title="lemma">le_zero_alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:16"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="psi_le_cons" class="idref" href="#psi_le_cons"><span class="id" title="lemma">psi_le_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a> <a id="beta:18" class="idref" href="#beta:18"><span class="id" title="binder">beta</span></a> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> <a id="gamma:20" class="idref" href="#gamma:20"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:17"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:18"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:18"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:20"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">psi_le_cons</span> <span class="id" title="var">le_zero_alpha</span>: <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_psi_term_le" class="idref" href="#le_psi_term_le"><span class="id" title="lemma">le_psi_term_le</span></a> <a id="alpha:21" class="idref" href="#alpha:21"><span class="id" title="binder">alpha</span></a> <a id="beta:22" class="idref" href="#beta:22"><span class="id" title="binder">beta</span></a>:  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:21"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:22"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#psi_term"><span class="id" title="definition">psi_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:21"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#psi_term"><span class="id" title="definition">psi_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:22"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_inv_nc" class="idref" href="#le_inv_nc"><span class="id" title="lemma">le_inv_nc</span></a> : <span class="id" title="keyword">∀</span> <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a> <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> <a id="c:26" class="idref" href="#c:26"><span class="id" title="binder">c</span></a>  <a id="n':27" class="idref" href="#n':27"><span class="id" title="binder">n'</span></a> <a id="c':28" class="idref" href="#c':28"><span class="id" title="binder">c'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:25"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:26"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':27"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c':28"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#n:25"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#n':27"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:25"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#n':27"><span class="id" title="variable">n'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:26"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c':28"><span class="id" title="variable">c'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_than_psi" class="idref" href="#lt_than_psi"><span class="id" title="lemma">lt_than_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> <a id="b:30" class="idref" href="#b:30"><span class="id" title="binder">b</span></a> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> <a id="c:32" class="idref" href="#c:32"><span class="id" title="binder">c</span></a> <a id="a':33" class="idref" href="#a':33"><span class="id" title="binder">a'</span></a> <a id="b':34" class="idref" href="#b':34"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:30"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:32"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a':33"><span class="id" title="variable">a'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b':34"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:29"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:30"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a':33"><span class="id" title="variable">a'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b':34"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ in order to establish trichotomy, we first use a measure on pair of
+    terms 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="lemmas_on_length" class="idref" href="#lemmas_on_length"><span class="id" title="section">lemmas_on_length</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_2" class="idref" href="#tricho_lt_2"><span class="id" title="lemma">tricho_lt_2</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:35" class="idref" href="#a1:35"><span class="id" title="binder">a1</span></a> <a id="a2:36" class="idref" href="#a2:36"><span class="id" title="binder">a2</span></a> <a id="b1:37" class="idref" href="#b1:37"><span class="id" title="binder">b1</span></a> <a id="b2:38" class="idref" href="#b2:38"><span class="id" title="binder">b2</span></a> <a id="n1:39" class="idref" href="#n1:39"><span class="id" title="binder">n1</span></a> <a id="n2:40" class="idref" href="#n2:40"><span class="id" title="binder">n2</span></a> <a id="r1:41" class="idref" href="#r1:41"><span class="id" title="binder">r1</span></a> <a id="r2:42" class="idref" href="#r2:42"><span class="id" title="binder">r2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:35"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:36"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:35"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:37"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:39"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r1:41"><span class="id" title="variable">r1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:36"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:38"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:40"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r2:42"><span class="id" title="variable">r2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_2'" class="idref" href="#tricho_lt_2'"><span class="id" title="lemma">tricho_lt_2'</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:43" class="idref" href="#a1:43"><span class="id" title="binder">a1</span></a> <a id="a2:44" class="idref" href="#a2:44"><span class="id" title="binder">a2</span></a> <a id="b1:45" class="idref" href="#b1:45"><span class="id" title="binder">b1</span></a> <a id="b2:46" class="idref" href="#b2:46"><span class="id" title="binder">b2</span></a> <a id="n1:47" class="idref" href="#n1:47"><span class="id" title="binder">n1</span></a> <a id="n2:48" class="idref" href="#n2:48"><span class="id" title="binder">n2</span></a> <a id="r1:49" class="idref" href="#r1:49"><span class="id" title="binder">r1</span></a> <a id="r2:50" class="idref" href="#r2:50"><span class="id" title="binder">r2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:45"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:44"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:46"><span class="id" title="variable">b2</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:43"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:45"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:47"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r1:49"><span class="id" title="variable">r1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:44"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:46"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:48"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r2:50"><span class="id" title="variable">r2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_3" class="idref" href="#tricho_lt_3"><span class="id" title="lemma">tricho_lt_3</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:51" class="idref" href="#a1:51"><span class="id" title="binder">a1</span></a> <a id="a2:52" class="idref" href="#a2:52"><span class="id" title="binder">a2</span></a> <a id="b1:53" class="idref" href="#b1:53"><span class="id" title="binder">b1</span></a> <a id="b2:54" class="idref" href="#b2:54"><span class="id" title="binder">b2</span></a> <a id="n1:55" class="idref" href="#n1:55"><span class="id" title="binder">n1</span></a> <a id="n2:56" class="idref" href="#n2:56"><span class="id" title="binder">n2</span></a> <a id="r1:57" class="idref" href="#r1:57"><span class="id" title="binder">r1</span></a> <a id="r2:58" class="idref" href="#r2:58"><span class="id" title="binder">r2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:53"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:54"><span class="id" title="variable">b2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>   <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:51"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:53"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:55"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r1:57"><span class="id" title="variable">r1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:52"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:54"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:56"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r2:58"><span class="id" title="variable">r2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_4" class="idref" href="#tricho_lt_4"><span class="id" title="lemma">tricho_lt_4</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:59" class="idref" href="#a1:59"><span class="id" title="binder">a1</span></a> <a id="a2:60" class="idref" href="#a2:60"><span class="id" title="binder">a2</span></a> <a id="b1:61" class="idref" href="#b1:61"><span class="id" title="binder">b1</span></a> <a id="b2:62" class="idref" href="#b2:62"><span class="id" title="binder">b2</span></a> <a id="n1:63" class="idref" href="#n1:63"><span class="id" title="binder">n1</span></a> <a id="n2:64" class="idref" href="#n2:64"><span class="id" title="binder">n2</span></a> <a id="r1:65" class="idref" href="#r1:65"><span class="id" title="binder">r1</span></a> <a id="r2:66" class="idref" href="#r2:66"><span class="id" title="binder">r2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:60"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:59"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:59"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:61"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:63"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r1:65"><span class="id" title="variable">r1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:60"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:62"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:64"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#r2:66"><span class="id" title="variable">r2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_4'" class="idref" href="#tricho_lt_4'"><span class="id" title="lemma">tricho_lt_4'</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:67" class="idref" href="#a1:67"><span class="id" title="binder">a1</span></a> <a id="a2:68" class="idref" href="#a2:68"><span class="id" title="binder">a2</span></a> <a id="b1:69" class="idref" href="#b1:69"><span class="id" title="binder">b1</span></a> <a id="b2:70" class="idref" href="#b2:70"><span class="id" title="binder">b2</span></a> <a id="n1:71" class="idref" href="#n1:71"><span class="id" title="binder">n1</span></a> <a id="n2:72" class="idref" href="#n2:72"><span class="id" title="binder">n2</span></a> <a id="c1:73" class="idref" href="#c1:73"><span class="id" title="binder">c1</span></a> <a id="c2:74" class="idref" href="#c2:74"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:67"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:69"><span class="id" title="variable">b1</span></a> 0 <a class="idref" href="hydras.Gamma0.Gamma0.html#c1:73"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:70"><span class="id" title="variable">b2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:67"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:69"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:71"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c1:73"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:68"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:70"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:72"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c2:74"><span class="id" title="variable">c2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_5" class="idref" href="#tricho_lt_5"><span class="id" title="lemma">tricho_lt_5</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:75" class="idref" href="#a1:75"><span class="id" title="binder">a1</span></a> <a id="a2:76" class="idref" href="#a2:76"><span class="id" title="binder">a2</span></a> <a id="b1:77" class="idref" href="#b1:77"><span class="id" title="binder">b1</span></a>  <a id="n1:78" class="idref" href="#n1:78"><span class="id" title="binder">n1</span></a> <a id="n2:79" class="idref" href="#n2:79"><span class="id" title="binder">n2</span></a> <a id="c1:80" class="idref" href="#c1:80"><span class="id" title="binder">c1</span></a> <a id="c2:81" class="idref" href="#c2:81"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:76"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:75"><span class="id" title="variable">a1</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:75"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:77"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:78"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c1:80"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:76"><span class="id" title="variable">a2</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:75"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:77"><span class="id" title="variable">b1</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)  <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:79"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c2:81"><span class="id" title="variable">c2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="tricho_lt_7" class="idref" href="#tricho_lt_7"><span class="id" title="lemma">tricho_lt_7</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:82" class="idref" href="#a1:82"><span class="id" title="binder">a1</span></a> <a id="b1:83" class="idref" href="#b1:83"><span class="id" title="binder">b1</span></a>  <a id="n1:84" class="idref" href="#n1:84"><span class="id" title="binder">n1</span></a>  <a id="c1:85" class="idref" href="#c1:85"><span class="id" title="binder">c1</span></a> <a id="c2:86" class="idref" href="#c2:86"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c1:85"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c2:86"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:82"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:83"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:84"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c1:85"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:82"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:83"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:84"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c2:86"><span class="id" title="variable">c2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#lemmas_on_length"><span class="id" title="section">lemmas_on_length</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">tricho_lt_7</span> <span class="id" title="var">tricho_lt_5</span> <span class="id" title="var">tricho_lt_4</span> <span class="id" title="var">tricho_lt_4'</span> <span class="id" title="var">tricho_lt_3</span> <span class="id" title="var">tricho_lt_2</span> <span class="id" title="var">tricho_lt_2</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="tricho_aux" class="idref" href="#tricho_aux"><span class="id" title="lemma">tricho_aux</span></a> (<a id="l:87" class="idref" href="#l:87"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">∀</span> <a id="t:88" class="idref" href="#t:88"><span class="id" title="binder">t</span></a> <a id="t':89" class="idref" href="#t':89"><span class="id" title="binder">t'</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:88"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':89"><span class="id" title="variable">t'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l:87"><span class="id" title="variable">l</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t:88"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':89"><span class="id" title="variable">t'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t:88"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':89"><span class="id" title="variable">t'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t':89"><span class="id" title="variable">t'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#t:88"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>. 
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> (<a id="t:90" class="idref" href="#t:90"><span class="id" title="binder">t</span></a> <a id="t':91" class="idref" href="#t':91"><span class="id" title="binder">t'</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t:90"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':91"><span class="id" title="variable">t'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t:90"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':91"><span class="id" title="variable">t'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t':91"><span class="id" title="variable">t'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#t:90"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt_ge_dec" class="idref" href="#lt_ge_dec"><span class="id" title="definition">lt_ge_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="t:92" class="idref" href="#t:92"><span class="id" title="binder">t</span></a> <a id="t':93" class="idref" href="#t':93"><span class="id" title="binder">t'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t:92"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':93"><span class="id" title="variable">t'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t':93"><span class="id" title="variable">t'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:92"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="compare_T2" class="idref" href="#compare_T2"><span class="id" title="instance">compare_T2</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> := <br/>
+<span class="id" title="keyword">fun</span> (<a id="t1:94" class="idref" href="#t1:94"><span class="id" title="binder">t1</span></a> <a id="t2:95" class="idref" href="#t2:95"><span class="id" title="binder">t2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) ⇒<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:94"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:95"><span class="id" title="variable">t2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inleft"><span class="id" title="constructor">inleft</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inleft"><span class="id" title="constructor">inleft</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inright"><span class="id" title="constructor">inright</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nfb" class="idref" href="#nfb"><span class="id" title="definition">nfb</span></a> (<a id="alpha:96" class="idref" href="#alpha:96"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:96"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nfb:97"><span class="id" title="definition">nfb</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#nfb:97"><span class="id" title="definition">nfb</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> ((<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">c</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><span class="id" title="var">a'</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nfb:97"><span class="id" title="definition">nfb</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nfb:97"><span class="id" title="definition">nfb</span></a> <span class="id" title="var">b</span>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#nfb:97"><span class="id" title="definition">nfb</span></a> <span class="id" title="var">c</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">tricho</span> <span class="id" title="var">t</span> <span class="id" title="var">t'</span> <span class="id" title="var">Hname</span> := <span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Gamma0.Gamma0.html#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> <span class="id" title="var">t</span> <span class="id" title="var">t'</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="tactic">intros</span> [<span class="id" title="var">Hname</span>|<span class="id" title="var">Hname</span>] | <span class="id" title="tactic">intro</span> <span class="id" title="var">Hname</span>].<br/>
+<span class="id" title="keyword">Tactic Notation</span> "tricho" <span class="id" title="keyword">constr</span>(<span class="id" title="var">t</span>) <span class="id" title="keyword">constr</span>(<span class="id" title="var">t'</span>) <span class="id" title="var">ident</span>(<span class="id" title="var">Hname</span>) := <span class="id" title="var">tricho</span> <span class="id" title="var">t</span> <span class="id" title="var">t'</span> <span class="id" title="var">Hname</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="trans_proof" class="idref" href="#trans_proof"><span class="id" title="section">trans_proof</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="trans_proof.a1" class="idref" href="#trans_proof.a1"><span class="id" title="variable">a1</span></a> <a id="trans_proof.b1" class="idref" href="#trans_proof.b1"><span class="id" title="variable">b1</span></a> <a id="trans_proof.c1" class="idref" href="#trans_proof.c1"><span class="id" title="variable">c1</span></a> <a id="trans_proof.a2" class="idref" href="#trans_proof.a2"><span class="id" title="variable">a2</span></a> <a id="trans_proof.b2" class="idref" href="#trans_proof.b2"><span class="id" title="variable">b2</span></a> <a id="trans_proof.c2" class="idref" href="#trans_proof.c2"><span class="id" title="variable">c2</span></a> <a id="trans_proof.a3" class="idref" href="#trans_proof.a3"><span class="id" title="variable">a3</span></a> <a id="trans_proof.b3" class="idref" href="#trans_proof.b3"><span class="id" title="variable">b3</span></a> <a id="trans_proof.c3" class="idref" href="#trans_proof.c3"><span class="id" title="variable">c3</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="trans_proof.n1" class="idref" href="#trans_proof.n1"><span class="id" title="variable">n1</span></a> <a id="trans_proof.n2" class="idref" href="#trans_proof.n2"><span class="id" title="variable">n2</span></a> <a id="trans_proof.n3" class="idref" href="#trans_proof.n3"><span class="id" title="variable">n3</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="trans_proof.H12" class="idref" href="#trans_proof.H12"><span class="id" title="variable">H12</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c2"><span class="id" title="variable">c2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="trans_proof.H23" class="idref" href="#trans_proof.H23"><span class="id" title="variable">H23</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c2"><span class="id" title="variable">c2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a3"><span class="id" title="variable">a3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b3"><span class="id" title="variable">b3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n3"><span class="id" title="variable">n3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c3"><span class="id" title="variable">c3</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="trans_proof.induc" class="idref" href="#trans_proof.induc"><span class="id" title="variable">induc</span></a> : <span class="id" title="keyword">∀</span> <a id="t:113" class="idref" href="#t:113"><span class="id" title="binder">t</span></a> <a id="t':114" class="idref" href="#t':114"><span class="id" title="binder">t'</span></a> <a id="t'':115" class="idref" href="#t'':115"><span class="id" title="binder">t''</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:113"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':114"><span class="id" title="variable">t'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t'':115"><span class="id" title="variable">t''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c1"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c2"><span class="id" title="variable">c2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a3"><span class="id" title="variable">a3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b3"><span class="id" title="variable">b3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n3"><span class="id" title="variable">n3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c3"><span class="id" title="variable">c3</span></a>))%<span class="id" title="var">nat</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:113"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':114"><span class="id" title="variable">t'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t':114"><span class="id" title="variable">t'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t'':115"><span class="id" title="variable">t''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:113"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t'':115"><span class="id" title="variable">t''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="trans_aux" class="idref" href="#trans_aux"><span class="id" title="lemma">trans_aux</span></a> :  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.a3"><span class="id" title="variable">a3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.b3"><span class="id" title="variable">b3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.n3"><span class="id" title="variable">n3</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof.c3"><span class="id" title="variable">c3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#trans_proof"><span class="id" title="section">trans_proof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_trans0" class="idref" href="#lt_trans0"><span class="id" title="lemma">lt_trans0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:117" class="idref" href="#n:117"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:118" class="idref" href="#t1:118"><span class="id" title="binder">t1</span></a> <a id="t2:119" class="idref" href="#t2:119"><span class="id" title="binder">t2</span></a> <a id="t3:120" class="idref" href="#t3:120"><span class="id" title="binder">t3</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:118"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:119"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t3:120"><span class="id" title="variable">t3</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:117"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:118"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:119"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:119"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t3:120"><span class="id" title="variable">t3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:118"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t3:120"><span class="id" title="variable">t3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_trans" class="idref" href="#lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:121" class="idref" href="#t1:121"><span class="id" title="binder">t1</span></a> <a id="t2:122" class="idref" href="#t2:122"><span class="id" title="binder">t2</span></a> <a id="t3:123" class="idref" href="#t3:123"><span class="id" title="binder">t3</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:121"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:122"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:122"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t3:123"><span class="id" title="variable">t3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:121"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t3:123"><span class="id" title="variable">t3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_lt_trans" class="idref" href="#le_lt_trans"><span class="id" title="lemma">le_lt_trans</span></a> <a id="alpha:124" class="idref" href="#alpha:124"><span class="id" title="binder">alpha</span></a> <a id="beta:125" class="idref" href="#beta:125"><span class="id" title="binder">beta</span></a> <a id="gamma:126" class="idref" href="#gamma:126"><span class="id" title="binder">gamma</span></a>:  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:124"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:125"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta:125"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:126"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:124"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:126"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span>  <a id="lt_le_trans" class="idref" href="#lt_le_trans"><span class="id" title="lemma">lt_le_trans</span></a>  <a id="alpha:127" class="idref" href="#alpha:127"><span class="id" title="binder">alpha</span></a> <a id="beta:128" class="idref" href="#beta:128"><span class="id" title="binder">beta</span></a> <a id="gamma:129" class="idref" href="#gamma:129"><span class="id" title="binder">gamma</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:127"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:128"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:128"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:129"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:127"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:129"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_trans" class="idref" href="#le_trans"><span class="id" title="lemma">le_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:130" class="idref" href="#alpha:130"><span class="id" title="binder">alpha</span></a> <a id="beta:131" class="idref" href="#beta:131"><span class="id" title="binder">beta</span></a> <a id="gamma:132" class="idref" href="#gamma:132"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:130"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:131"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta:131"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:132"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:130"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:132"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="psi_lt_head" class="idref" href="#psi_lt_head"><span class="id" title="lemma">psi_lt_head</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:133" class="idref" href="#alpha:133"><span class="id" title="binder">alpha</span></a> <a id="beta:134" class="idref" href="#beta:134"><span class="id" title="binder">beta</span></a> <a id="n:135" class="idref" href="#n:135"><span class="id" title="binder">n</span></a> <a id="gamma:136" class="idref" href="#gamma:136"><span class="id" title="binder">gamma</span></a>  <a id="alpha':137" class="idref" href="#alpha':137"><span class="id" title="binder">alpha'</span></a> <a id="beta':138" class="idref" href="#beta':138"><span class="id" title="binder">beta'</span></a> <a id="n':139" class="idref" href="#n':139"><span class="id" title="binder">n'</span></a> <a id="gamma':140" class="idref" href="#gamma':140"><span class="id" title="binder">gamma'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:133"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:134"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':137"><span class="id" title="variable">alpha'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta':138"><span class="id" title="variable">beta'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:133"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:134"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:135"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:136"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':137"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta':138"><span class="id" title="variable">beta'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':139"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma':140"><span class="id" title="variable">gamma'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_inv_tail" class="idref" href="#nf_inv_tail"><span class="id" title="lemma">nf_inv_tail</span></a> : <span class="id" title="keyword">∀</span> <a id="a:141" class="idref" href="#a:141"><span class="id" title="binder">a</span></a> <a id="b:142" class="idref" href="#b:142"><span class="id" title="binder">b</span></a> <a id="n:143" class="idref" href="#n:143"><span class="id" title="binder">n</span></a> <a id="c:144" class="idref" href="#c:144"><span class="id" title="binder">c</span></a> , <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:141"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:142"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:143"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:144"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#c:144"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:141"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:142"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_beta_psi" class="idref" href="#lt_beta_psi"><span class="id" title="lemma">lt_beta_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:145" class="idref" href="#beta:145"><span class="id" title="binder">beta</span></a> <a id="alpha:146" class="idref" href="#alpha:146"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:145"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:146"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:145"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_beta_cons" class="idref" href="#lt_beta_cons"><span class="id" title="lemma">lt_beta_cons</span></a> :  <span class="id" title="keyword">∀</span> <a id="alpha:147" class="idref" href="#alpha:147"><span class="id" title="binder">alpha</span></a> <a id="beta:148" class="idref" href="#beta:148"><span class="id" title="binder">beta</span></a> <a id="n:149" class="idref" href="#n:149"><span class="id" title="binder">n</span></a> <a id="gamma:150" class="idref" href="#gamma:150"><span class="id" title="binder">gamma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta:148"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:147"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:148"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:149"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:150"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_alpha_psi" class="idref" href="#lt_alpha_psi"><span class="id" title="lemma">lt_alpha_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:151" class="idref" href="#alpha:151"><span class="id" title="binder">alpha</span></a> <a id="beta:152" class="idref" href="#beta:152"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:151"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:151"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:152"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_alpha_cons" class="idref" href="#lt_alpha_cons"><span class="id" title="lemma">lt_alpha_cons</span></a> :  <span class="id" title="keyword">∀</span> <a id="alpha:153" class="idref" href="#alpha:153"><span class="id" title="binder">alpha</span></a> <a id="beta:154" class="idref" href="#beta:154"><span class="id" title="binder">beta</span></a> <a id="n:155" class="idref" href="#n:155"><span class="id" title="binder">n</span></a> <a id="gamma:156" class="idref" href="#gamma:156"><span class="id" title="binder">gamma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:153"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:153"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:154"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:155"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:156"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">lt_beta_cons</span> <span class="id" title="var">lt_alpha_cons</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_cons_tail" class="idref" href="#le_cons_tail"><span class="id" title="lemma">le_cons_tail</span></a> <a id="alpha:157" class="idref" href="#alpha:157"><span class="id" title="binder">alpha</span></a> <a id="beta:158" class="idref" href="#beta:158"><span class="id" title="binder">beta</span></a> <a id="n:159" class="idref" href="#n:159"><span class="id" title="binder">n</span></a> <a id="gamma:160" class="idref" href="#gamma:160"><span class="id" title="binder">gamma</span></a> <a id="gamma':161" class="idref" href="#gamma':161"><span class="id" title="binder">gamma'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:160"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma':161"><span class="id" title="variable">gamma'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:158"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:159"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:160"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:157"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:158"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:159"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma':161"><span class="id" title="variable">gamma'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab113"></a><h2 class="section">terms in normal form</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_fin_inv" class="idref" href="#nf_fin_inv"><span class="id" title="lemma">nf_fin_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="gamma:162" class="idref" href="#gamma:162"><span class="id" title="binder">gamma</span></a> <a id="n:163" class="idref" href="#n:163"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:163"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:162"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:162"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_tail0" class="idref" href="#lt_tail0"><span class="id" title="lemma">lt_tail0</span></a>: <span class="id" title="keyword">∀</span> <a id="c:164" class="idref" href="#c:164"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:164"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:164"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gtail"><span class="id" title="definition">gtail</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:164"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:164"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_tail" class="idref" href="#lt_tail"><span class="id" title="lemma">lt_tail</span></a>: <span class="id" title="keyword">∀</span> <a id="a:165" class="idref" href="#a:165"><span class="id" title="binder">a</span></a> <a id="b:166" class="idref" href="#b:166"><span class="id" title="binder">b</span></a> <a id="n:167" class="idref" href="#n:167"><span class="id" title="binder">n</span></a> <a id="c:168" class="idref" href="#c:168"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:165"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:166"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:167"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:168"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#c:168"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:165"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:166"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:167"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:168"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_one_cons" class="idref" href="#le_one_cons"><span class="id" title="lemma">le_one_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="a:169" class="idref" href="#a:169"><span class="id" title="binder">a</span></a> <a id="b:170" class="idref" href="#b:170"><span class="id" title="binder">b</span></a> <a id="n:171" class="idref" href="#n:171"><span class="id" title="binder">n</span></a> <a id="c:172" class="idref" href="#c:172"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:169"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:170"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:171"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:172"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">le_one_cons</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fin_lt_omega" class="idref" href="#fin_lt_omega"><span class="id" title="lemma">fin_lt_omega</span></a> : <span class="id" title="keyword">∀</span> <a id="n:173" class="idref" href="#n:173"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_lt_epsilon0" class="idref" href="#omega_lt_epsilon0"><span class="id" title="lemma">omega_lt_epsilon0</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_lt_epsilon" class="idref" href="#omega_lt_epsilon"><span class="id" title="lemma">omega_lt_epsilon</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:174" class="idref" href="#alpha:174"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:174"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_one_inv" class="idref" href="#lt_one_inv"><span class="id" title="lemma">lt_one_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:175" class="idref" href="#alpha:175"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:175"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:175"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_cons_omega_inv" class="idref" href="#lt_cons_omega_inv"><span class="id" title="lemma">lt_cons_omega_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:176" class="idref" href="#alpha:176"><span class="id" title="binder">alpha</span></a> <a id="beta:177" class="idref" href="#beta:177"><span class="id" title="binder">beta</span></a> <a id="n:178" class="idref" href="#n:178"><span class="id" title="binder">n</span></a> <a id="gamma:179" class="idref" href="#gamma:179"><span class="id" title="binder">gamma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:176"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:177"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:178"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:179"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:176"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:177"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:178"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:179"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:176"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:177"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:179"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_inv" class="idref" href="#lt_omega_inv"><span class="id" title="lemma">lt_omega_inv</span></a> <a id="alpha:180" class="idref" href="#alpha:180"><span class="id" title="binder">alpha</span></a> :  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:180"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:180"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:181" class="idref" href="#n:181"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:180"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:181"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_is_fin" class="idref" href="#lt_omega_is_fin"><span class="id" title="lemma">lt_omega_is_fin</span></a> <a id="alpha:182" class="idref" href="#alpha:182"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:182"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:182"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_finite"><span class="id" title="inductive">is_finite</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:182"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_compat" class="idref" href="#lt_compat"><span class="id" title="lemma">lt_compat</span></a> (<a id="n:183" class="idref" href="#n:183"><span class="id" title="binder">n</span></a> <a id="p:184" class="idref" href="#p:184"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):   <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:183"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:184"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#n:183"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:184"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_compatR" class="idref" href="#lt_compatR"><span class="id" title="lemma">lt_compatR</span></a> (<a id="n:185" class="idref" href="#n:185"><span class="id" title="binder">n</span></a> <a id="p:186" class="idref" href="#p:186"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):  (<a class="idref" href="hydras.Gamma0.Gamma0.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#p:186"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:185"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:186"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_is_finite" class="idref" href="#finite_is_finite"><span class="id" title="lemma">finite_is_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="n:187" class="idref" href="#n:187"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#is_finite"><span class="id" title="inductive">is_finite</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:187"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_finite_finite" class="idref" href="#is_finite_finite"><span class="id" title="lemma">is_finite_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:188" class="idref" href="#alpha:188"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#is_finite"><span class="id" title="inductive">is_finite</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:188"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:189" class="idref" href="#n:189"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:188"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:189"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> <a id="alpha:190" class="idref" href="#alpha:190"><span class="id" title="binder">alpha</span></a> <a id="beta:191" class="idref" href="#beta:191"><span class="id" title="binder">beta</span></a> : <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:190"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:191"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:190"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:191"><span class="id" title="variable">beta</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:190"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:191"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:191"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:190"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> <a id="alpha:192" class="idref" href="#alpha:192"><span class="id" title="binder">alpha</span></a> <a id="beta:193" class="idref" href="#beta:193"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompareSpec"><span class="id" title="inductive">CompareSpec</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:192"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:193"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:192"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:193"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:193"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:192"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:192"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:193"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_Lt" class="idref" href="#compare_Lt"><span class="id" title="lemma">compare_Lt</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:194" class="idref" href="#alpha:194"><span class="id" title="binder">alpha</span></a> <a id="beta:195" class="idref" href="#beta:195"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:194"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:195"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:194"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:195"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Compare with the proof of <span class="inlinecode"><span class="id" title="var">T2.Ex6</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex6" class="idref" href="#Ex6"><span class="id" title="definition">Ex6</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> 1 0 12 <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>0<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_Eq" class="idref" href="#compare_Eq"><span class="id" title="lemma">compare_Eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="alpha:196" class="idref" href="#alpha:196"><span class="id" title="binder">alpha</span></a> <a id="beta:197" class="idref" href="#beta:197"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>), <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:196"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:197"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:196"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:197"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_Gt" class="idref" href="#compare_Gt"><span class="id" title="lemma">compare_Gt</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:198" class="idref" href="#alpha:198"><span class="id" title="binder">alpha</span></a> <a id="beta:199" class="idref" href="#beta:199"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:198"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:199"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta:199"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:198"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#compare_Gt"><span class="id" title="lemma">compare_Gt</span></a> [<span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>].<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#compare_Lt"><span class="id" title="lemma">compare_Lt</span></a> [<span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>].<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#compare_Eq"><span class="id" title="lemma">compare_Eq</span></a> [<span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>].<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">compare_Eq</span> <span class="id" title="var">compare_Lt</span> <span class="id" title="var">compare_Gt</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_rw_lt" class="idref" href="#compare_rw_lt"><span class="id" title="lemma">compare_rw_lt</span></a>  <a id="alpha:200" class="idref" href="#alpha:200"><span class="id" title="binder">alpha</span></a> <a id="beta:201" class="idref" href="#beta:201"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:200"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:201"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:200"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:201"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_rw_eq" class="idref" href="#compare_rw_eq"><span class="id" title="lemma">compare_rw_eq</span></a> (<a id="alpha:202" class="idref" href="#alpha:202"><span class="id" title="binder">alpha</span></a> <a id="beta:203" class="idref" href="#beta:203"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>):  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:202"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:203"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:202"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:203"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_rw_gt" class="idref" href="#compare_rw_gt"><span class="id" title="lemma">compare_rw_gt</span></a>  <a id="alpha:204" class="idref" href="#alpha:204"><span class="id" title="binder">alpha</span></a> <a id="beta:205" class="idref" href="#beta:205"><span class="id" title="binder">beta</span></a>:  <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:205"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:204"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:204"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:205"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ plus is defined here, because it requires decidable comparison 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="plus" class="idref" href="#plus"><span class="id" title="definition">plus</span></a> (<a id="t1:206" class="idref" href="#t1:206"><span class="id" title="binder">t1</span></a> <a id="t2:207" class="idref" href="#t2:207"><span class="id" title="binder">t2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t1</span>} : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:206"><span class="id" title="variable">t1</span></a>,<a class="idref" href="hydras.Gamma0.Gamma0.html#t2:207"><span class="id" title="variable">t2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>, <span class="id" title="var">y</span>  ⇒ <span class="id" title="var">y</span><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">x</span>, <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span>, <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<span class="id" title="var">c</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">b'</span> <span class="id" title="var">n'</span> <span class="id" title="var">c'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>  ⇒ <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>(<span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><span class="id" title="var">n'</span>)) <span class="id" title="var">c'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">where</span> <a id="43589bbdaffd7406c3f135d34490f8e8" class="idref" href="#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">&quot;</span></a>alpha + beta" := (<a class="idref" href="hydras.Gamma0.Gamma0.html#plus:208"><span class="id" title="definition">plus</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>): <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex7" class="idref" href="#Ex7"><span class="id" title="definition">Ex7</span></a> : 3 <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.   
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_alpha_0" class="idref" href="#plus_alpha_0"><span class="id" title="lemma">plus_alpha_0</span></a> (<a id="alpha:211" class="idref" href="#alpha:211"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>): <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:211"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:211"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ" class="idref" href="#lt_succ"><span class="id" title="lemma">lt_succ</span></a> (<a id="alpha:212" class="idref" href="#alpha:212"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>): <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:212"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:212"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lt_succ_le" class="idref" href="#lt_succ_le"><span class="id" title="lemma">lt_succ_le</span></a> <a id="alpha:213" class="idref" href="#alpha:213"><span class="id" title="binder">alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:214" class="idref" href="#beta:214"><span class="id" title="binder">beta</span></a>,  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:213"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:214"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:214"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:213"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:214"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_lt_le" class="idref" href="#succ_lt_le"><span class="id" title="lemma">succ_lt_le</span></a> : <span class="id" title="keyword">∀</span> <a id="a:215" class="idref" href="#a:215"><span class="id" title="binder">a</span></a> <a id="b:216" class="idref" href="#b:216"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:215"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:216"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:215"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:216"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:215"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:216"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_of_cons" class="idref" href="#succ_of_cons"><span class="id" title="lemma">succ_of_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="a:217" class="idref" href="#a:217"><span class="id" title="binder">a</span></a> <a id="b:218" class="idref" href="#b:218"><span class="id" title="binder">b</span></a> <a id="n:219" class="idref" href="#n:219"><span class="id" title="binder">n</span></a> <a id="c:220" class="idref" href="#c:220"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:217"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:218"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:217"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:218"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:219"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:220"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:217"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:218"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:219"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:220"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab114"></a><h1 class="section">Well-foundedness (with rpo) (Evelyne Contejean)</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="subterm" class="idref" href="#subterm"><span class="id" title="inductive">subterm</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="subterm_a" class="idref" href="#subterm_a"><span class="id" title="constructor">subterm_a</span></a> : <span class="id" title="keyword">∀</span> <a id="a:223" class="idref" href="#a:223"><span class="id" title="binder">a</span></a> <a id="b:224" class="idref" href="#b:224"><span class="id" title="binder">b</span></a> <a id="n:225" class="idref" href="#n:225"><span class="id" title="binder">n</span></a> <a id="c:226" class="idref" href="#c:226"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:223"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#a:223"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:224"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:225"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:226"><span class="id" title="variable">c</span></a>)<br/>
+| <a id="subterm_b" class="idref" href="#subterm_b"><span class="id" title="constructor">subterm_b</span></a> : <span class="id" title="keyword">∀</span> <a id="a:227" class="idref" href="#a:227"><span class="id" title="binder">a</span></a> <a id="b:228" class="idref" href="#b:228"><span class="id" title="binder">b</span></a> <a id="n:229" class="idref" href="#n:229"><span class="id" title="binder">n</span></a> <a id="c:230" class="idref" href="#c:230"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:228"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:227"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:228"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:229"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:230"><span class="id" title="variable">c</span></a>)<br/>
+| <a id="subterm_c" class="idref" href="#subterm_c"><span class="id" title="constructor">subterm_c</span></a> : <span class="id" title="keyword">∀</span> <a id="a:231" class="idref" href="#a:231"><span class="id" title="binder">a</span></a> <a id="b:232" class="idref" href="#b:232"><span class="id" title="binder">b</span></a> <a id="n:233" class="idref" href="#n:233"><span class="id" title="binder">n</span></a> <a id="c:234" class="idref" href="#c:234"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:234"><span class="id" title="variable">c</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:231"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:232"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:233"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:234"><span class="id" title="variable">c</span></a>)<br/>
+| <a id="subterm_trans" class="idref" href="#subterm_trans"><span class="id" title="constructor">subterm_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="t:235" class="idref" href="#t:235"><span class="id" title="binder">t</span></a> <a id="t1:236" class="idref" href="#t1:236"><span class="id" title="binder">t1</span></a> <a id="t2:237" class="idref" href="#t2:237"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:235"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:236"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:236"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:237"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#subterm:221"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:235"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:237"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_subterm" class="idref" href="#nf_subterm"><span class="id" title="lemma">nf_subterm</span></a> <a id="alpha:238" class="idref" href="#alpha:238"><span class="id" title="binder">alpha</span></a> <a id="beta:239" class="idref" href="#beta:239"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:238"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:239"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:239"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:238"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="subterm_lt" class="idref" href="#subterm_lt"><span class="id" title="lemma">subterm_lt</span></a> <a id="alpha:240" class="idref" href="#alpha:240"><span class="id" title="binder">alpha</span></a> <a id="beta:241" class="idref" href="#beta:241"><span class="id" title="binder">beta</span></a>:  <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:240"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:241"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:241"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:240"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:241"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">subtermtac</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;[|- <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> ?<span class="id" title="var">t1</span> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">t2</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">t3</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span> 1<br/>
+&nbsp;&nbsp;| [|- <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> ?<span class="id" title="var">t2</span> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">t2</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">t3</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span> 2<br/>
+&nbsp;&nbsp;| [|- <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> ?<span class="id" title="var">t3</span> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">t2</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">t3</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span> 3<br/>
+&nbsp;&nbsp;| [|- <a class="idref" href="hydras.Gamma0.Gamma0.html#subterm"><span class="id" title="inductive">subterm</span></a> ?<span class="id" title="var">t4</span> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> ?<span class="id" title="var">t1</span> ?<span class="id" title="var">t2</span> ?<span class="id" title="var">n</span> ?<span class="id" title="var">t3</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;((<span class="id" title="tactic">constructor</span> 4 <span class="id" title="keyword">with</span> <span class="id" title="var">t1</span>; <span class="id" title="var">subtermtac</span>)     ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">constructor</span> 4 <span class="id" title="keyword">with</span> <span class="id" title="var">t2</span>; <span class="id" title="var">subtermtac</span>)       ||<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">constructor</span> 4 <span class="id" title="keyword">with</span> <span class="id" title="var">t3</span>; <span class="id" title="var">subtermtac</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span>  <a id="Gamma0_sig" class="idref" href="#Gamma0_sig"><span class="id" title="module">Gamma0_sig</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Signature"><span class="id" title="module">Signature</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Gamma0_sig.symb0" class="idref" href="#Gamma0_sig.symb0"><span class="id" title="inductive">symb0</span></a> : <span class="id" title="keyword">Set</span> := <a id="Gamma0_sig.nat_0" class="idref" href="#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a> | <a id="Gamma0_sig.nat_S" class="idref" href="#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> | <a id="Gamma0_sig.ord_zero" class="idref" href="#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> | <a id="Gamma0_sig.ord_psi" class="idref" href="#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a> | <a id="Gamma0_sig.ord_cons" class="idref" href="#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Gamma0_sig.symb" class="idref" href="#Gamma0_sig.symb"><span class="id" title="definition">symb</span></a> := <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb0"><span class="id" title="inductive">symb0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Gamma0_sig.eq_symbol_dec" class="idref" href="#Gamma0_sig.eq_symbol_dec"><span class="id" title="lemma">eq_symbol_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="f1:244" class="idref" href="#f1:244"><span class="id" title="binder">f1</span></a> <a id="f2:245" class="idref" href="#f2:245"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb"><span class="id" title="definition">symb</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#f1:244"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#f2:245"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#f1:244"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#f2:245"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The arity of a symbol contains also the information about built-in theories as in CiME 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Gamma0_sig.arity_type" class="idref" href="#Gamma0_sig.arity_type"><span class="id" title="inductive">arity_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Gamma0_sig.AC" class="idref" href="#Gamma0_sig.AC"><span class="id" title="constructor">AC</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#arity_type:246"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0_sig.C" class="idref" href="#Gamma0_sig.C"><span class="id" title="constructor">C</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#arity_type:246"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0_sig.Free" class="idref" href="#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#arity_type:246"><span class="id" title="inductive">arity_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Gamma0_sig.arity" class="idref" href="#Gamma0_sig.arity"><span class="id" title="definition">arity</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb"><span class="id" title="definition">symb</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.arity_type"><span class="id" title="inductive">arity_type</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="f:248" class="idref" href="#f:248"><span class="id" title="binder">f</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#f:248"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free"><span class="id" title="constructor">Free</span></a> 3<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig"><span class="id" title="module">Gamma0_sig</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab115"></a><h2 class="section">Module Type Variables.</h2>
+
+ There are almost no assumptions, except a decidable equality. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Vars" class="idref" href="#Vars"><span class="id" title="module">Vars</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Variables"><span class="id" title="module">Variables</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Vars.empty_set" class="idref" href="#Vars.empty_set"><span class="id" title="inductive">empty_set</span></a> : <span class="id" title="keyword">Set</span> := .<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Vars.var" class="idref" href="#Vars.var"><span class="id" title="definition">var</span></a> := <a class="idref" href="hydras.Gamma0.Gamma0.html#Vars.empty_set"><span class="id" title="inductive">empty_set</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Vars.eq_variable_dec" class="idref" href="#Vars.eq_variable_dec"><span class="id" title="lemma">eq_variable_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="v1:252" class="idref" href="#v1:252"><span class="id" title="binder">v1</span></a> <a id="v2:253" class="idref" href="#v2:253"><span class="id" title="binder">v2</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#Vars.var"><span class="id" title="definition">var</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#v1:252"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#v2:253"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#v1:252"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#v2:253"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#Vars"><span class="id" title="module">Vars</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span>  <a id="Gamma0_prec" class="idref" href="#Gamma0_prec"><span class="id" title="module">Gamma0_prec</span></a> &lt;: <a class="idref" href="hydras.rpo.rpo.html#Precedence"><span class="id" title="module">Precedence</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Gamma0_prec.A" class="idref" href="#Gamma0_prec.A"><span class="id" title="definition">A</span></a> : <span class="id" title="keyword">Set</span> := <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb"><span class="id" title="definition">Gamma0_sig.symb</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig"><span class="id" title="module">Gamma0_sig</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Gamma0_prec.prec" class="idref" href="#Gamma0_prec.prec"><span class="id" title="definition">prec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.A"><span class="id" title="definition">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="f:254" class="idref" href="#f:254"><span class="id" title="binder">f</span></a> <a id="g:255" class="idref" href="#g:255"><span class="id" title="binder">g</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#f:254"><span class="id" title="variable">f</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#g:255"><span class="id" title="variable">g</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0"><span class="id" title="constructor">nat_0</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero"><span class="id" title="constructor">ord_zero</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S"><span class="id" title="constructor">nat_S</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons"><span class="id" title="constructor">ord_cons</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi"><span class="id" title="constructor">ord_psi</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Gamma0_prec.status_type" class="idref" href="#Gamma0_prec.status_type"><span class="id" title="inductive">status_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Gamma0_prec.Lex" class="idref" href="#Gamma0_prec.Lex"><span class="id" title="constructor">Lex</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#status_type:258"><span class="id" title="inductive">status_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Gamma0_prec.Mul" class="idref" href="#Gamma0_prec.Mul"><span class="id" title="constructor">Mul</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#status_type:258"><span class="id" title="inductive">status_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Gamma0_prec.status" class="idref" href="#Gamma0_prec.status"><span class="id" title="definition">status</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.A"><span class="id" title="definition">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.status_type"><span class="id" title="inductive">status_type</span></a> := <span class="id" title="keyword">fun</span> <a id="f:260" class="idref" href="#f:260"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.Lex"><span class="id" title="constructor">Lex</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Gamma0_prec.prec_dec" class="idref" href="#Gamma0_prec.prec_dec"><span class="id" title="lemma">prec_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:261" class="idref" href="#a1:261"><span class="id" title="binder">a1</span></a> <a id="a2:262" class="idref" href="#a2:262"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.A"><span class="id" title="definition">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:261"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:262"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a1:261"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a2:262"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Gamma0_prec.prec_antisym" class="idref" href="#Gamma0_prec.prec_antisym"><span class="id" title="lemma">prec_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="s:263" class="idref" href="#s:263"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec"><span class="id" title="definition">prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#s:263"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#s:263"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Gamma0_prec.prec_transitive" class="idref" href="#Gamma0_prec.prec_transitive"><span class="id" title="lemma">prec_transitive</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec"><span class="id" title="definition">prec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec"><span class="id" title="module">Gamma0_prec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Gamma0_alg" class="idref" href="#Gamma0_alg"><span class="id" title="module">Gamma0_alg</span></a> &lt;: <a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">Term</span></a> := <a class="idref" href="hydras.rpo.term.html#Make"><span class="id" title="module">term.Make</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig"><span class="id" title="module">Gamma0_sig</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#Vars"><span class="id" title="module">Vars</span></a>).<br/>
+<span class="id" title="keyword">Module</span> <a id="Gamma0_rpo" class="idref" href="#Gamma0_rpo"><span class="id" title="module">Gamma0_rpo</span></a> &lt;: <a class="idref" href="hydras.rpo.rpo.html#RPO"><span class="id" title="module">RPO</span></a> := <a class="idref" href="hydras.rpo.rpo.html#Make"><span class="id" title="module">rpo.Make</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_alg"><span class="id" title="module">Gamma0_alg</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_prec"><span class="id" title="module">Gamma0_prec</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_alg"><span class="id" title="module">Gamma0_alg</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_rpo"><span class="id" title="module">Gamma0_rpo</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#Gamma0_sig"><span class="id" title="module">Gamma0_sig</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nat_2_term" class="idref" href="#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> (<a id="n:264" class="idref" href="#n:264"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Gamma0.Gamma0.html#term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:264"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#nat_0"><span class="id" title="constructor">nat_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#nat_S"><span class="id" title="constructor">nat_S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term:265"><span class="id" title="definition">nat_2_term</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab116"></a><h2 class="section">Every (representation of a) natural number is less than</h2>
+
+ a non zero ordinal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_lt_cons" class="idref" href="#nat_lt_cons"><span class="id" title="lemma">nat_lt_cons</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:267" class="idref" href="#n:267"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="t:268" class="idref" href="#t:268"><span class="id" title="binder">t</span></a> <a id="p:269" class="idref" href="#p:269"><span class="id" title="binder">p</span></a>  <a id="c:270" class="idref" href="#c:270"><span class="id" title="binder">c</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:267"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#t:268"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#p:269"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#c:270"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_lt_psi" class="idref" href="#nat_lt_psi"><span class="id" title="lemma">nat_lt_psi</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:271" class="idref" href="#n:271"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="a:272" class="idref" href="#a:272"><span class="id" title="binder">a</span></a> <a id="b:273" class="idref" href="#b:273"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:271"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#a:272"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:273"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="rpo_trans" class="idref" href="#rpo_trans"><span class="id" title="lemma">rpo_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="t:274" class="idref" href="#t:274"><span class="id" title="binder">t</span></a> <a id="t1:275" class="idref" href="#t1:275"><span class="id" title="binder">t1</span></a> <a id="t2:276" class="idref" href="#t2:276"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:274"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:275"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:275"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:276"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t:274"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t2:276"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T2_2_term" class="idref" href="#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> (<a id="a:277" class="idref" href="#a:277"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="hydras.Gamma0.Gamma0.html#term"><span class="id" title="inductive">term</span></a> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:277"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_zero"><span class="id" title="constructor">ord_zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;|<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term:278"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:277"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term:278"><span class="id" title="definition">T2_2_term</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;|<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term:278"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:277"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term:278"><span class="id" title="definition">T2_2_term</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term:278"><span class="id" title="definition">T2_2_term</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T2_size" class="idref" href="#T2_size"><span class="id" title="definition">T2_size</span></a> (<a id="o:280" class="idref" href="#o:280"><span class="id" title="binder">o</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>):<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:280"><span class="id" title="variable">o</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size:281"><span class="id" title="definition">T2_size</span></a> <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size:281"><span class="id" title="definition">T2_size</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size:281"><span class="id" title="definition">T2_size</span></a> <span class="id" title="var">c</span>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T2_size1" class="idref" href="#T2_size1"><span class="id" title="lemma">T2_size1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:283" class="idref" href="#a:283"><span class="id" title="binder">a</span></a> <a id="b:284" class="idref" href="#b:284"><span class="id" title="binder">b</span></a> <a id="n:285" class="idref" href="#n:285"><span class="id" title="binder">n</span></a> <a id="c:286" class="idref" href="#c:286"><span class="id" title="binder">c</span></a>, (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:283"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:284"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:285"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:286"><span class="id" title="variable">c</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T2_size2" class="idref" href="#T2_size2"><span class="id" title="lemma">T2_size2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:287" class="idref" href="#a:287"><span class="id" title="binder">a</span></a> <a id="b:288" class="idref" href="#b:288"><span class="id" title="binder">b</span></a> <a id="n:289" class="idref" href="#n:289"><span class="id" title="binder">n</span></a> <a id="c:290" class="idref" href="#c:290"><span class="id" title="binder">c</span></a> , (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:287"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:287"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:288"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:289"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:290"><span class="id" title="variable">c</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T2_size3" class="idref" href="#T2_size3"><span class="id" title="lemma">T2_size3</span></a> : <span class="id" title="keyword">∀</span> <a id="a:291" class="idref" href="#a:291"><span class="id" title="binder">a</span></a> <a id="b:292" class="idref" href="#b:292"><span class="id" title="binder">b</span></a> <a id="n:293" class="idref" href="#n:293"><span class="id" title="binder">n</span></a> <a id="c:294" class="idref" href="#c:294"><span class="id" title="binder">c</span></a> , (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:292"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:291"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:292"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:293"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:294"><span class="id" title="variable">c</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T2_size4" class="idref" href="#T2_size4"><span class="id" title="lemma">T2_size4</span></a> : <span class="id" title="keyword">∀</span> <a id="a:295" class="idref" href="#a:295"><span class="id" title="binder">a</span></a> <a id="b:296" class="idref" href="#b:296"><span class="id" title="binder">b</span></a> <a id="n:297" class="idref" href="#n:297"><span class="id" title="binder">n</span></a> <a id="c:298" class="idref" href="#c:298"><span class="id" title="binder">c</span></a> , (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:298"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:295"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:296"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:297"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:298"><span class="id" title="variable">c</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T2_size1</span> <span class="id" title="var">T2_size2</span> <span class="id" title="var">T2_size3</span> <span class="id" title="var">T2_size4</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+let us recall subterm properties on T2 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_subterm1" class="idref" href="#lt_subterm1"><span class="id" title="lemma">lt_subterm1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:299" class="idref" href="#a:299"><span class="id" title="binder">a</span></a> <a id="a':300" class="idref" href="#a':300"><span class="id" title="binder">a'</span></a>  <a id="n':301" class="idref" href="#n':301"><span class="id" title="binder">n'</span></a>  <a id="b':302" class="idref" href="#b':302"><span class="id" title="binder">b'</span></a> <a id="c':303" class="idref" href="#c':303"><span class="id" title="binder">c'</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#a:299"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a':300"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:299"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a':300"><span class="id" title="variable">a'</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#b':302"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':301"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c':303"><span class="id" title="variable">c'</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nat_lt_cons</span> <span class="id" title="var">lt_subterm1</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_2_term_mono" class="idref" href="#nat_2_term_mono"><span class="id" title="lemma">nat_2_term_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="n:304" class="idref" href="#n:304"><span class="id" title="binder">n</span></a> <a id="n':305" class="idref" href="#n':305"><span class="id" title="binder">n'</span></a>, (<a class="idref" href="hydras.Gamma0.Gamma0.html#n:304"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':305"><span class="id" title="variable">n'</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:304"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':305"><span class="id" title="variable">n'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T2_size_psi" class="idref" href="#T2_size_psi"><span class="id" title="lemma">T2_size_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="a:306" class="idref" href="#a:306"><span class="id" title="binder">a</span></a> <a id="b:307" class="idref" href="#b:307"><span class="id" title="binder">b</span></a> <a id="n:308" class="idref" href="#n:308"><span class="id" title="binder">n</span></a> <a id="c:309" class="idref" href="#c:309"><span class="id" title="binder">c</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:306"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:307"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:306"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:307"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:308"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:309"><span class="id" title="variable">c</span></a>))%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_2_2" class="idref" href="#rpo_2_2"><span class="id" title="lemma">rpo_2_2</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:310" class="idref" href="#ta1:310"><span class="id" title="binder">ta1</span></a> <a id="ta2:311" class="idref" href="#ta2:311"><span class="id" title="binder">ta2</span></a> <a id="tb1:312" class="idref" href="#tb1:312"><span class="id" title="binder">tb1</span></a> <a id="tb2:313" class="idref" href="#tb2:313"><span class="id" title="binder">tb2</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:310"><span class="id" title="variable">ta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:311"><span class="id" title="variable">ta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:312"><span class="id" title="variable">tb1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:311"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:313"><span class="id" title="variable">tb2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:310"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:312"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:311"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:313"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_2_3" class="idref" href="#rpo_2_3"><span class="id" title="lemma">rpo_2_3</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:314" class="idref" href="#ta1:314"><span class="id" title="binder">ta1</span></a> <a id="ta2:315" class="idref" href="#ta2:315"><span class="id" title="binder">ta2</span></a> <a id="tb1:316" class="idref" href="#tb1:316"><span class="id" title="binder">tb1</span></a> <a id="tb2:317" class="idref" href="#tb2:317"><span class="id" title="binder">tb2</span></a> <a id="n1:318" class="idref" href="#n1:318"><span class="id" title="binder">n1</span></a> <a id="tc1:319" class="idref" href="#tc1:319"><span class="id" title="binder">tc1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:314"><span class="id" title="variable">ta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:315"><span class="id" title="variable">ta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:316"><span class="id" title="variable">tb1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:315"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:317"><span class="id" title="variable">tb2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:319"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:314"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:316"><span class="id" title="variable">tb1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:314"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:316"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:318"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:319"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:315"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:317"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_2_1" class="idref" href="#rpo_2_1"><span class="id" title="lemma">rpo_2_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:320" class="idref" href="#ta1:320"><span class="id" title="binder">ta1</span></a> <a id="ta2:321" class="idref" href="#ta2:321"><span class="id" title="binder">ta2</span></a> <a id="tb1:322" class="idref" href="#tb1:322"><span class="id" title="binder">tb1</span></a> <a id="tb2:323" class="idref" href="#tb2:323"><span class="id" title="binder">tb2</span></a> <a id="n1:324" class="idref" href="#n1:324"><span class="id" title="binder">n1</span></a> <a id="n2:325" class="idref" href="#n2:325"><span class="id" title="binder">n2</span></a> <a id="tc1:326" class="idref" href="#tc1:326"><span class="id" title="binder">tc1</span></a> <a id="tc2:327" class="idref" href="#tc2:327"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:320"><span class="id" title="variable">ta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:321"><span class="id" title="variable">ta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:322"><span class="id" title="variable">tb1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:321"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:323"><span class="id" title="variable">tb2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:326"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:320"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:322"><span class="id" title="variable">tb1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:320"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:322"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:324"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:326"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:321"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:323"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:325"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:327"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_2_4" class="idref" href="#rpo_2_4"><span class="id" title="lemma">rpo_2_4</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:328" class="idref" href="#ta1:328"><span class="id" title="binder">ta1</span></a> <a id="ta2:329" class="idref" href="#ta2:329"><span class="id" title="binder">ta2</span></a> <a id="tb1:330" class="idref" href="#tb1:330"><span class="id" title="binder">tb1</span></a> <a id="tb2:331" class="idref" href="#tb2:331"><span class="id" title="binder">tb2</span></a>  <a id="n2:332" class="idref" href="#n2:332"><span class="id" title="binder">n2</span></a>  <a id="tc2:333" class="idref" href="#tc2:333"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:328"><span class="id" title="variable">ta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:329"><span class="id" title="variable">ta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:330"><span class="id" title="variable">tb1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:329"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:331"><span class="id" title="variable">tb2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:328"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:330"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:329"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:331"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:332"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:333"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_3_2" class="idref" href="#rpo_3_2"><span class="id" title="lemma">rpo_3_2</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:334" class="idref" href="#ta1:334"><span class="id" title="binder">ta1</span></a>  <a id="tb1:335" class="idref" href="#tb1:335"><span class="id" title="binder">tb1</span></a> <a id="tb2:336" class="idref" href="#tb2:336"><span class="id" title="binder">tb2</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:335"><span class="id" title="variable">tb1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:336"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:334"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:335"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:334"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:336"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_3_3" class="idref" href="#rpo_3_3"><span class="id" title="lemma">rpo_3_3</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:337" class="idref" href="#ta1:337"><span class="id" title="binder">ta1</span></a>  <a id="tb1:338" class="idref" href="#tb1:338"><span class="id" title="binder">tb1</span></a> <a id="tb2:339" class="idref" href="#tb2:339"><span class="id" title="binder">tb2</span></a> <a id="n1:340" class="idref" href="#n1:340"><span class="id" title="binder">n1</span></a> <a id="tc1:341" class="idref" href="#tc1:341"><span class="id" title="binder">tc1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:338"><span class="id" title="variable">tb1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:339"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:341"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:337"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:338"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:337"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:338"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:340"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:341"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:337"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:339"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_3_1" class="idref" href="#rpo_3_1"><span class="id" title="lemma">rpo_3_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:342" class="idref" href="#ta1:342"><span class="id" title="binder">ta1</span></a>  <a id="tb1:343" class="idref" href="#tb1:343"><span class="id" title="binder">tb1</span></a> <a id="tb2:344" class="idref" href="#tb2:344"><span class="id" title="binder">tb2</span></a> <a id="n1:345" class="idref" href="#n1:345"><span class="id" title="binder">n1</span></a> <a id="n2:346" class="idref" href="#n2:346"><span class="id" title="binder">n2</span></a> <a id="tc1:347" class="idref" href="#tc1:347"><span class="id" title="binder">tc1</span></a> <a id="tc2:348" class="idref" href="#tc2:348"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:343"><span class="id" title="variable">tb1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:344"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:347"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:342"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:343"><span class="id" title="variable">tb1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:342"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:343"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:345"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:347"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:342"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:344"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:346"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:348"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_3_4" class="idref" href="#rpo_3_4"><span class="id" title="lemma">rpo_3_4</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:349" class="idref" href="#ta1:349"><span class="id" title="binder">ta1</span></a>  <a id="tb1:350" class="idref" href="#tb1:350"><span class="id" title="binder">tb1</span></a> <a id="tb2:351" class="idref" href="#tb2:351"><span class="id" title="binder">tb2</span></a>  <a id="n2:352" class="idref" href="#n2:352"><span class="id" title="binder">n2</span></a>  <a id="tc2:353" class="idref" href="#tc2:353"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:350"><span class="id" title="variable">tb1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:351"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:349"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:350"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:349"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:351"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:352"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:353"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_4_2" class="idref" href="#rpo_4_2"><span class="id" title="lemma">rpo_4_2</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:354" class="idref" href="#ta1:354"><span class="id" title="binder">ta1</span></a> <a id="ta2:355" class="idref" href="#ta2:355"><span class="id" title="binder">ta2</span></a>  <a id="tb1:356" class="idref" href="#tb1:356"><span class="id" title="binder">tb1</span></a> <a id="tb2:357" class="idref" href="#tb2:357"><span class="id" title="binder">tb2</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:354"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:356"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:357"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:354"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:356"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:355"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:357"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_4_3" class="idref" href="#rpo_4_3"><span class="id" title="lemma">rpo_4_3</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:358" class="idref" href="#ta1:358"><span class="id" title="binder">ta1</span></a>  <a id="ta2:359" class="idref" href="#ta2:359"><span class="id" title="binder">ta2</span></a> <a id="tb1:360" class="idref" href="#tb1:360"><span class="id" title="binder">tb1</span></a> <a id="tb2:361" class="idref" href="#tb2:361"><span class="id" title="binder">tb2</span></a> <a id="n1:362" class="idref" href="#n1:362"><span class="id" title="binder">n1</span></a> <a id="tc1:363" class="idref" href="#tc1:363"><span class="id" title="binder">tc1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:358"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:360"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:361"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:363"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:358"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:360"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:358"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:360"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:362"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:363"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:359"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:361"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_4_1" class="idref" href="#rpo_4_1"><span class="id" title="lemma">rpo_4_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:364" class="idref" href="#ta1:364"><span class="id" title="binder">ta1</span></a>  <a id="ta2:365" class="idref" href="#ta2:365"><span class="id" title="binder">ta2</span></a> <a id="tb1:366" class="idref" href="#tb1:366"><span class="id" title="binder">tb1</span></a> <a id="tb2:367" class="idref" href="#tb2:367"><span class="id" title="binder">tb2</span></a> <a id="n1:368" class="idref" href="#n1:368"><span class="id" title="binder">n1</span></a> <a id="n2:369" class="idref" href="#n2:369"><span class="id" title="binder">n2</span></a> <a id="tc1:370" class="idref" href="#tc1:370"><span class="id" title="binder">tc1</span></a> <a id="tc2:371" class="idref" href="#tc2:371"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:364"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:366"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:367"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:370"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:364"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:366"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:364"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:366"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:368"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:370"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:365"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:367"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:369"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:371"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_4_4" class="idref" href="#rpo_4_4"><span class="id" title="lemma">rpo_4_4</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:372" class="idref" href="#ta1:372"><span class="id" title="binder">ta1</span></a>  <a id="ta2:373" class="idref" href="#ta2:373"><span class="id" title="binder">ta2</span></a> <a id="tb1:374" class="idref" href="#tb1:374"><span class="id" title="binder">tb1</span></a> <a id="tb2:375" class="idref" href="#tb2:375"><span class="id" title="binder">tb2</span></a>  <a id="n2:376" class="idref" href="#n2:376"><span class="id" title="binder">n2</span></a>  <a id="tc2:377" class="idref" href="#tc2:377"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:372"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:374"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:375"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:372"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:374"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:373"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb2:375"><span class="id" title="variable">tb2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:376"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:377"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_5_2" class="idref" href="#rpo_5_2"><span class="id" title="lemma">rpo_5_2</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="ta1:378" class="idref" href="#ta1:378"><span class="id" title="binder">ta1</span></a> <a id="ta2:379" class="idref" href="#ta2:379"><span class="id" title="binder">ta2</span></a>  <a id="tb1:380" class="idref" href="#tb1:380"><span class="id" title="binder">tb1</span></a>  ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:378"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:380"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:379"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:378"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:380"><span class="id" title="variable">tb1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_5_3" class="idref" href="#rpo_5_3"><span class="id" title="lemma">rpo_5_3</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:381" class="idref" href="#ta1:381"><span class="id" title="binder">ta1</span></a>  <a id="ta2:382" class="idref" href="#ta2:382"><span class="id" title="binder">ta2</span></a> <a id="tb1:383" class="idref" href="#tb1:383"><span class="id" title="binder">tb1</span></a>  <a id="n1:384" class="idref" href="#n1:384"><span class="id" title="binder">n1</span></a> <a id="tc1:385" class="idref" href="#tc1:385"><span class="id" title="binder">tc1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:385"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:381"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:383"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:381"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:383"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:384"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:385"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:382"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:381"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:383"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_5_1" class="idref" href="#rpo_5_1"><span class="id" title="lemma">rpo_5_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:386" class="idref" href="#ta1:386"><span class="id" title="binder">ta1</span></a>  <a id="ta2:387" class="idref" href="#ta2:387"><span class="id" title="binder">ta2</span></a> <a id="tb1:388" class="idref" href="#tb1:388"><span class="id" title="binder">tb1</span></a>  <a id="n1:389" class="idref" href="#n1:389"><span class="id" title="binder">n1</span></a> <a id="n2:390" class="idref" href="#n2:390"><span class="id" title="binder">n2</span></a> <a id="tc1:391" class="idref" href="#tc1:391"><span class="id" title="binder">tc1</span></a> <a id="tc2:392" class="idref" href="#tc2:392"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:391"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:386"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:388"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:386"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:388"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:389"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:391"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:387"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:386"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:388"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:390"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:392"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_5_4" class="idref" href="#rpo_5_4"><span class="id" title="lemma">rpo_5_4</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:393" class="idref" href="#ta1:393"><span class="id" title="binder">ta1</span></a>  <a id="ta2:394" class="idref" href="#ta2:394"><span class="id" title="binder">ta2</span></a> <a id="tb1:395" class="idref" href="#tb1:395"><span class="id" title="binder">tb1</span></a>  <a id="n2:396" class="idref" href="#n2:396"><span class="id" title="binder">n2</span></a>  <a id="tc2:397" class="idref" href="#tc2:397"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:393"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:395"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta2:394"><span class="id" title="variable">ta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:393"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:395"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:396"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:397"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_6_1" class="idref" href="#rpo_6_1"><span class="id" title="lemma">rpo_6_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:398" class="idref" href="#ta1:398"><span class="id" title="binder">ta1</span></a> <a id="tb1:399" class="idref" href="#tb1:399"><span class="id" title="binder">tb1</span></a> <a id="n1:400" class="idref" href="#n1:400"><span class="id" title="binder">n1</span></a> <a id="n2:401" class="idref" href="#n2:401"><span class="id" title="binder">n2</span></a> <a id="tc1:402" class="idref" href="#tc1:402"><span class="id" title="binder">tc1</span></a> <a id="tc2:403" class="idref" href="#tc2:403"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:402"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:398"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:399"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#n1:400"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:401"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:398"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:399"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:400"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:402"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:398"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:399"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:401"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:403"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_6_4" class="idref" href="#rpo_6_4"><span class="id" title="lemma">rpo_6_4</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:404" class="idref" href="#ta1:404"><span class="id" title="binder">ta1</span></a> <a id="tb1:405" class="idref" href="#tb1:405"><span class="id" title="binder">tb1</span></a>  <a id="n2:406" class="idref" href="#n2:406"><span class="id" title="binder">n2</span></a>  <a id="tc2:407" class="idref" href="#tc2:407"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:406"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:404"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:405"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:404"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:405"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n2:406"><span class="id" title="variable">n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:407"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rpo_7_1" class="idref" href="#rpo_7_1"><span class="id" title="lemma">rpo_7_1</span></a> : <span class="id" title="keyword">∀</span> <a id="ta1:408" class="idref" href="#ta1:408"><span class="id" title="binder">ta1</span></a> <a id="tb1:409" class="idref" href="#tb1:409"><span class="id" title="binder">tb1</span></a> <a id="n1:410" class="idref" href="#n1:410"><span class="id" title="binder">n1</span></a> <a id="tc1:411" class="idref" href="#tc1:411"><span class="id" title="binder">tc1</span></a> <a id="tc2:412" class="idref" href="#tc2:412"><span class="id" title="binder">tc2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:411"><span class="id" title="variable">tc1</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:408"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:409"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:411"><span class="id" title="variable">tc1</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:412"><span class="id" title="variable">tc2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:408"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:409"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:410"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc1:411"><span class="id" title="variable">tc1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#ta1:408"><span class="id" title="variable">ta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#tb1:409"><span class="id" title="variable">tb1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n1:410"><span class="id" title="variable">n1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#tc2:412"><span class="id" title="variable">tc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab117"></a><h2 class="section">inclusion of the order <span class="inlinecode"><span class="id" title="var">lt</span></span> in the rpo</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="lt_incl_rpo" class="idref" href="#lt_incl_rpo"><span class="id" title="section">lt_incl_rpo</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="lt_incl_rpo.s" class="idref" href="#lt_incl_rpo.s"><span class="id" title="variable">s</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="lt_incl_rpo.a1" class="idref" href="#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a> <a id="lt_incl_rpo.b1" class="idref" href="#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a> <a id="lt_incl_rpo.c1" class="idref" href="#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a> <a id="lt_incl_rpo.a2" class="idref" href="#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a> <a id="lt_incl_rpo.b2" class="idref" href="#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a> <a id="lt_incl_rpo.c2" class="idref" href="#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>)(<a id="lt_incl_rpo.n1" class="idref" href="#lt_incl_rpo.n1"><span class="id" title="variable">n1</span></a> <a id="lt_incl_rpo.n2" class="idref" href="#lt_incl_rpo.n2"><span class="id" title="variable">n2</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="lt_incl_rpo.Hsize" class="idref" href="#lt_incl_rpo.Hsize"><span class="id" title="variable">Hsize</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.s"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="lt_incl_rpo.Hrec" class="idref" href="#lt_incl_rpo.Hrec"><span class="id" title="variable">Hrec</span></a> :   <span class="id" title="keyword">∀</span> <a id="o':423" class="idref" href="#o':423"><span class="id" title="binder">o'</span></a> <a id="o:424" class="idref" href="#o:424"><span class="id" title="binder">o</span></a>, (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:424"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':423"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.s"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#o:424"><span class="id" title="variable">o</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':423"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:424"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':423"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:424"><span class="id" title="variable">o</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':423"><span class="id" title="variable">o'</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="lt_incl_rpo.nf1" class="idref" href="#lt_incl_rpo.nf1"><span class="id" title="variable">nf1</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="lt_incl_rpo.nf2" class="idref" href="#lt_incl_rpo.nf2"><span class="id" title="variable">nf2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_a1" class="idref" href="#nf_a1"><span class="id" title="lemma">nf_a1</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_a2" class="idref" href="#nf_a2"><span class="id" title="lemma">nf_a2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_b1" class="idref" href="#nf_b1"><span class="id" title="lemma">nf_b1</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_b2" class="idref" href="#nf_b2"><span class="id" title="lemma">nf_b2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf1</span> <span class="id" title="var">nf2</span> <span class="id" title="var">nf_a1</span> <span class="id" title="var">nf_a2</span> <span class="id" title="var">nf_b1</span> <span class="id" title="var">nf_b2</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_c1" class="idref" href="#nf_c1"><span class="id" title="lemma">nf_c1</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="nf_c2" class="idref" href="#nf_c2"><span class="id" title="lemma">nf_c2</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_c1</span> <span class="id" title="var">nf_c2</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="lt_incl_rpo.H" class="idref" href="#lt_incl_rpo.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="cons_rw" class="idref" href="#cons_rw"><span class="id" title="lemma">cons_rw</span></a> : <span class="id" title="keyword">∀</span> <a id="a:429" class="idref" href="#a:429"><span class="id" title="binder">a</span></a> <a id="b:430" class="idref" href="#b:430"><span class="id" title="binder">b</span></a> <a id="n:431" class="idref" href="#n:431"><span class="id" title="binder">n</span></a> <a id="c:432" class="idref" href="#c:432"><span class="id" title="binder">c</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#n:431"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:432"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:429"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:430"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:431"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:432"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:429"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:430"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:429"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:430"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:431"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:432"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:429"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:430"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#nat_2_term"><span class="id" title="definition">nat_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:431"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:432"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_rpo_cons_cons" class="idref" href="#lt_rpo_cons_cons"><span class="id" title="lemma">lt_rpo_cons_cons</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1"><span class="id" title="variable">c1</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2"><span class="id" title="variable">c2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_incl_rpo"><span class="id" title="section">lt_incl_rpo</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_inc_rpo_0" class="idref" href="#lt_inc_rpo_0"><span class="id" title="lemma">lt_inc_rpo_0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:433" class="idref" href="#n:433"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="o':434" class="idref" href="#o':434"><span class="id" title="binder">o'</span></a> <a id="o:435" class="idref" href="#o:435"><span class="id" title="binder">o</span></a>, (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:435"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#T2_size"><span class="id" title="definition">T2_size</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':434"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:433"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#o:435"><span class="id" title="variable">o</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':434"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:435"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':434"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:435"><span class="id" title="variable">o</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':434"><span class="id" title="variable">o'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#nat_0"><span class="id" title="constructor">nat_0</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R1</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_zero"><span class="id" title="constructor">ord_zero</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R2</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#nat_S"><span class="id" title="constructor">nat_S</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R3</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R4" class="idref" href="#R4"><span class="id" title="lemma">R4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_cons"><span class="id" title="constructor">ord_cons</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R4</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#prec"><span class="id" title="axiom">P.prec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#ord_psi"><span class="id" title="constructor">ord_psi</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">R5</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="well_founded_rpo" class="idref" href="#well_founded_rpo"><span class="id" title="lemma">well_founded_rpo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span>  <a id="well_founded" class="idref" href="#well_founded"><span class="id" title="section">well_founded</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="well_founded.R" class="idref" href="#well_founded.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <span class="id" title="var">R</span> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R_inc_rpo" class="idref" href="#R_inc_rpo"><span class="id" title="lemma">R_inc_rpo</span></a> : <span class="id" title="keyword">∀</span> <a id="o:436" class="idref" href="#o:436"><span class="id" title="binder">o</span></a> <a id="o':437" class="idref" href="#o':437"><span class="id" title="binder">o'</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#well_founded.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:436"><span class="id" title="variable">o</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':437"><span class="id" title="variable">o'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:436"><span class="id" title="variable">o</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#T2_2_term"><span class="id" title="definition">T2_2_term</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o':437"><span class="id" title="variable">o'</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="nf_Wf" class="idref" href="#nf_Wf"><span class="id" title="lemma">nf_Wf</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#well_founded"><span class="id" title="section">well_founded</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transfinite_induction" class="idref" href="#transfinite_induction"><span class="id" title="definition">transfinite_induction</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:442" class="idref" href="#P:442"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:443" class="idref" href="#x:443"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:443"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:444" class="idref" href="#y:444"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:444"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#y:444"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:443"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:442"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:444"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:442"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:443"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:445" class="idref" href="#a:445"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:445"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:442"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:445"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="transfinite_induction_Q" class="idref" href="#transfinite_induction_Q"><span class="id" title="definition">transfinite_induction_Q</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:446" class="idref" href="#P:446"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) (<a id="Q:447" class="idref" href="#Q:447"><span class="id" title="binder">Q</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:448" class="idref" href="#x:448"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Q:447"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:448"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:448"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:449" class="idref" href="#y:449"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#Q:447"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:449"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:449"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#y:449"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:448"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:446"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:449"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:446"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:448"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:450" class="idref" href="#a:450"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:450"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Q:447"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:450"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:446"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:450"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab118"></a><h1 class="section">the Veblen function phi</h1>
+
+<div class="paragraph"> </div>
+
+ See Schutte.Critical.phi  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="phi" class="idref" href="#phi"><span class="id" title="definition">phi</span></a> (<a id="alpha:453" class="idref" href="#alpha:453"><span class="id" title="binder">alpha</span></a> <a id="beta:454" class="idref" href="#beta:454"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:454"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:454"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><span class="id" title="var">b1</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b2</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">b1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><span class="id" title="var">b1</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b2</span> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,[</span></a><span class="id" title="var">b1</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b2</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a><br/>
+     <span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> 0 (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>  <span class="id" title="var">n</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>) ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">b1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <span class="id" title="var">n</span>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒  <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>))<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;| <span class="id" title="var">any_beta</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:453"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">any_beta</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex8" class="idref" href="#Ex8"><span class="id" title="definition">Ex8</span></a>:  <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> 1 (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>0<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> 1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.   
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ All epsilons are fixpoints of phi 0 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="epsilon_fxp" class="idref" href="#epsilon_fxp"><span class="id" title="lemma">epsilon_fxp</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:456" class="idref" href="#beta:456"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:456"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:456"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="epsilon0_fxp" class="idref" href="#epsilon0_fxp"><span class="id" title="lemma">epsilon0_fxp</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_of_psi" class="idref" href="#phi_of_psi"><span class="id" title="lemma">phi_of_psi</span></a>  : <span class="id" title="keyword">∀</span> <a id="a:457" class="idref" href="#a:457"><span class="id" title="binder">a</span></a> <a id="b1:458" class="idref" href="#b1:458"><span class="id" title="binder">b1</span></a> <a id="b2:459" class="idref" href="#b2:459"><span class="id" title="binder">b2</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:457"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b1:458"><span class="id" title="variable">b1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:459"><span class="id" title="variable">b2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="hydras.Gamma0.Gamma0.html#lt_ge_dec"><span class="id" title="definition">lt_ge_dec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:457"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:458"><span class="id" title="variable">b1</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b1:458"><span class="id" title="variable">b1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:459"><span class="id" title="variable">b2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:457"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b1:458"><span class="id" title="variable">b1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:459"><span class="id" title="variable">b2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi" class="idref" href="#phi_to_psi"><span class="id" title="lemma">phi_to_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:460" class="idref" href="#alpha:460"><span class="id" title="binder">alpha</span></a> <a id="beta:461" class="idref" href="#beta:461"><span class="id" title="binder">beta</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="alpha':462" class="idref" href="#alpha':462"><span class="id" title="binder">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta':463" class="idref" href="#beta':463"><span class="id" title="binder">beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:460"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:461"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':462"><span class="id" title="variable">alpha'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta':463"><span class="id" title="variable">beta'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_principal" class="idref" href="#phi_principal"><span class="id" title="lemma">phi_principal</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:464" class="idref" href="#alpha:464"><span class="id" title="binder">alpha</span></a> <a id="beta:465" class="idref" href="#beta:465"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#ap"><span class="id" title="inductive">ap</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:464"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:465"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_alpha_zero" class="idref" href="#phi_alpha_zero"><span class="id" title="lemma">phi_alpha_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:466" class="idref" href="#alpha:466"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:466"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:466"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_of_psi_succ" class="idref" href="#phi_of_psi_succ"><span class="id" title="lemma">phi_of_psi_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="a:467" class="idref" href="#a:467"><span class="id" title="binder">a</span></a> <a id="b1:468" class="idref" href="#b1:468"><span class="id" title="binder">b1</span></a> <a id="b2:469" class="idref" href="#b2:469"><span class="id" title="binder">b2</span></a> <a id="n:470" class="idref" href="#n:470"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:467"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:468"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:469"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:470"><span class="id" title="variable">n</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#lt_ge_dec"><span class="id" title="definition">lt_ge_dec</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:467"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:468"><span class="id" title="variable">b1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:467"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:468"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:469"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:470"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:467"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:468"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:469"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:470"><span class="id" title="variable">n</span></a>))<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="phi_cases_aux" class="idref" href="#phi_cases_aux"><span class="id" title="lemma">phi_cases_aux</span></a> : <span class="id" title="keyword">∀</span> <a id="P:471" class="idref" href="#P:471"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#P:471"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b1:472" class="idref" href="#b1:472"><span class="id" title="binder">b1</span></a> <a id="b2:473" class="idref" href="#b2:473"><span class="id" title="binder">b2</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:472"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:473"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:471"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b1:472"><span class="id" title="variable">b1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:473"><span class="id" title="variable">b2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b1:474" class="idref" href="#b1:474"><span class="id" title="binder">b1</span></a> <a id="b2:475" class="idref" href="#b2:475"><span class="id" title="binder">b2</span></a> <a id="n:476" class="idref" href="#n:476"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:474"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:475"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#P:471"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:474"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:475"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:476"><span class="id" title="variable">n</span></a>)))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b1:477" class="idref" href="#b1:477"><span class="id" title="binder">b1</span></a> <a id="b2:478" class="idref" href="#b2:478"><span class="id" title="binder">b2</span></a> <a id="n:479" class="idref" href="#n:479"><span class="id" title="binder">n</span></a> <a id="c:480" class="idref" href="#c:480"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:477"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:478"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:479"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:480"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:480"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:479"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#P:471"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:477"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:478"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:479"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:480"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:481" class="idref" href="#alpha:481"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:481"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:471"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:481"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_cases'" class="idref" href="#phi_cases'"><span class="id" title="lemma">phi_cases'</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:482" class="idref" href="#a:482"><span class="id" title="binder">a</span></a> <a id="b:483" class="idref" href="#b:483"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="b1:484" class="idref" href="#b1:484"><span class="id" title="binder">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b2:485" class="idref" href="#b2:485"><span class="id" title="binder">b2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b1:484"><span class="id" title="variable">b1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:485"><span class="id" title="variable">b2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:484"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="b1:486" class="idref" href="#b1:486"><span class="id" title="binder">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="b2:487" class="idref" href="#b2:487"><span class="id" title="binder">b2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:488" class="idref" href="#n:488"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:486"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:487"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:488"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:486"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:483"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:482"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:486"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:487"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:488"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_cases" class="idref" href="#phi_cases"><span class="id" title="lemma">phi_cases</span></a> : <span class="id" title="keyword">∀</span> <a id="a:489" class="idref" href="#a:489"><span class="id" title="binder">a</span></a> <a id="b:490" class="idref" href="#b:490"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:489"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:489"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:489"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b':491" class="idref" href="#b':491"><span class="id" title="binder">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b':491"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:489"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:489"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b':491"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#b':491"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:490"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_nf" class="idref" href="#phi_nf"><span class="id" title="lemma">phi_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:492" class="idref" href="#alpha:492"><span class="id" title="binder">alpha</span></a> <a id="beta:493" class="idref" href="#beta:493"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:492"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:493"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:492"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:493"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_of_any_cons" class="idref" href="#phi_of_any_cons"><span class="id" title="lemma">phi_of_any_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:494" class="idref" href="#alpha:494"><span class="id" title="binder">alpha</span></a> <a id="beta1:495" class="idref" href="#beta1:495"><span class="id" title="binder">beta1</span></a> <a id="beta2:496" class="idref" href="#beta2:496"><span class="id" title="binder">beta2</span></a> <a id="n:497" class="idref" href="#n:497"><span class="id" title="binder">n</span></a> <a id="gamma:498" class="idref" href="#gamma:498"><span class="id" title="binder">gamma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:498"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:497"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:494"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:495"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:496"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:497"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:498"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:494"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:495"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:496"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:497"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:498"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_fix" class="idref" href="#phi_fix"><span class="id" title="lemma">phi_fix</span></a> <a id="alpha:499" class="idref" href="#alpha:499"><span class="id" title="binder">alpha</span></a> <a id="beta:500" class="idref" href="#beta:500"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:499"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:500"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:500"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="beta1:501" class="idref" href="#beta1:501"><span class="id" title="binder">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta2:502" class="idref" href="#beta2:502"><span class="id" title="binder">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:500"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:501"><span class="id" title="variable">beta1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:502"><span class="id" title="variable">beta2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:499"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:501"><span class="id" title="variable">beta1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_le" class="idref" href="#phi_le"><span class="id" title="lemma">phi_le</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:503" class="idref" href="#alpha:503"><span class="id" title="binder">alpha</span></a> <a id="beta:504" class="idref" href="#beta:504"><span class="id" title="binder">beta</span></a> <a id="alpha':505" class="idref" href="#alpha':505"><span class="id" title="binder">alpha'</span></a> <a id="beta':506" class="idref" href="#beta':506"><span class="id" title="binder">beta'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:504"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:503"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:504"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':505"><span class="id" title="variable">alpha'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta':506"><span class="id" title="variable">beta'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:503"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':505"><span class="id" title="variable">alpha'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_le_ge" class="idref" href="#phi_le_ge"><span class="id" title="lemma">phi_le_ge</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:507" class="idref" href="#alpha:507"><span class="id" title="binder">alpha</span></a> <a id="beta:508" class="idref" href="#beta:508"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:507"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:508"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="alpha':509" class="idref" href="#alpha':509"><span class="id" title="binder">alpha'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta':510" class="idref" href="#beta':510"><span class="id" title="binder">beta'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:507"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:508"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':509"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#beta':510"><span class="id" title="variable">beta'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:507"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':509"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta':510"><span class="id" title="variable">beta'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:508"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ <span class="inlinecode"><span class="id" title="var">phi</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span> is a common fixpoint of all the <span class="inlinecode"><span class="id" title="var">phi</span></span> <span class="inlinecode"><span class="id" title="var">gamma</span></span>, 
+for any <span class="inlinecode"><span class="id" title="var">gamma</span></span> <span class="inlinecode"><span class="id" title="var">t2</span>&lt;</span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> as specified by Schutte 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_spec1" class="idref" href="#phi_spec1"><span class="id" title="lemma">phi_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:511" class="idref" href="#alpha:511"><span class="id" title="binder">alpha</span></a> <a id="beta:512" class="idref" href="#beta:512"><span class="id" title="binder">beta</span></a> <a id="gamma:513" class="idref" href="#gamma:513"><span class="id" title="binder">gamma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:511"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:512"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:513"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:513"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:511"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:513"><span class="id" title="variable">gamma</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:511"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:512"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:511"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:512"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="phi_principalR" class="idref" href="#phi_principalR"><span class="id" title="lemma">phi_principalR</span></a> <a id="alpha:514" class="idref" href="#alpha:514"><span class="id" title="binder">alpha</span></a> <a id="beta:515" class="idref" href="#beta:515"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:514"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:515"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="gamma:516" class="idref" href="#gamma:516"><span class="id" title="binder">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:514"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:515"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:516"><span class="id" title="variable">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_alpha_zero_gt_alpha" class="idref" href="#phi_alpha_zero_gt_alpha"><span class="id" title="lemma">phi_alpha_zero_gt_alpha</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:517" class="idref" href="#alpha:517"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:517"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:517"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="le_b_phi_ab" class="idref" href="#le_b_phi_ab"><span class="id" title="lemma">le_b_phi_ab</span></a> : <span class="id" title="keyword">∀</span> <a id="a:518" class="idref" href="#a:518"><span class="id" title="binder">a</span></a> <a id="b:519" class="idref" href="#b:519"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:518"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:519"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <a class="idref" href="hydras.Gamma0.Gamma0.html#b:519"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:518"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:519"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_of_psi_plus_fin" class="idref" href="#phi_of_psi_plus_fin"><span class="id" title="lemma">phi_of_psi_plus_fin</span></a> : <span class="id" title="keyword">∀</span> <a id="a:520" class="idref" href="#a:520"><span class="id" title="binder">a</span></a> <a id="b1:521" class="idref" href="#b1:521"><span class="id" title="binder">b1</span></a> <a id="b2:522" class="idref" href="#b2:522"><span class="id" title="binder">b2</span></a> <a id="n:523" class="idref" href="#n:523"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:520"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:521"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:520"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:521"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:522"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:523"><span class="id" title="variable">n</span></a>)) <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:520"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:521"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:522"><span class="id" title="variable">b2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:523"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_mono_r" class="idref" href="#phi_mono_r"><span class="id" title="lemma">phi_mono_r</span></a> : <span class="id" title="keyword">∀</span> <a id="a:524" class="idref" href="#a:524"><span class="id" title="binder">a</span></a> <a id="b:525" class="idref" href="#b:525"><span class="id" title="binder">b</span></a> <a id="c:526" class="idref" href="#c:526"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:524"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:525"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:526"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#b:525"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:526"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:524"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:525"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:524"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:526"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_mono_weak_r" class="idref" href="#phi_mono_weak_r"><span class="id" title="lemma">phi_mono_weak_r</span></a> : <span class="id" title="keyword">∀</span> <a id="a:527" class="idref" href="#a:527"><span class="id" title="binder">a</span></a> <a id="b:528" class="idref" href="#b:528"><span class="id" title="binder">b</span></a> <a id="c:529" class="idref" href="#c:529"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:527"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:528"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:529"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#b:528"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:529"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:527"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:528"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:527"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:529"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_inj_r" class="idref" href="#phi_inj_r"><span class="id" title="lemma">phi_inj_r</span></a> : <span class="id" title="keyword">∀</span> <a id="a:530" class="idref" href="#a:530"><span class="id" title="binder">a</span></a> <a id="b:531" class="idref" href="#b:531"><span class="id" title="binder">b</span></a> <a id="c:532" class="idref" href="#c:532"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:530"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:531"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:532"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:530"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:531"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:530"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:532"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:531"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:532"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_a_phi_ab" class="idref" href="#lt_a_phi_ab"><span class="id" title="lemma">lt_a_phi_ab</span></a> : <span class="id" title="keyword">∀</span> <a id="a:533" class="idref" href="#a:533"><span class="id" title="binder">a</span></a> <a id="b:534" class="idref" href="#b:534"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:533"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:534"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:533"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:533"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:534"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_not_lim" class="idref" href="#zero_not_lim"><span class="id" title="lemma">zero_not_lim</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="F_not_lim" class="idref" href="#F_not_lim"><span class="id" title="lemma">F_not_lim</span></a> : <span class="id" title="keyword">∀</span> <a id="n:535" class="idref" href="#n:535"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a>  (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:535"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succb_not_lim" class="idref" href="#succb_not_lim"><span class="id" title="lemma">succb_not_lim</span></a>  <a id="alpha:536" class="idref" href="#alpha:536"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#is_successor"><span class="id" title="inductive">is_successor</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:536"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:536"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_not_succ" class="idref" href="#is_limit_not_succ"><span class="id" title="lemma">is_limit_not_succ</span></a> <a id="alpha:537" class="idref" href="#alpha:537"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:537"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_successor"><span class="id" title="inductive">is_successor</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:537"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+   limit_plus_fin alpha n beta  means :
+   beta = alpha + fin n and (alpha is limit or alpha = zero)
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="limit_plus_fin" class="idref" href="#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="limit_plus_F_0" class="idref" href="#limit_plus_F_0"><span class="id" title="constructor">limit_plus_F_0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:540" class="idref" href="#p:540"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin:538"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:540"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:540"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;|<a id="limit_plus_F_cons" class="idref" href="#limit_plus_F_cons"><span class="id" title="constructor">limit_plus_F_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="beta1:541" class="idref" href="#beta1:541"><span class="id" title="binder">beta1</span></a> <a id="beta2:542" class="idref" href="#beta2:542"><span class="id" title="binder">beta2</span></a> <a id="n:543" class="idref" href="#n:543"><span class="id" title="binder">n</span></a> <a id="gamma0:544" class="idref" href="#gamma0:544"><span class="id" title="binder">gamma0</span></a> <a id="gamma:545" class="idref" href="#gamma:545"><span class="id" title="binder">gamma</span></a> <a id="p:546" class="idref" href="#p:546"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:541"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:542"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin:538"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma0:544"><span class="id" title="variable">gamma0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:546"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:545"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin:538"><span class="id" title="inductive">limit_plus_fin</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:541"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:542"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:543"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma0:544"><span class="id" title="variable">gamma0</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#p:546"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:541"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:542"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:543"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:545"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_plus_fin_plus" class="idref" href="#limit_plus_fin_plus"><span class="id" title="lemma">limit_plus_fin_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:547" class="idref" href="#alpha:547"><span class="id" title="binder">alpha</span></a> <a id="alpha':548" class="idref" href="#alpha':548"><span class="id" title="binder">alpha'</span></a> <a id="p:549" class="idref" href="#p:549"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:547"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:549"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':548"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:547"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':548"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:547"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:549"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_plus_fin_lim" class="idref" href="#limit_plus_fin_lim"><span class="id" title="lemma">limit_plus_fin_lim</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:550" class="idref" href="#alpha:550"><span class="id" title="binder">alpha</span></a> <a id="alpha':551" class="idref" href="#alpha':551"><span class="id" title="binder">alpha'</span></a> <a id="p:552" class="idref" href="#p:552"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:550"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#p:552"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha':551"><span class="id" title="variable">alpha'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:550"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:550"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:550"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_plus_fin_inv0" class="idref" href="#limit_plus_fin_inv0"><span class="id" title="lemma">limit_plus_fin_inv0</span></a>  <a id="alpha:553" class="idref" href="#alpha:553"><span class="id" title="binder">alpha</span></a> <a id="beta:554" class="idref" href="#beta:554"><span class="id" title="binder">beta</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:553"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:554"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:553"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:553"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:554"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_cons_inv" class="idref" href="#is_limit_cons_inv"><span class="id" title="lemma">is_limit_cons_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="b1:555" class="idref" href="#b1:555"><span class="id" title="binder">b1</span></a> <a id="b2:556" class="idref" href="#b2:556"><span class="id" title="binder">b2</span></a> <a id="n:557" class="idref" href="#n:557"><span class="id" title="binder">n</span></a> <a id="c:558" class="idref" href="#c:558"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:555"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:556"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:557"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:558"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:555"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:556"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:557"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:558"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:558"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:558"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_intro" class="idref" href="#is_limit_intro"><span class="id" title="lemma">is_limit_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="b1:559" class="idref" href="#b1:559"><span class="id" title="binder">b1</span></a> <a id="b2:560" class="idref" href="#b2:560"><span class="id" title="binder">b2</span></a> <a id="n:561" class="idref" href="#n:561"><span class="id" title="binder">n</span></a> , <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:559"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:560"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:559"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:560"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a>  (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b1:559"><span class="id" title="variable">b1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b2:560"><span class="id" title="variable">b2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:561"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_epsilon0_ok" class="idref" href="#lt_epsilon0_ok"><span class="id" title="lemma">lt_epsilon0_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:562" class="idref" href="#alpha:562"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:562"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:562"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:562"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Derive</span> <span class="id" title="var">Inversion_clear</span> <span class="id" title="var">lt_01</span> <span class="id" title="keyword">with</span> (<span class="id" title="keyword">∀</span> (<a id="a:563" class="idref" href="#a:563"><span class="id" title="binder">a</span></a> <a id="b:564" class="idref" href="#b:564"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:563"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:564"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>) <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Derive</span> <span class="id" title="var">Inversion_clear</span> <span class="id" title="var">lt_02</span> <span class="id" title="keyword">with</span> (<span class="id" title="keyword">∀</span> (<a id="a:565" class="idref" href="#a:565"><span class="id" title="binder">a</span></a> <a id="b:566" class="idref" href="#b:566"><span class="id" title="binder">b</span></a> <a id="c:567" class="idref" href="#c:567"><span class="id" title="binder">c</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>)(<a id="n:568" class="idref" href="#n:568"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:565"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:566"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:568"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:567"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>) <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="psi_lt_epsilon0" class="idref" href="#psi_lt_epsilon0"><span class="id" title="lemma">psi_lt_epsilon0</span></a> : <span class="id" title="keyword">∀</span> <a id="a:569" class="idref" href="#a:569"><span class="id" title="binder">a</span></a> <a id="b:570" class="idref" href="#b:570"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:569"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:570"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:569"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:570"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cons_lt_epsilon0" class="idref" href="#cons_lt_epsilon0"><span class="id" title="lemma">cons_lt_epsilon0</span></a> : <span class="id" title="keyword">∀</span> <a id="a:571" class="idref" href="#a:571"><span class="id" title="binder">a</span></a> <a id="b:572" class="idref" href="#b:572"><span class="id" title="binder">b</span></a> <a id="n:573" class="idref" href="#n:573"><span class="id" title="binder">n</span></a> <a id="c:574" class="idref" href="#c:574"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:571"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:572"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:573"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:574"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:571"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:572"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:573"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:574"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#a:571"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:572"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:574"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_epsilon0_okR" class="idref" href="#lt_epsilon0_okR"><span class="id" title="lemma">lt_epsilon0_okR</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:575" class="idref" href="#alpha:575"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:575"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:575"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:575"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1_to_T2_inj" class="idref" href="#T1_to_T2_inj"><span class="id" title="lemma">T1_to_T2_inj</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:576" class="idref" href="#alpha:576"><span class="id" title="binder">alpha</span></a> <a id="beta:577" class="idref" href="#beta:577"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:576"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:577"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:576"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:577"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1_to_T2_lt" class="idref" href="#T1_to_T2_lt"><span class="id" title="lemma">T1_to_T2_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="c:578" class="idref" href="#c:578"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:578"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="T1_to_T2_R" class="idref" href="#T1_to_T2_R"><span class="id" title="definition">T1_to_T2_R</span></a> : <span class="id" title="keyword">∀</span> <a id="a:579" class="idref" href="#a:579"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:579"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="c:580" class="idref" href="#c:580"><span class="id" title="binder">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:580"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:579"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1_to_T2_mono" class="idref" href="#T1_to_T2_mono"><span class="id" title="lemma">T1_to_T2_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:581" class="idref" href="#alpha:581"><span class="id" title="binder">alpha</span></a> <a id="beta:582" class="idref" href="#beta:582"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:581"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:582"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:581"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:582"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="T1_to_T2_monoR" class="idref" href="#T1_to_T2_monoR"><span class="id" title="lemma">T1_to_T2_monoR</span></a> : <span class="id" title="keyword">∀</span> <a id="c:583" class="idref" href="#c:583"><span class="id" title="binder">c</span></a> <a id="c':584" class="idref" href="#c':584"><span class="id" title="binder">c'</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:583"><span class="id" title="variable">c</span></a>) (<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c':584"><span class="id" title="variable">c'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:583"><span class="id" title="variable">c</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#c':584"><span class="id" title="variable">c'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_epsilon0_trans" class="idref" href="#lt_epsilon0_trans"><span class="id" title="lemma">lt_epsilon0_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:585" class="idref" href="#a:585"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:585"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:585"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:586" class="idref" href="#b:586"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:586"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:585"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:586"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:586"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_coeff_irrelevance" class="idref" href="#nf_coeff_irrelevance"><span class="id" title="lemma">nf_coeff_irrelevance</span></a> : <span class="id" title="keyword">∀</span> <a id="a:587" class="idref" href="#a:587"><span class="id" title="binder">a</span></a> <a id="b:588" class="idref" href="#b:588"><span class="id" title="binder">b</span></a> <a id="n:589" class="idref" href="#n:589"><span class="id" title="binder">n</span></a> <a id="n':590" class="idref" href="#n':590"><span class="id" title="binder">n'</span></a> <a id="c:591" class="idref" href="#c:591"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:587"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:588"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:589"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:591"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:587"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:588"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n':590"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:591"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="psi_principal" class="idref" href="#psi_principal"><span class="id" title="lemma">psi_principal</span></a> : <span class="id" title="keyword">∀</span> <a id="a:592" class="idref" href="#a:592"><span class="id" title="binder">a</span></a> <a id="b:593" class="idref" href="#b:593"><span class="id" title="binder">b</span></a> <a id="c:594" class="idref" href="#c:594"><span class="id" title="binder">c</span></a> <a id="d:595" class="idref" href="#d:595"><span class="id" title="binder">d</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:594"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:594"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:592"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:593"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#d:595"><span class="id" title="variable">d</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:592"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:593"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#c:594"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#d:595"><span class="id" title="variable">d</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:592"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:593"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_intro" class="idref" href="#nf_intro"><span class="id" title="lemma">nf_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="a:596" class="idref" href="#a:596"><span class="id" title="binder">a</span></a> <a id="b:597" class="idref" href="#b:597"><span class="id" title="binder">b</span></a> <a id="n:598" class="idref" href="#n:598"><span class="id" title="binder">n</span></a> <a id="c:599" class="idref" href="#c:599"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:596"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:597"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#c:599"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:596"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:597"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:599"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:596"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:597"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:598"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:599"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_nf" class="idref" href="#plus_nf"><span class="id" title="lemma">plus_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:600" class="idref" href="#alpha:600"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:600"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="beta:601" class="idref" href="#beta:601"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:601"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:600"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:601"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_as_plus" class="idref" href="#succ_as_plus"><span class="id" title="lemma">succ_as_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:602" class="idref" href="#alpha:602"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:602"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:602"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:602"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_nf" class="idref" href="#succ_nf"><span class="id" title="lemma">succ_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:603" class="idref" href="#alpha:603"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:603"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:603"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_epsilon0_succ" class="idref" href="#lt_epsilon0_succ"><span class="id" title="lemma">lt_epsilon0_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="a:604" class="idref" href="#a:604"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:604"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a>  (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:604"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span>  <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+<span class="id" title="keyword">Theorem</span> <a id="epsilon0_as_lub" class="idref" href="#epsilon0_as_lub"><span class="id" title="lemma">epsilon0_as_lub</span></a> : <span class="id" title="keyword">∀</span> <a id="b:605" class="idref" href="#b:605"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:605"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:606" class="idref" href="#a:606"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:606"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:606"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:605"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#le"><span class="id" title="definition">le</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:605"><span class="id" title="variable">b</span></a>.<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">epsilon0</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lub" class="idref" href="#lub"><span class="id" title="definition">lub</span></a> (<a id="P:607" class="idref" href="#P:607"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)(<a id="x:608" class="idref" href="#x:608"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:608"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:609" class="idref" href="#y:609"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#P:607"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:608"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:610" class="idref" href="#y:610"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:611" class="idref" href="#x:611"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#P:607"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:611"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:611"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#x:611"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:610"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:610"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#x:608"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#y:610"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lub_unicity" class="idref" href="#lub_unicity"><span class="id" title="lemma">lub_unicity</span></a> : <span class="id" title="keyword">∀</span> <a id="P:612" class="idref" href="#P:612"><span class="id" title="binder">P</span></a> <a id="l:613" class="idref" href="#l:613"><span class="id" title="binder">l</span></a> <a id="l':614" class="idref" href="#l':614"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#lub"><span class="id" title="definition">lub</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:612"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l:613"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lub"><span class="id" title="definition">lub</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:612"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l':614"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l:613"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l':614"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lub_mono" class="idref" href="#lub_mono"><span class="id" title="lemma">lub_mono</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:615" class="idref" href="#P:615"><span class="id" title="binder">P</span></a> <a id="Q:616" class="idref" href="#Q:616"><span class="id" title="binder">Q</span></a> :<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="l:617" class="idref" href="#l:617"><span class="id" title="binder">l</span></a> <a id="l':618" class="idref" href="#l':618"><span class="id" title="binder">l'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="o:619" class="idref" href="#o:619"><span class="id" title="binder">o</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:619"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:615"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:619"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Q:616"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:619"><span class="id" title="variable">o</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#lub"><span class="id" title="definition">lub</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#P:615"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l:617"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#lub"><span class="id" title="definition">lub</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#Q:616"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l':618"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l:617"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#l':618"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_limit_dec" class="idref" href="#succ_limit_dec"><span class="id" title="lemma">succ_limit_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a:620" class="idref" href="#a:620"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:620"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#a:620"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+{</span></a><a class="idref" href="hydras.Gamma0.T2.html#is_successor"><span class="id" title="inductive">is_successor</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#a:620"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+{</span></a><a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:620"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_plus_r" class="idref" href="#le_plus_r"><span class="id" title="lemma">le_plus_r</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:621" class="idref" href="#alpha:621"><span class="id" title="binder">alpha</span></a> <a id="beta:622" class="idref" href="#beta:622"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:621"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:622"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:621"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:621"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:622"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_plus_l" class="idref" href="#le_plus_l"><span class="id" title="lemma">le_plus_l</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:623" class="idref" href="#alpha:623"><span class="id" title="binder">alpha</span></a> <a id="beta:624" class="idref" href="#beta:624"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:623"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:624"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:623"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:624"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:623"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_r" class="idref" href="#plus_mono_r"><span class="id" title="lemma">plus_mono_r</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:625" class="idref" href="#alpha:625"><span class="id" title="binder">alpha</span></a> , <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:625"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="beta:626" class="idref" href="#beta:626"><span class="id" title="binder">beta</span></a> <a id="gamma:627" class="idref" href="#gamma:627"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:626"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:627"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:626"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:627"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:625"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:626"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:625"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:627"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_l_weak" class="idref" href="#plus_mono_l_weak"><span class="id" title="lemma">plus_mono_l_weak</span></a>: <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="o:628" class="idref" href="#o:628"><span class="id" title="binder">o</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:628"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:629" class="idref" href="#alpha:629"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:629"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:629"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:628"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:630" class="idref" href="#beta:630"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:630"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:630"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#o:628"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="gamma:631" class="idref" href="#gamma:631"><span class="id" title="binder">gamma</span></a> , <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:631"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:629"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:630"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:629"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:631"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:630"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:631"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R_pred_Sn" class="idref" href="#R_pred_Sn"><span class="id" title="lemma">R_pred_Sn</span></a> : <span class="id" title="keyword">∀</span> <a id="n:632" class="idref" href="#n:632"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:632"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:632"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_cons" class="idref" href="#pred_of_cons"><span class="id" title="lemma">pred_of_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="a:633" class="idref" href="#a:633"><span class="id" title="binder">a</span></a> <a id="b:634" class="idref" href="#b:634"><span class="id" title="binder">b</span></a> <a id="n:635" class="idref" href="#n:635"><span class="id" title="binder">n</span></a> <a id="c:636" class="idref" href="#c:636"><span class="id" title="binder">c</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:633"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:634"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:633"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:634"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#n:635"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:636"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:636"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">c'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:633"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:634"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:635"><span class="id" title="variable">n</span></a> <span class="id" title="var">c'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_cons'" class="idref" href="#pred_of_cons'"><span class="id" title="lemma">pred_of_cons'</span></a> : <span class="id" title="keyword">∀</span> <a id="a:637" class="idref" href="#a:637"><span class="id" title="binder">a</span></a> <a id="b:638" class="idref" href="#b:638"><span class="id" title="binder">b</span></a> <a id="n:639" class="idref" href="#n:639"><span class="id" title="binder">n</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:637"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:638"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:637"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:638"><span class="id" title="variable">b</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#n:639"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_ab" class="idref" href="#is_limit_ab"><span class="id" title="lemma">is_limit_ab</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:640" class="idref" href="#alpha:640"><span class="id" title="binder">alpha</span></a> <a id="beta:641" class="idref" href="#beta:641"><span class="id" title="binder">beta</span></a> <a id="n:642" class="idref" href="#n:642"><span class="id" title="binder">n</span></a> <a id="gamma:643" class="idref" href="#gamma:643"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:640"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:641"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:642"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:643"><span class="id" title="variable">gamma</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:640"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:641"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_limit" class="idref" href="#pred_of_limit"><span class="id" title="lemma">pred_of_limit</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:644" class="idref" href="#alpha:644"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:644"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:644"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:644"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_succ" class="idref" href="#pred_of_succ"><span class="id" title="lemma">pred_of_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:645" class="idref" href="#alpha:645"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:645"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:645"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:645"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_plus_fin_ok" class="idref" href="#limit_plus_fin_ok"><span class="id" title="lemma">limit_plus_fin_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:646" class="idref" href="#alpha:646"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Gamma0.T2.html#is_limit"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:647" class="idref" href="#n:647"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:647"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:646"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:647"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="phi_to_psi" class="idref" href="#phi_to_psi"><span class="id" title="section">phi_to_psi</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="phi_to_psi.alpha" class="idref" href="#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_1" class="idref" href="#phi_to_psi_1"><span class="id" title="lemma">phi_to_psi_1</span></a> : <span class="id" title="keyword">∀</span> <a id="beta1:649" class="idref" href="#beta1:649"><span class="id" title="binder">beta1</span></a> <a id="beta2:650" class="idref" href="#beta2:650"><span class="id" title="binder">beta2</span></a> <a id="n:651" class="idref" href="#n:651"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:649"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:649"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:650"><span class="id" title="variable">beta2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:651"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:649"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:650"><span class="id" title="variable">beta2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:651"><span class="id" title="variable">n</span></a>))).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_2" class="idref" href="#phi_to_psi_2"><span class="id" title="lemma">phi_to_psi_2</span></a> : <span class="id" title="keyword">∀</span> <a id="beta1:652" class="idref" href="#beta1:652"><span class="id" title="binder">beta1</span></a> <a id="beta2:653" class="idref" href="#beta2:653"><span class="id" title="binder">beta2</span></a> <a id="n:654" class="idref" href="#n:654"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:652"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:652"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:653"><span class="id" title="variable">beta2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:654"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">)]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:652"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:653"><span class="id" title="variable">beta2</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:654"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_3" class="idref" href="#phi_to_psi_3"><span class="id" title="lemma">phi_to_psi_3</span></a> : <span class="id" title="keyword">∀</span>  <a id="beta1:655" class="idref" href="#beta1:655"><span class="id" title="binder">beta1</span></a> <a id="beta2:656" class="idref" href="#beta2:656"><span class="id" title="binder">beta2</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:655"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:655"><span class="id" title="variable">beta1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:656"><span class="id" title="variable">beta2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:655"><span class="id" title="variable">beta1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:656"><span class="id" title="variable">beta2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_4" class="idref" href="#phi_to_psi_4"><span class="id" title="lemma">phi_to_psi_4</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_5" class="idref" href="#phi_to_psi_5"><span class="id" title="lemma">phi_to_psi_5</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta1:657" class="idref" href="#beta1:657"><span class="id" title="binder">beta1</span></a> <a id="beta2:658" class="idref" href="#beta2:658"><span class="id" title="binder">beta2</span></a> <a id="n:659" class="idref" href="#n:659"><span class="id" title="binder">n</span></a> <a id="gamma:660" class="idref" href="#gamma:660"><span class="id" title="binder">gamma</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:660"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:659"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:657"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:658"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:659"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:660"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:657"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:658"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:659"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:660"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_to_psi_6" class="idref" href="#phi_to_psi_6"><span class="id" title="lemma">phi_to_psi_6</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:661" class="idref" href="#beta:661"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:661"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:661"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:661"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:661"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:661"><span class="id" title="variable">beta</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_psi" class="idref" href="#phi_psi"><span class="id" title="lemma">phi_psi</span></a> : <span class="id" title="keyword">∀</span>  <a id="beta:662" class="idref" href="#beta:662"><span class="id" title="binder">beta</span></a> <a id="gamma:663" class="idref" href="#gamma:663"><span class="id" title="binder">gamma</span></a> <a id="n:664" class="idref" href="#n:664"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:663"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:662"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:664"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:663"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:662"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:662"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:663"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:663"><span class="id" title="variable">gamma</span></a>). 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="th_14_5" class="idref" href="#th_14_5"><span class="id" title="lemma">th_14_5</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:665" class="idref" href="#alpha1:665"><span class="id" title="binder">alpha1</span></a> <a id="beta1:666" class="idref" href="#beta1:666"><span class="id" title="binder">beta1</span></a> <a id="alpha2:667" class="idref" href="#alpha2:667"><span class="id" title="binder">alpha2</span></a> <a id="beta2:668" class="idref" href="#beta2:668"><span class="id" title="binder">beta2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:666"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:668"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:666"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:668"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:666"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:668"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:666"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:668"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:667"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:665"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:666"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:668"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_not_gt" class="idref" href="#lt_not_gt"><span class="id" title="lemma">lt_not_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="a:669" class="idref" href="#a:669"><span class="id" title="binder">a</span></a> <a id="b:670" class="idref" href="#b:670"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#a:669"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:670"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#b:670"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:669"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi_mono_RR" class="idref" href="#phi_mono_RR"><span class="id" title="lemma">phi_mono_RR</span></a> : <span class="id" title="keyword">∀</span> <a id="a:671" class="idref" href="#a:671"><span class="id" title="binder">a</span></a> <a id="b:672" class="idref" href="#b:672"><span class="id" title="binder">b</span></a> <a id="c:673" class="idref" href="#c:673"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:671"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:672"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:673"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:671"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:672"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:671"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:673"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:672"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:673"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="th_14_6" class="idref" href="#th_14_6"><span class="id" title="lemma">th_14_6</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:674" class="idref" href="#alpha1:674"><span class="id" title="binder">alpha1</span></a> <a id="beta1:675" class="idref" href="#beta1:675"><span class="id" title="binder">beta1</span></a> <a id="alpha2:676" class="idref" href="#alpha2:676"><span class="id" title="binder">alpha2</span></a> <a id="beta2:677" class="idref" href="#beta2:677"><span class="id" title="binder">beta2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:675"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:677"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:675"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:677"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:675"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:677"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:675"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:677"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#alpha2:676"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha1:674"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta1:675"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta2:677"><span class="id" title="variable">beta2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="moser_lepper" class="idref" href="#moser_lepper"><span class="id" title="definition">moser_lepper</span></a> (<a id="beta0:678" class="idref" href="#beta0:678"><span class="id" title="binder">beta0</span></a> <a id="beta:679" class="idref" href="#beta:679"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>)(<a id="n:680" class="idref" href="#n:680"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#limit_plus_fin"><span class="id" title="inductive">limit_plus_fin</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:678"><span class="id" title="variable">beta0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:680"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:679"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:678"><span class="id" title="variable">beta0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:678"><span class="id" title="variable">beta0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ml_psi" class="idref" href="#ml_psi"><span class="id" title="lemma">ml_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="beta0:681" class="idref" href="#beta0:681"><span class="id" title="binder">beta0</span></a> <a id="beta:682" class="idref" href="#beta:682"><span class="id" title="binder">beta</span></a> <a id="n:683" class="idref" href="#n:683"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.Gamma0.html#moser_lepper"><span class="id" title="definition">moser_lepper</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:681"><span class="id" title="variable">beta0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:682"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:683"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="t1:684" class="idref" href="#t1:684"><span class="id" title="binder">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="t2:685" class="idref" href="#t2:685"><span class="id" title="binder">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:681"><span class="id" title="variable">beta0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t1:684"><span class="id" title="variable">t1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#t2:685"><span class="id" title="variable">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#t1:684"><span class="id" title="variable">t1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ml_1" class="idref" href="#ml_1"><span class="id" title="lemma">ml_1</span></a> : <span class="id" title="keyword">∀</span> <a id="beta0:686" class="idref" href="#beta0:686"><span class="id" title="binder">beta0</span></a> <a id="beta:687" class="idref" href="#beta:687"><span class="id" title="binder">beta</span></a> <a id="n:688" class="idref" href="#n:688"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.Gamma0.html#moser_lepper"><span class="id" title="definition">moser_lepper</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:686"><span class="id" title="variable">beta0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:687"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:688"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:687"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta0:686"><span class="id" title="variable">beta0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:687"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:687"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi_to_psi"><span class="id" title="section">phi_to_psi</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex9" class="idref" href="#Ex9"><span class="id" title="definition">Ex9</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> 2 <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> 4<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> 0 (<a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> 2 <a class="idref" href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8"><span class="id" title="notation">+</span></a> 5).   
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex10" class="idref" href="#Ex10"><span class="id" title="definition">Ex10</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> 5<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> 5<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.   
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="G0" class="idref" href="#G0"><span class="id" title="module">G0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="G0.LT" class="idref" href="#G0.LT"><span class="id" title="definition">LT</span></a> := <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.Lt_wf" class="idref" href="#G0.Lt_wf"><span class="id" title="lemma">Lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.LT"><span class="id" title="definition">LT</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab119"></a><h2 class="section">Temporary compatibility  nf/nfb</h2>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.zero_nfb" class="idref" href="#G0.zero_nfb"><span class="id" title="lemma">zero_nfb</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.nfb_a" class="idref" href="#G0.nfb_a"><span class="id" title="lemma">nfb_a</span></a> <a id="a:689" class="idref" href="#a:689"><span class="id" title="binder">a</span></a> <a id="b:690" class="idref" href="#b:690"><span class="id" title="binder">b</span></a> <a id="n:691" class="idref" href="#n:691"><span class="id" title="binder">n</span></a> <a id="c:692" class="idref" href="#c:692"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:689"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#b:690"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#n:691"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#c:692"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#a:689"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.nfb_equiv" class="idref" href="#G0.nfb_equiv"><span class="id" title="lemma">nfb_equiv</span></a> <a id="gamma:693" class="idref" href="#gamma:693"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:693"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#gamma:693"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.nfb_proof_unicity" class="idref" href="#G0.nfb_proof_unicity"><span class="id" title="lemma">nfb_proof_unicity</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="alpha:694" class="idref" href="#alpha:694"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) (<a id="H:695" class="idref" href="#H:695"><span class="id" title="binder">H</span></a> <a id="H':696" class="idref" href="#H':696"><span class="id" title="binder">H'</span></a>: <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:694"><span class="id" title="variable">alpha</span></a>), <a class="idref" href="hydras.Gamma0.Gamma0.html#H:695"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#H':696"><span class="id" title="variable">H'</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a id="G0.G0" class="idref" href="#G0.G0"><span class="id" title="record">G0</span></a> := <a id="G0.mkg0" class="idref" href="#G0.mkg0"><span class="id" title="constructor">mkg0</span></a> {<a id="G0.vnf" class="idref" href="#G0.vnf"><span class="id" title="projection">vnf</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>; <a id="G0.vnf_ok" class="idref" href="#G0.vnf_ok"><span class="id" title="projection">vnf_ok</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#nfb"><span class="id" title="definition">nfb</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#vnf:698"><span class="id" title="method">vnf</span></a>}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="G0.lt" class="idref" href="#G0.lt"><span class="id" title="definition">lt</span></a> (<a id="alpha:700" class="idref" href="#alpha:700"><span class="id" title="binder">alpha</span></a> <a id="beta:701" class="idref" href="#beta:701"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>) := <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">T2.lt</span></a> (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:700"><span class="id" title="variable">alpha</span></a>) (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:701"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.compare_G0" class="idref" href="#G0.compare_G0"><span class="id" title="instance">compare_G0</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:702" class="idref" href="#alpha:702"><span class="id" title="binder">alpha</span></a> <a id="beta:703" class="idref" href="#beta:703"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:702"><span class="id" title="variable">alpha</span></a>) (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:703"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.lt_LT_incl" class="idref" href="#G0.lt_LT_incl"><span class="id" title="lemma">lt_LT_incl</span></a>  <a id="alpha:704" class="idref" href="#alpha:704"><span class="id" title="binder">alpha</span></a> <a id="beta:705" class="idref" href="#beta:705"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:704"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:705"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.LT"><span class="id" title="definition">LT</span></a> (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:704"><span class="id" title="variable">alpha</span></a>) (@<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.vnf"><span class="id" title="method">vnf</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:705"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.lt_sto" class="idref" href="#G0.lt_sto"><span class="id" title="instance">lt_sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a>. 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="G0.lt_wf" class="idref" href="#G0.lt_wf"><span class="id" title="lemma">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="G0.compare_correct" class="idref" href="#G0.compare_correct"><span class="id" title="lemma">compare_correct</span></a> <a id="alpha:710" class="idref" href="#alpha:710"><span class="id" title="binder">alpha</span></a> <a id="beta:711" class="idref" href="#beta:711"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompareSpec"><span class="id" title="inductive">CompareSpec</span></a> (<a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:710"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:711"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:710"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:711"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:711"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:710"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#alpha:710"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#beta:711"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span>, <span class="id" title="var">program</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.zero" class="idref" href="#G0.zero"><span class="id" title="instance">zero</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a> := @<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.mkg0"><span class="id" title="constructor">mkg0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">T2.zero</span></a> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span>, <span class="id" title="var">program</span>] <span class="id" title="keyword">Instance</span> <a id="G0.Omega" class="idref" href="#G0.Omega"><span class="id" title="instance">Omega</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a> := @<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.mkg0"><span class="id" title="constructor">mkg0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="G0.omega" class="idref" href="#G0.omega"><span class="id" title="abbreviation">omega</span></a> := <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Omega"><span class="id" title="instance">Omega</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="G0.le" class="idref" href="#G0.le"><span class="id" title="definition">le</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.Gamma0_comp" class="idref" href="#G0.Gamma0_comp"><span class="id" title="instance">Gamma0_comp</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>. 
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.Gamma0" class="idref" href="#G0.Gamma0"><span class="id" title="instance">Gamma0</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>. 
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.Finite" class="idref" href="#G0.Finite"><span class="id" title="instance">Finite</span></a> (<a id="n:712" class="idref" href="#n:712"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.Plus" class="idref" href="#G0.Plus"><span class="id" title="instance">Plus</span></a> (<a id="alpha:713" class="idref" href="#alpha:713"><span class="id" title="binder">alpha</span></a> <a id="beta:714" class="idref" href="#beta:714"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>) : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="f8274dee76f0f0f5010365f2cfa67705" class="idref" href="#f8274dee76f0f0f5010365f2cfa67705"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Plus"><span class="id" title="instance">Plus</span></a> : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="G0.Phi" class="idref" href="#G0.Phi"><span class="id" title="instance">Phi</span></a> (<a id="alpha:715" class="idref" href="#alpha:715"><span class="id" title="binder">alpha</span></a> <a id="beta:716" class="idref" href="#beta:716"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>) : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.G0"><span class="id" title="class">G0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="G0.phi" class="idref" href="#G0.phi"><span class="id" title="abbreviation">phi</span></a> := <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Phi"><span class="id" title="instance">Phi</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="G0.phi0" class="idref" href="#G0.phi0"><span class="id" title="abbreviation">phi0</span></a> := (<a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Phi"><span class="id" title="instance">Phi</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.zero"><span class="id" title="instance">zero</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="e70a85d390e2078e674210edfffae3cb" class="idref" href="#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">&quot;</span></a>'omega^'" := <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.phi0"><span class="id" title="abbreviation">phi0</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Finite"><span class="id" title="instance">Finite</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Finite"><span class="id" title="instance">:</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Finite"><span class="id" title="instance">nat</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Finite"><span class="id" title="instance">&gt;-&gt;</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.Finite"><span class="id" title="instance">G0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">g0_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="G0.Ex42" class="idref" href="#G0.Ex42"><span class="id" title="definition">Ex42</span></a> : <a class="idref" href="hydras.Gamma0.Gamma0.html#G0.omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#f8274dee76f0f0f5010365f2cfa67705"><span class="id" title="notation">+</span></a> 42 <a class="idref" href="hydras.Gamma0.Gamma0.html#f8274dee76f0f0f5010365f2cfa67705"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb"><span class="id" title="notation">2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.Gamma0.html#G0"><span class="id" title="module">G0</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Gamma0.T2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Gamma0.T2</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Data-type for Veblen normal form 
+   (ordinals below Gamma0)   
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a>  <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Restriction</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Prelude.OrdNotations.html#"><span class="id" title="library">OrdNotations</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab109"></a><h1 class="section">Definitions</h1>
+
+<div class="paragraph"> </div>
+
+  <span class="inlinecode"><span class="id" title="var">gcons</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">gamma</span></span> is : <span class="inlinecode"><span class="id" title="var">psi</span>(<span class="id" title="var">alpha</span>,<span class="id" title="keyword">beta</span>)*(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)+</span> <span class="inlinecode"><span class="id" title="var">gamma</span></span>  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">T2_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">T2_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">t2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="T2" class="idref" href="#T2"><span class="id" title="inductive">T2</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+| <a id="zero" class="idref" href="#zero"><span class="id" title="constructor">zero</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2:1"><span class="id" title="inductive">T2</span></a><br/>
+| <a id="gcons" class="idref" href="#gcons"><span class="id" title="constructor">gcons</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2:1"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2:1"><span class="id" title="inductive">T2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2:1"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2:1"><span class="id" title="inductive">T2</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="19285cf1708f341b0870d51774920007" class="idref" href="#19285cf1708f341b0870d51774920007"><span class="id" title="notation">&quot;</span></a>[ alpha , beta ]" := (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0): <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="psi" class="idref" href="#psi"><span class="id" title="definition">psi</span></a> <a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> <a id="beta:4" class="idref" href="#beta:4"><span class="id" title="binder">beta</span></a>  := <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#alpha:3"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:4"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="psi_term" class="idref" href="#psi_term"><span class="id" title="definition">psi_term</span></a> <a id="alpha:5" class="idref" href="#alpha:5"><span class="id" title="binder">alpha</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#alpha:5"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="psi_eq" class="idref" href="#psi_eq"><span class="id" title="lemma">psi_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#psi"><span class="id" title="definition">psi</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:8"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#a:7"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#b:8"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">fold_psi</span> :=  <span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="hydras.Gamma0.T2.html#psi_eq"><span class="id" title="lemma">psi_eq</span></a>.<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">fold_psis</span> := <span class="id" title="tactic">repeat</span> <span class="id" title="var">fold_psi</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span>  <a id="one" class="idref" href="#one"><span class="id" title="abbreviation">one</span></a>  := <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="FS" class="idref" href="#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fin" class="idref" href="#fin"><span class="id" title="definition">fin</span></a> (<a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">T2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="omega" class="idref" href="#omega"><span class="id" title="abbreviation">omega</span></a>  := <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="epsilon0" class="idref" href="#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>  := <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="epsilon" class="idref" href="#epsilon"><span class="id" title="definition">epsilon</span></a> <a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> := <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:11"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="is_finite" class="idref" href="#is_finite"><span class="id" title="inductive">is_finite</span></a>:  <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="keyword">Set</span> := <br/>
+&nbsp;<a id="zero_finite" class="idref" href="#zero_finite"><span class="id" title="constructor">zero_finite</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#is_finite:12"><span class="id" title="inductive">is_finite</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+|<a id="succ_finite" class="idref" href="#succ_finite"><span class="id" title="constructor">succ_finite</span></a> : <span class="id" title="keyword">∀</span> <a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#is_finite:12"><span class="id" title="inductive">is_finite</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Gamma0.T2.html#is_finite"><span class="id" title="inductive">is_finite</span></a> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="T1_to_T2" class="idref" href="#T1_to_T2"><span class="id" title="definition">T1_to_T2</span></a> (<a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> :<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#alpha:15"><span class="id" title="variable">alpha</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2:16"><span class="id" title="definition">T1_to_T2</span></a> <span class="id" title="var">a</span>) <span class="id" title="var">n</span> (<a class="idref" href="hydras.Gamma0.T2.html#T1_to_T2:16"><span class="id" title="definition">T1_to_T2</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+additive principals 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="ap" class="idref" href="#ap"><span class="id" title="inductive">ap</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+<a id="ap_intro" class="idref" href="#ap_intro"><span class="id" title="constructor">ap_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a> <a id="beta:21" class="idref" href="#beta:21"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#ap:18"><span class="id" title="inductive">ap</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#alpha:20"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:21"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+strict order on terms 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&gt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&gt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;=' y 't2&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;=' y 't2&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;' y 't2&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 't2&lt;' y 't2&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="lt" class="idref" href="#lt"><span class="id" title="inductive">lt</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <br/>
+&nbsp;<a id="lt_1" class="idref" href="#lt_1"><span class="id" title="constructor">lt_1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:24" class="idref" href="#alpha:24"><span class="id" title="binder">alpha</span></a> <a id="beta:25" class="idref" href="#beta:25"><span class="id" title="binder">beta</span></a> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> <a id="gamma:27" class="idref" href="#gamma:27"><span class="id" title="binder">gamma</span></a>,  <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:25"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma:27"><span class="id" title="variable">gamma</span></a><br/>
+| <br/>
+&nbsp;<a id="lt_2" class="idref" href="#lt_2"><span class="id" title="constructor">lt_2</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:28" class="idref" href="#alpha1:28"><span class="id" title="binder">alpha1</span></a> <a id="alpha2:29" class="idref" href="#alpha2:29"><span class="id" title="binder">alpha2</span></a> <a id="beta1:30" class="idref" href="#beta1:30"><span class="id" title="binder">beta1</span></a> <a id="beta2:31" class="idref" href="#beta2:31"><span class="id" title="binder">beta2</span></a> <a id="n1:32" class="idref" href="#n1:32"><span class="id" title="binder">n1</span></a> <a id="n2:33" class="idref" href="#n2:33"><span class="id" title="binder">n2</span></a> <a id="gamma1:34" class="idref" href="#gamma1:34"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:35" class="idref" href="#gamma2:35"><span class="id" title="binder">gamma2</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#alpha1:28"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha2:29"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#beta1:30"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha2:29"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:31"><span class="id" title="variable">beta2</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:28"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:30"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:32"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:34"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha2:29"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:31"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:33"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:35"><span class="id" title="variable">gamma2</span></a><br/>
+| <br/>
+&nbsp;<a id="lt_3" class="idref" href="#lt_3"><span class="id" title="constructor">lt_3</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:36" class="idref" href="#alpha1:36"><span class="id" title="binder">alpha1</span></a>  <a id="beta1:37" class="idref" href="#beta1:37"><span class="id" title="binder">beta1</span></a> <a id="beta2:38" class="idref" href="#beta2:38"><span class="id" title="binder">beta2</span></a> <a id="n1:39" class="idref" href="#n1:39"><span class="id" title="binder">n1</span></a> <a id="n2:40" class="idref" href="#n2:40"><span class="id" title="binder">n2</span></a> <a id="gamma1:41" class="idref" href="#gamma1:41"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:42" class="idref" href="#gamma2:42"><span class="id" title="binder">gamma2</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#beta1:37"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:38"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:36"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:37"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:39"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:41"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:36"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:38"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:40"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:42"><span class="id" title="variable">gamma2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+| <br/>
+<a id="lt_4" class="idref" href="#lt_4"><span class="id" title="constructor">lt_4</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:43" class="idref" href="#alpha1:43"><span class="id" title="binder">alpha1</span></a> <a id="alpha2:44" class="idref" href="#alpha2:44"><span class="id" title="binder">alpha2</span></a> <a id="beta1:45" class="idref" href="#beta1:45"><span class="id" title="binder">beta1</span></a> <a id="beta2:46" class="idref" href="#beta2:46"><span class="id" title="binder">beta2</span></a> <a id="n1:47" class="idref" href="#n1:47"><span class="id" title="binder">n1</span></a> <a id="n2:48" class="idref" href="#n2:48"><span class="id" title="binder">n2</span></a> <a id="gamma1:49" class="idref" href="#gamma1:49"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:50" class="idref" href="#gamma2:50"><span class="id" title="binder">gamma2</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#alpha2:44"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:43"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#alpha1:43"><span class="id" title="variable">alpha1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:45"><span class="id" title="variable">beta1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:46"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:43"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:45"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:47"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:49"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha2:44"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta2:46"><span class="id" title="variable">beta2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:48"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:50"><span class="id" title="variable">gamma2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+| <br/>
+<a id="lt_5" class="idref" href="#lt_5"><span class="id" title="constructor">lt_5</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:51" class="idref" href="#alpha1:51"><span class="id" title="binder">alpha1</span></a> <a id="alpha2:52" class="idref" href="#alpha2:52"><span class="id" title="binder">alpha2</span></a> <a id="beta1:53" class="idref" href="#beta1:53"><span class="id" title="binder">beta1</span></a> <a id="n1:54" class="idref" href="#n1:54"><span class="id" title="binder">n1</span></a> <a id="n2:55" class="idref" href="#n2:55"><span class="id" title="binder">n2</span></a> <a id="gamma1:56" class="idref" href="#gamma1:56"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:57" class="idref" href="#gamma2:57"><span class="id" title="binder">gamma2</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#alpha2:52"><span class="id" title="variable">alpha2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:51"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:51"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:53"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:54"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:56"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha2:52"><span class="id" title="variable">alpha2</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#alpha1:51"><span class="id" title="variable">alpha1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:53"><span class="id" title="variable">beta1</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:55"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:57"><span class="id" title="variable">gamma2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+| <br/>
+<a id="lt_6" class="idref" href="#lt_6"><span class="id" title="constructor">lt_6</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:58" class="idref" href="#alpha1:58"><span class="id" title="binder">alpha1</span></a> <a id="beta1:59" class="idref" href="#beta1:59"><span class="id" title="binder">beta1</span></a>  <a id="n1:60" class="idref" href="#n1:60"><span class="id" title="binder">n1</span></a>  <a id="n2:61" class="idref" href="#n2:61"><span class="id" title="binder">n2</span></a> <a id="gamma1:62" class="idref" href="#gamma1:62"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:63" class="idref" href="#gamma2:63"><span class="id" title="binder">gamma2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Gamma0.T2.html#n1:60"><span class="id" title="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:61"><span class="id" title="variable">n2</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:58"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:59"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:60"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:62"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:58"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:59"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n2:61"><span class="id" title="variable">n2</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:63"><span class="id" title="variable">gamma2</span></a><br/>
+<br/>
+| <br/>
+<a id="lt_7" class="idref" href="#lt_7"><span class="id" title="constructor">lt_7</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha1:64" class="idref" href="#alpha1:64"><span class="id" title="binder">alpha1</span></a> <a id="beta1:65" class="idref" href="#beta1:65"><span class="id" title="binder">beta1</span></a> <a id="n1:66" class="idref" href="#n1:66"><span class="id" title="binder">n1</span></a>   <a id="gamma1:67" class="idref" href="#gamma1:67"><span class="id" title="binder">gamma1</span></a> <a id="gamma2:68" class="idref" href="#gamma2:68"><span class="id" title="binder">gamma2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gamma1:67"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:68"><span class="id" title="variable">gamma2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:64"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:65"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:66"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma1:67"><span class="id" title="variable">gamma1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha1:64"><span class="id" title="variable">alpha1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta1:65"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n1:66"><span class="id" title="variable">n1</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma2:68"><span class="id" title="variable">gamma2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+<span class="id" title="keyword">where</span>  <a id="::T2_scope:x_'t2&lt;'_x" class="idref" href="#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&quot;</span></a>o1 t2&lt; o2" := (<a class="idref" href="hydras.Gamma0.T2.html#lt:23"><span class="id" title="inductive">lt</span></a> <span class="id" title="var">o1</span> <span class="id" title="var">o2</span>): <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Gamma0.T2.html#lt"><span class="id" title="inductive">lt</span></a> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="le" class="idref" href="#le"><span class="id" title="definition">le</span></a> <a id="t:69" class="idref" href="#t:69"><span class="id" title="binder">t</span></a> <a id="t':70" class="idref" href="#t':70"><span class="id" title="binder">t'</span></a> := <a class="idref" href="hydras.Gamma0.T2.html#t:69"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t':70"><span class="id" title="variable">t'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t:69"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t':70"><span class="id" title="variable">t'</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Gamma0.T2.html#le"><span class="id" title="definition">le</span></a> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="700fc75f60b054fa3b8ffac2d7cf60a3" class="idref" href="#700fc75f60b054fa3b8ffac2d7cf60a3"><span class="id" title="notation">&quot;</span></a>o1 t2&lt;= o2" := (<a class="idref" href="hydras.Gamma0.T2.html#le"><span class="id" title="definition">le</span></a> <span class="id" title="var">o1</span> <span class="id" title="var">o2</span>): <span class="id" title="var">T2_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab110"></a><h3 class="section">Examples</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a>: 0 <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+  
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex3" class="idref" href="#Ex3"><span class="id" title="definition">Ex3</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> 8 12 56 <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> 8 12 57.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex4" class="idref" href="#Ex4"><span class="id" title="definition">Ex4</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex5" class="idref" href="#Ex5"><span class="id" title="definition">Ex5</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>3<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex6" class="idref" href="#Ex6"><span class="id" title="definition">Ex6</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> 1 0 12 <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>0<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex7" class="idref" href="#Ex7"><span class="id" title="definition">Ex7</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> 2 1 42 <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a>2<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a>1<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]]</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="gtail" class="idref" href="#gtail"><span class="id" title="definition">gtail</span></a> <a id="c:71" class="idref" href="#c:71"><span class="id" title="binder">c</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#c:71"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#c:71"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="nf" class="idref" href="#nf"><span class="id" title="inductive">nf</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="zero_nf" class="idref" href="#zero_nf"><span class="id" title="constructor">zero_nf</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+| <a id="single_nf" class="idref" href="#single_nf"><span class="id" title="constructor">single_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="a:75" class="idref" href="#a:75"><span class="id" title="binder">a</span></a> <a id="b:76" class="idref" href="#b:76"><span class="id" title="binder">b</span></a> <a id="n:77" class="idref" href="#n:77"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:77"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+| <a id="gcons_nf" class="idref" href="#gcons_nf"><span class="id" title="constructor">gcons_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="a:78" class="idref" href="#a:78"><span class="id" title="binder">a</span></a> <a id="b:79" class="idref" href="#b:79"><span class="id" title="binder">b</span></a> <a id="n:80" class="idref" href="#n:80"><span class="id" title="binder">n</span></a> <a id="a':81" class="idref" href="#a':81"><span class="id" title="binder">a'</span></a> <a id="b':82" class="idref" href="#b':82"><span class="id" title="binder">b'</span></a> <a id="n':83" class="idref" href="#n':83"><span class="id" title="binder">n'</span></a> <a id="c':84" class="idref" href="#c':84"><span class="id" title="binder">c'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#a':81"><span class="id" title="variable">a'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b':82"><span class="id" title="variable">b'</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#a:78"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:79"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:79"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a>(<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a':81"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b':82"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n':83"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c':84"><span class="id" title="variable">c'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf:73"><span class="id" title="inductive">nf</span></a>(<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:79"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:80"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a':81"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b':82"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n':83"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c':84"><span class="id" title="variable">c'</span></a>)).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> : <span class="id" title="var">T2</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="nf_fin" class="idref" href="#nf_fin"><span class="id" title="lemma">nf_fin</span></a> <a id="i:85" class="idref" href="#i:85"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.Gamma0.T2.html#i:85"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_omega" class="idref" href="#nf_omega"><span class="id" title="lemma">nf_omega</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_epsilon0" class="idref" href="#nf_epsilon0"><span class="id" title="lemma">nf_epsilon0</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_epsilon" class="idref" href="#nf_epsilon"><span class="id" title="lemma">nf_epsilon</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:86" class="idref" href="#alpha:86"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:86"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:86"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex8" class="idref" href="#Ex8"><span class="id" title="definition">Ex8</span></a>: <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> 2 1 42 <a class="idref" href="hydras.Gamma0.T2.html#epsilon0"><span class="id" title="abbreviation">epsilon0</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="is_successor" class="idref" href="#is_successor"><span class="id" title="inductive">is_successor</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="finite_succ" class="idref" href="#finite_succ"><span class="id" title="constructor">finite_succ</span></a> : <span class="id" title="keyword">∀</span>  <a id="n:89" class="idref" href="#n:89"><span class="id" title="binder">n</span></a>  , <a class="idref" href="hydras.Gamma0.T2.html#is_successor:87"><span class="id" title="inductive">is_successor</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:89"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;|<a id="cons_succ" class="idref" href="#cons_succ"><span class="id" title="constructor">cons_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="a:90" class="idref" href="#a:90"><span class="id" title="binder">a</span></a> <a id="b:91" class="idref" href="#b:91"><span class="id" title="binder">b</span></a> <a id="n:92" class="idref" href="#n:92"><span class="id" title="binder">n</span></a> <a id="c:93" class="idref" href="#c:93"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:92"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:93"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#is_successor:87"><span class="id" title="inductive">is_successor</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:93"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_successor:87"><span class="id" title="inductive">is_successor</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a>  <a class="idref" href="hydras.Gamma0.T2.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:92"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:93"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="is_limit" class="idref" href="#is_limit"><span class="id" title="inductive">is_limit</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="is_limit_0" class="idref" href="#is_limit_0"><span class="id" title="constructor">is_limit_0</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:96" class="idref" href="#alpha:96"><span class="id" title="binder">alpha</span></a> <a id="beta:97" class="idref" href="#beta:97"><span class="id" title="binder">beta</span></a> <a id="n:98" class="idref" href="#n:98"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">t2</span></a><a class="idref" href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:97"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:97"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit:94"><span class="id" title="inductive">is_limit</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:96"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:97"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:98"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+| <a id="is_limit_cons" class="idref" href="#is_limit_cons"><span class="id" title="constructor">is_limit_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:99" class="idref" href="#alpha:99"><span class="id" title="binder">alpha</span></a>  <a id="beta:100" class="idref" href="#beta:100"><span class="id" title="binder">beta</span></a> <a id="n:101" class="idref" href="#n:101"><span class="id" title="binder">n</span></a> <a id="gamma:102" class="idref" href="#gamma:102"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit:94"><span class="id" title="inductive">is_limit</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma:102"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nf"><span class="id" title="inductive">nf</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:99"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:100"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:101"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma:102"><span class="id" title="variable">gamma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#is_limit:94"><span class="id" title="inductive">is_limit</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#alpha:99"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Gamma0.T2.html#beta:100"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:101"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#gamma:102"><span class="id" title="variable">gamma</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="succ" class="idref" href="#succ"><span class="id" title="definition">succ</span></a> (<a id="a:103" class="idref" href="#a:103"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#a:103"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#one"><span class="id" title="abbreviation">one</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#fin"><span class="id" title="definition">fin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:103"><span class="id" title="variable">a</span></a> <span class="id" title="var">b</span> <span class="id" title="var">n</span> (<a class="idref" href="hydras.Gamma0.T2.html#succ:104"><span class="id" title="definition">succ</span></a> <span class="id" title="var">c</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="pred" class="idref" href="#pred"><span class="id" title="definition">pred</span></a>  (<a id="a:106" class="idref" href="#a:106"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#a:106"><span class="id" title="variable">a</span></a>  <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ⇒ (<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Gamma0.T2.html#pred:107"><span class="id" title="definition">pred</span></a> <span class="id" title="var">c</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">c'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:106"><span class="id" title="variable">a</span></a> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">c'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="lt_epsilon0" class="idref" href="#lt_epsilon0"><span class="id" title="inductive">lt_epsilon0</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="zero_lt_e0" class="idref" href="#zero_lt_e0"><span class="id" title="constructor">zero_lt_e0</span></a> : <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0:109"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <br/>
+| <a id="gcons_lt_e0" class="idref" href="#gcons_lt_e0"><span class="id" title="constructor">gcons_lt_e0</span></a> : <span class="id" title="keyword">∀</span>  <a id="b:111" class="idref" href="#b:111"><span class="id" title="binder">b</span></a> <a id="n:112" class="idref" href="#n:112"><span class="id" title="binder">n</span></a> <a id="c:113" class="idref" href="#c:113"><span class="id" title="binder">c</span></a>,  <a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0:109"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:111"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0:109"><span class="id" title="inductive">lt_epsilon0</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:113"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#lt_epsilon0:109"><span class="id" title="inductive">lt_epsilon0</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:111"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:112"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:113"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab111"></a><h2 class="section">Length (as in Schutte)</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="on_length" class="idref" href="#on_length"><span class="id" title="section">on_length</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="nbterms" class="idref" href="#nbterms"><span class="id" title="definition">nbterms</span></a> (<a id="t:114" class="idref" href="#t:114"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#t:114"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#nbterms:115"><span class="id" title="definition">nbterms</span></a> <span class="id" title="var">v</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="t2_length" class="idref" href="#t2_length"><span class="id" title="definition">t2_length</span></a> (<a id="t:117" class="idref" href="#t:117"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#t:117"><span class="id" title="variable">t</span></a>  <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#nbterms"><span class="id" title="definition">nbterms</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#t2_length:119"><span class="id" title="definition">t2_length</span></a> <span class="id" title="var">a</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#t2_length:119"><span class="id" title="definition">t2_length</span></a> <span class="id" title="var">b</span>) (<a class="idref" href="hydras.Gamma0.T2.html#t2_length_aux:120"><span class="id" title="definition">t2_length_aux</span></a> <span class="id" title="var">v</span>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">with</span> <a id="t2_length_aux" class="idref" href="#t2_length_aux"><span class="id" title="definition">t2_length_aux</span></a> (<a id="t:118" class="idref" href="#t:118"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.Gamma0.T2.html#T2"><span class="id" title="inductive">T2</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Gamma0.T2.html#t:118"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">n</span> <span class="id" title="var">v</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#t2_length:119"><span class="id" title="definition">t2_length</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#t2_length:119"><span class="id" title="definition">t2_length</span></a> <span class="id" title="var">b</span>) (<a class="idref" href="hydras.Gamma0.T2.html#t2_length_aux:120"><span class="id" title="definition">t2_length_aux</span></a> <span class="id" title="var">v</span>))<br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_a" class="idref" href="#length_a"><span class="id" title="lemma">length_a</span></a> : <span class="id" title="keyword">∀</span> <a id="a:123" class="idref" href="#a:123"><span class="id" title="binder">a</span></a> <a id="b:124" class="idref" href="#b:124"><span class="id" title="binder">b</span></a> <a id="n:125" class="idref" href="#n:125"><span class="id" title="binder">n</span></a> <a id="v:126" class="idref" href="#v:126"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:123"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:123"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:124"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:125"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#v:126"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_b" class="idref" href="#length_b"><span class="id" title="lemma">length_b</span></a> : <span class="id" title="keyword">∀</span> <a id="a:127" class="idref" href="#a:127"><span class="id" title="binder">a</span></a> <a id="b:128" class="idref" href="#b:128"><span class="id" title="binder">b</span></a> <a id="n:129" class="idref" href="#n:129"><span class="id" title="binder">n</span></a> <a id="v:130" class="idref" href="#v:130"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:128"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:127"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:128"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:129"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#v:130"><span class="id" title="variable">v</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_c" class="idref" href="#length_c"><span class="id" title="lemma">length_c</span></a> : <span class="id" title="keyword">∀</span> <a id="a:131" class="idref" href="#a:131"><span class="id" title="binder">a</span></a> <a id="b:132" class="idref" href="#b:132"><span class="id" title="binder">b</span></a> <a id="n:133" class="idref" href="#n:133"><span class="id" title="binder">n</span></a> <a id="v:134" class="idref" href="#v:134"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#v:134"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:131"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:132"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:133"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#v:134"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_n" class="idref" href="#length_n"><span class="id" title="lemma">length_n</span></a> : <span class="id" title="keyword">∀</span> <a id="a:135" class="idref" href="#a:135"><span class="id" title="binder">a</span></a> <a id="b:136" class="idref" href="#b:136"><span class="id" title="binder">b</span></a> <a id="r:137" class="idref" href="#r:137"><span class="id" title="binder">r</span></a> <a id="n:138" class="idref" href="#n:138"><span class="id" title="binder">n</span></a> <a id="p:139" class="idref" href="#p:139"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Gamma0.T2.html#n:138"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Gamma0.T2.html#p:139"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:135"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:136"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:138"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#r:137"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:135"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:136"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#p:139"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Gamma0.T2.html#r:137"><span class="id" title="variable">r</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_psi" class="idref" href="#length_psi"><span class="id" title="lemma">length_psi</span></a> : <span class="id" title="keyword">∀</span> <a id="a:140" class="idref" href="#a:140"><span class="id" title="binder">a</span></a> <a id="b:141" class="idref" href="#b:141"><span class="id" title="binder">b</span></a> <a id="n:142" class="idref" href="#n:142"><span class="id" title="binder">n</span></a> <a id="c:143" class="idref" href="#c:143"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Gamma0.T2.html#a:140"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:141"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:140"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:141"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:142"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:143"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_ab" class="idref" href="#length_ab"><span class="id" title="lemma">length_ab</span></a> : <span class="id" title="keyword">∀</span> <a id="a:144" class="idref" href="#a:144"><span class="id" title="binder">a</span></a> <a id="b:145" class="idref" href="#b:145"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:144"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:145"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:144"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:145"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="hydras.Gamma0.T2.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_abnc" class="idref" href="#length_abnc"><span class="id" title="lemma">length_abnc</span></a> : <span class="id" title="keyword">∀</span> <a id="a:146" class="idref" href="#a:146"><span class="id" title="binder">a</span></a> <a id="b:147" class="idref" href="#b:147"><span class="id" title="binder">b</span></a> <a id="n:148" class="idref" href="#n:148"><span class="id" title="binder">n</span></a> <a id="c:149" class="idref" href="#c:149"><span class="id" title="binder">c</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:146"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:147"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Gamma0.T2.html#t2_length"><span class="id" title="definition">t2_length</span></a> (<a class="idref" href="hydras.Gamma0.T2.html#gcons"><span class="id" title="constructor">gcons</span></a> <a class="idref" href="hydras.Gamma0.T2.html#a:146"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Gamma0.T2.html#b:147"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Gamma0.T2.html#n:148"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Gamma0.T2.html#c:149"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Gamma0.T2.html#on_length"><span class="id" title="section">on_length</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Battle_length.html b/theories/html/hydras.Hydra.Battle_length.html
new file mode 100644
index 00000000..d1ab34bb
--- /dev/null
+++ b/theories/html/hydras.Hydra.Battle_length.html
@@ -0,0 +1,119 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Battle_length</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Battle_length</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+   "Computing" standard battle length
+
+<div class="paragraph"> </div>
+
+Pierre Castéran, LaBRI, University of Bordeaux
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.L_alpha.html#"><span class="id" title="library">L_alpha</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#"><span class="id" title="library">Epsilon0_Needed_Free</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#"><span class="id" title="library">Epsilon0_Needed_Std</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#"><span class="id" title="library">Hydra_Termination</span></a> <a class="idref" href="hydras.Hydra.O2H.html#"><span class="id" title="library">O2H</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="module">E0</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="module">Large_Sets</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#"><span class="id" title="module">Hprime</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="module">Paths</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="module">MoreLists</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Battle_length" class="idref" href="#Battle_length"><span class="id" title="section">Battle_length</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Battle_length.alpha" class="idref" href="#Battle_length.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Battle_length.Halpha" class="idref" href="#Battle_length.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Battle_length.k" class="idref" href="#Battle_length.k"><span class="id" title="variable">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Battle_length.Hk" class="idref" href="#Battle_length.Hk"><span class="id" title="variable">Hk</span></a> : (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Battle_length.l" class="idref" href="#Battle_length.l"><span class="id" title="variable">l</span></a> := <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R0" class="idref" href="#R0"><span class="id" title="lemma">R0</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="hydras.Epsilon0.Large_Sets.html#mlarge"><span class="id" title="definition">mlarge</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>)  (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4" class="idref" href="#R4"><span class="id" title="lemma">R4</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">O2H.iota</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">O2H.iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>)).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">O2H.iota</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a>  <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">O2H.iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R6" class="idref" href="#R6"><span class="id" title="lemma">R6</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a>1)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L06" class="idref" href="#L06"><span class="id" title="lemma">L06</span></a>:  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a>1)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="battle_length_std" class="idref" href="#battle_length_std"><span class="id" title="lemma">battle_length_std</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_length"><span class="id" title="definition">battle_length</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.alpha"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a><a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length.k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>. 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Battle_length.html#Battle_length"><span class="id" title="section">Battle_length</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="l_std" class="idref" href="#l_std"><span class="id" title="definition">l_std</span></a> <a id="alpha:5" class="idref" href="#alpha:5"><span class="id" title="binder">alpha</span></a> <a id="k:6" class="idref" href="#k:6"><span class="id" title="binder">k</span></a> := (<a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#alpha:5"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#k:6"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#k:6"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="l_std_ok" class="idref" href="#l_std_ok"><span class="id" title="lemma">l_std_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Battle_length.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:8" class="idref" href="#k:8"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#k:8"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_length"><span class="id" title="definition">battle_length</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#k:8"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#alpha:7"><span class="id" title="variable">alpha</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Battle_length.html#l_std"><span class="id" title="definition">l_std</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#k:8"><span class="id" title="variable">k</span></a>).  
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.BigBattle.html b/theories/html/hydras.Hydra.BigBattle.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Hydra.BigBattle.html
@@ -0,0 +1,501 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.BigBattle</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.BigBattle</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab217"></a><h1 class="section">A long hydra battle</h1>
+
+
+<div class="paragraph"> </div>
+
+  Pierre Castéran, LaBRI, Univ Bordeaux 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="library">Hydra_Definitions</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us consider a small hydra <span class="inlinecode"><span class="id" title="var">hinit</span></span> 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span>  <a id="h3" class="idref" href="#h3"><span class="id" title="abbreviation">h3</span></a> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 3).<br/>
+<span class="id" title="keyword">Definition</span> <a id="hinit" class="idref" href="#hinit"><span class="id" title="definition">hinit</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd3"><span class="id" title="abbreviation">hyd3</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#h3"><span class="id" title="abbreviation">h3</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+the first steps of a standard battle (Hercules chops off the rightmost head)
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_0_2" class="idref" href="#L_0_2"><span class="id" title="lemma">L_0_2</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 0 <a class="idref" href="hydras.Hydra.BigBattle.html#hinit"><span class="id" title="definition">hinit</span></a> 2 (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#h3"><span class="id" title="abbreviation">h3</span></a>).<br/>
+  
+<br/>
+</div>
+
+<div class="doc">
+In the next round, there is a replication of h3 into 3 copies of h2 
+
+<div class="paragraph"> </div>
+
+More generally,  all the future hydras will be composed of several copies
+of h2, then several copies of h1, then several heads 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="h2" class="idref" href="#h2"><span class="id" title="abbreviation">h2</span></a> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2).<br/>
+<span class="id" title="keyword">Notation</span> <a id="h1" class="idref" href="#h1"><span class="id" title="abbreviation">h1</span></a> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hyd" class="idref" href="#hyd"><span class="id" title="abbreviation">hyd</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> := <br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#h2"><span class="id" title="abbreviation">h2</span></a>  <span class="id" title="var">a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#h1"><span class="id" title="abbreviation">h1</span></a>  <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <span class="id" title="var">c</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)))).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_2_3" class="idref" href="#L_2_3"><span class="id" title="lemma">L_2_3</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 2 (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#h3"><span class="id" title="abbreviation">h3</span></a>)  3 (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 3 0 0).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L_0_3" class="idref" href="#L_0_3"><span class="id" title="lemma">L_0_3</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 0 <a class="idref" href="hydras.Hydra.BigBattle.html#hinit"><span class="id" title="definition">hinit</span></a> 3 (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 3 0 0).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+From now on, we abstract the configurations of the battle
+  as tuples (round number, n2, n1, nh) where n2 (resp. n1, nh) is the number of daughters of type h2 <span class="inlinecode"><span class="id" title="var">resp</span>.</span> <span class="inlinecode"><span class="id" title="var">h1</span>,</span> <span class="inlinecode"><span class="id" title="var">heads</span></span> 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="state" class="idref" href="#state"><span class="id" title="record">state</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;&nbsp;<a id="mks" class="idref" href="#mks"><span class="id" title="constructor">mks</span></a> {<a id="round" class="idref" href="#round"><span class="id" title="projection">round</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ; <a id="n2" class="idref" href="#n2"><span class="id" title="projection">n2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ; <a id="n1" class="idref" href="#n1"><span class="id" title="projection">n1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ; <a id="nh" class="idref" href="#nh"><span class="id" title="projection">nh</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="next" class="idref" href="#next"><span class="id" title="definition">next</span></a> (<a id="s:6" class="idref" href="#s:6"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#state"><span class="id" title="record">state</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.BigBattle.html#s:6"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#round"><span class="id" title="projection">round</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">c</span>) ⇒ <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">round</span>) <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#round"><span class="id" title="projection">round</span></a> <span class="id" title="var">a</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">b</span>) 0 ⇒ <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">round</span>) <span class="id" title="var">a</span> <span class="id" title="var">b</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">round</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#round"><span class="id" title="projection">round</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">a</span>) 0 0 ⇒ <a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">round</span>) <span class="id" title="var">a</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">round</span>) 0<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Hydra.BigBattle.html#s:6"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+returns the state at the n-th round 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="test" class="idref" href="#test"><span class="id" title="definition">test</span></a> <a id="n:8" class="idref" href="#n:8"><span class="id" title="binder">n</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#next"><span class="id" title="definition">next</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#n:8"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a>3)  (<a class="idref" href="hydras.Hydra.BigBattle.html#mks"><span class="id" title="constructor">mks</span></a> 3 3 0 0).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="doubleS" class="idref" href="#doubleS"><span class="id" title="definition">doubleS</span></a> (<a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:9"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+OK, instead of tests based on the function next, we consider now
+    relations (which allow us to consider accessibility predicates)  
+<div class="paragraph"> </div>
+
+ steps  i a b c j a' b' c' has  almost the same meaning as
+    iterate test (j - i) (mks t a b c) = (mks t' a' b' c')
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="one_step" class="idref" href="#one_step"><span class="id" title="inductive">one_step</span></a> (<a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="step1" class="idref" href="#step1"><span class="id" title="constructor">step1</span></a>: <span class="id" title="keyword">∀</span> <a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a> <a id="b:14" class="idref" href="#b:14"><span class="id" title="binder">b</span></a> <a id="c:15" class="idref" href="#c:15"><span class="id" title="binder">c</span></a>,  <a class="idref" href="hydras.Hydra.BigBattle.html#one_step:11"><span class="id" title="inductive">one_step</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:10"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:14"><span class="id" title="variable">b</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:15"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:14"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:15"><span class="id" title="variable">c</span></a><br/>
+| <a id="step2" class="idref" href="#step2"><span class="id" title="constructor">step2</span></a>: <span class="id" title="keyword">∀</span> <a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a> <a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a> ,  <a class="idref" href="hydras.Hydra.BigBattle.html#one_step:11"><span class="id" title="inductive">one_step</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:10"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:16"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:17"><span class="id" title="variable">b</span></a>) 0 <a class="idref" href="hydras.Hydra.BigBattle.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:17"><span class="id" title="variable">b</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:10"><span class="id" title="variable">i</span></a>)<br/>
+| <a id="step3" class="idref" href="#step3"><span class="id" title="constructor">step3</span></a>: <span class="id" title="keyword">∀</span> <a id="a:18" class="idref" href="#a:18"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Hydra.BigBattle.html#one_step:11"><span class="id" title="inductive">one_step</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:10"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:18"><span class="id" title="variable">a</span></a>) 0 0 <a class="idref" href="hydras.Hydra.BigBattle.html#a:18"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:10"><span class="id" title="variable">i</span></a>) 0.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="step_rounds" class="idref" href="#step_rounds"><span class="id" title="lemma">step_rounds</span></a> : <span class="id" title="keyword">∀</span> <a id="i:19" class="idref" href="#i:19"><span class="id" title="binder">i</span></a> <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> <a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a> <a id="c:22" class="idref" href="#c:22"><span class="id" title="binder">c</span></a> <a id="a':23" class="idref" href="#a':23"><span class="id" title="binder">a'</span></a> <a id="b':24" class="idref" href="#b':24"><span class="id" title="binder">b'</span></a> <a id="c':25" class="idref" href="#c':25"><span class="id" title="binder">c'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#one_step"><span class="id" title="inductive">one_step</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:19"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:22"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':23"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':24"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':25"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:19"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a>  <a class="idref" href="hydras.Hydra.BigBattle.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:22"><span class="id" title="variable">c</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:19"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':23"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':24"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':25"><span class="id" title="variable">c'</span></a>). 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="steps" class="idref" href="#steps"><span class="id" title="inductive">steps</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="steps1" class="idref" href="#steps1"><span class="id" title="constructor">steps1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a> <a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> <a id="b:30" class="idref" href="#b:30"><span class="id" title="binder">b</span></a> <a id="c:31" class="idref" href="#c:31"><span class="id" title="binder">c</span></a> <a id="a':32" class="idref" href="#a':32"><span class="id" title="binder">a'</span></a> <a id="b':33" class="idref" href="#b':33"><span class="id" title="binder">b'</span></a> <a id="c':34" class="idref" href="#c':34"><span class="id" title="binder">c'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#one_step"><span class="id" title="inductive">one_step</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:28"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:30"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:31"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':32"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':33"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':34"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#steps:26"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:28"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:30"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:31"><span class="id" title="variable">c</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:28"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a':32"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':33"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':34"><span class="id" title="variable">c'</span></a><br/>
+| <a id="steps_S" class="idref" href="#steps_S"><span class="id" title="constructor">steps_S</span></a> : <span class="id" title="keyword">∀</span> <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a> <a id="a:36" class="idref" href="#a:36"><span class="id" title="binder">a</span></a> <a id="b:37" class="idref" href="#b:37"><span class="id" title="binder">b</span></a> <a id="c:38" class="idref" href="#c:38"><span class="id" title="binder">c</span></a> <a id="j:39" class="idref" href="#j:39"><span class="id" title="binder">j</span></a> <a id="a':40" class="idref" href="#a':40"><span class="id" title="binder">a'</span></a> <a id="b':41" class="idref" href="#b':41"><span class="id" title="binder">b'</span></a> <a id="c':42" class="idref" href="#c':42"><span class="id" title="binder">c'</span></a> <a id="k:43" class="idref" href="#k:43"><span class="id" title="binder">k</span></a> <a id="a'':44" class="idref" href="#a'':44"><span class="id" title="binder">a''</span></a> <a id="b'':45" class="idref" href="#b'':45"><span class="id" title="binder">b''</span></a> <a id="c'':46" class="idref" href="#c'':46"><span class="id" title="binder">c''</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#steps:26"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:37"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:38"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#j:39"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':40"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':41"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':42"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#steps:26"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#j:39"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':40"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':41"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':42"><span class="id" title="variable">c'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#k:43"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a'':44"><span class="id" title="variable">a''</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b'':45"><span class="id" title="variable">b''</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c'':46"><span class="id" title="variable">c''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#steps:26"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:37"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:38"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#k:43"><span class="id" title="variable">k</span></a>  <a class="idref" href="hydras.Hydra.BigBattle.html#a'':44"><span class="id" title="variable">a''</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b'':45"><span class="id" title="variable">b''</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c'':46"><span class="id" title="variable">c''</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ reachability (for i &gt; 0) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="reachable" class="idref" href="#reachable"><span class="id" title="definition">reachable</span></a> (<a id="i:47" class="idref" href="#i:47"><span class="id" title="binder">i</span></a> <a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="b:49" class="idref" href="#b:49"><span class="id" title="binder">b</span></a> <a id="c:50" class="idref" href="#c:50"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#steps"><span class="id" title="inductive">steps</span></a> 3 3 0 0 <a class="idref" href="hydras.Hydra.BigBattle.html#i:47"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:49"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:50"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="steps_rounds" class="idref" href="#steps_rounds"><span class="id" title="lemma">steps_rounds</span></a> : <span class="id" title="keyword">∀</span> <a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a> <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a> <a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a> <a id="c:54" class="idref" href="#c:54"><span class="id" title="binder">c</span></a> <a id="j:55" class="idref" href="#j:55"><span class="id" title="binder">j</span></a>  <a id="a':56" class="idref" href="#a':56"><span class="id" title="binder">a'</span></a> <a id="b':57" class="idref" href="#b':57"><span class="id" title="binder">b'</span></a> <a id="c':58" class="idref" href="#c':58"><span class="id" title="binder">c'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#steps"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:54"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#j:55"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':56"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':57"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':58"><span class="id" title="variable">c'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:51"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a>  <a class="idref" href="hydras.Hydra.BigBattle.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:54"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#j:55"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a':56"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b':57"><span class="id" title="variable">b'</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c':58"><span class="id" title="variable">c'</span></a>). 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ From now on, we play again the same tests as above, but instead of plain uses of Compute, we prove and register lemmas that we will be used later 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L4" class="idref" href="#L4"><span class="id" title="lemma">L4</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 4 2 4 0.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Now we prove some laws we observed in our test phase 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LS" class="idref" href="#LS"><span class="id" title="lemma">LS</span></a> : <span class="id" title="keyword">∀</span> <a id="c:59" class="idref" href="#c:59"><span class="id" title="binder">c</span></a> <a id="a:60" class="idref" href="#a:60"><span class="id" title="binder">a</span></a> <a id="b:61" class="idref" href="#b:61"><span class="id" title="binder">b</span></a> <a id="i:62" class="idref" href="#i:62"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Hydra.BigBattle.html#steps"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:61"><span class="id" title="variable">b</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:59"><span class="id" title="variable">c</span></a>) (<a class="idref" href="hydras.Hydra.BigBattle.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#c:59"><span class="id" title="variable">c</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:61"><span class="id" title="variable">b</span></a> 0.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+  The law that relates two consecutive events with  (nh = 0)  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="doubleS_law" class="idref" href="#doubleS_law"><span class="id" title="lemma">doubleS_law</span></a> : <span class="id" title="keyword">∀</span>  <a id="a:63" class="idref" href="#a:63"><span class="id" title="binder">a</span></a> <a id="b:64" class="idref" href="#b:64"><span class="id" title="binder">b</span></a> <a id="i:65" class="idref" href="#i:65"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Hydra.BigBattle.html#steps"><span class="id" title="inductive">steps</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:65"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:63"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:64"><span class="id" title="variable">b</span></a>) 0 (<a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:65"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a:63"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:64"><span class="id" title="variable">b</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reachable_S" class="idref" href="#reachable_S"><span class="id" title="lemma">reachable_S</span></a>  : <span class="id" title="keyword">∀</span> <a id="i:66" class="idref" href="#i:66"><span class="id" title="binder">i</span></a> <a id="a:67" class="idref" href="#a:67"><span class="id" title="binder">a</span></a> <a id="b:68" class="idref" href="#b:68"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:66"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:67"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:68"><span class="id" title="variable">b</span></a>) 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:66"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:68"><span class="id" title="variable">b</span></a> 0.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L10" class="idref" href="#L10"><span class="id" title="lemma">L10</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 10 2 3 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L22" class="idref" href="#L22"><span class="id" title="lemma">L22</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 22 2 2 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L46" class="idref" href="#L46"><span class="id" title="lemma">L46</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 46 2 1 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L94" class="idref" href="#L94"><span class="id" title="lemma">L94</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 94 2 0 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L95" class="idref" href="#L95"><span class="id" title="lemma">L95</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> 95 1 95 0.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L0_95" class="idref" href="#L0_95"><span class="id" title="lemma">L0_95</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 3 (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 3 0 0) 95 (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 1 95 0).<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+No more tests ! we are going to build bigger transitions 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Bigstep" class="idref" href="#Bigstep"><span class="id" title="lemma">Bigstep</span></a> : <span class="id" title="keyword">∀</span> <a id="b:69" class="idref" href="#b:69"><span class="id" title="binder">b</span></a> <a id="i:70" class="idref" href="#i:70"><span class="id" title="binder">i</span></a> <a id="a:71" class="idref" href="#a:71"><span class="id" title="binder">a</span></a> , <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:70"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:69"><span class="id" title="variable">b</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#b:69"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#i:70"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.BigBattle.html#a:71"><span class="id" title="variable">a</span></a> 0 0.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+From all the lemmas above, we now get a pretty big step 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="M" class="idref" href="#M"><span class="id" title="definition">M</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> 95 95.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2_95" class="idref" href="#L2_95"><span class="id" title="lemma">L2_95</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a> 1 0 0.   
+<br/>
+</div>
+
+<div class="doc">
+the next step creates (M+1) copies of h1 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2_95_S" class="idref" href="#L2_95_S"><span class="id" title="lemma">L2_95_S</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>) 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>) 0.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="N" class="idref" href="#N"><span class="id" title="definition">N</span></a> :=   <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span>   <a id="SuperbigStep" class="idref" href="#SuperbigStep"><span class="id" title="lemma">SuperbigStep</span></a> : <a class="idref" href="hydras.Hydra.BigBattle.html#reachable"><span class="id" title="definition">reachable</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#N"><span class="id" title="definition">N</span></a>  0 0 0.   
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We can  ow apply our study based on abstract states to "real" hydras 
+<div class="paragraph"> </div>
+
+ transforming our relation one_step into standard battles 
+<div class="paragraph"> </div>
+
+ We get now statements about hydras and battles 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Almost_done" class="idref" href="#Almost_done"><span class="id" title="lemma">Almost_done</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 3 (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 3 0 0) <a class="idref" href="hydras.Hydra.BigBattle.html#N"><span class="id" title="definition">N</span></a> (<a class="idref" href="hydras.Hydra.BigBattle.html#hyd"><span class="id" title="abbreviation">hyd</span></a> 0 0 0).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Done" class="idref" href="#Done"><span class="id" title="lemma">Done</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 0 <a class="idref" href="hydras.Hydra.BigBattle.html#hinit"><span class="id" title="definition">hinit</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#N"><span class="id" title="definition">N</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The natural number N is expressed in terms of the (iterate doubleS)
+   function.  The rest of this file is dedicated to get an intuition of how it is huge .
+     Our idea is to get a minoration by exponentials of base 2 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minoration_0" class="idref" href="#minoration_0"><span class="id" title="lemma">minoration_0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:72" class="idref" href="#n:72"><span class="id" title="binder">n</span></a>,  2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:72"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:72"><span class="id" title="variable">n</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minoration_1" class="idref" href="#minoration_1"><span class="id" title="lemma">minoration_1</span></a> : <span class="id" title="keyword">∀</span> <a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> <a id="x:74" class="idref" href="#x:74"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#x:74"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#doubleS"><span class="id" title="definition">doubleS</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#x:74"><span class="id" title="variable">x</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minoration_2" class="idref" href="#minoration_2"><span class="id" title="lemma">minoration_2</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> 95 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 95 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minoration_3" class="idref" href="#minoration_3"><span class="id" title="lemma">minoration_3</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#M"><span class="id" title="definition">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#N"><span class="id" title="definition">N</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_mono1" class="idref" href="#exp2_mono1"><span class="id" title="lemma">exp2_mono1</span></a> : <span class="id" title="keyword">∀</span> <a id="n:75" class="idref" href="#n:75"><span class="id" title="binder">n</span></a> <a id="p:76" class="idref" href="#p:76"><span class="id" title="binder">p</span></a>,  <a class="idref" href="hydras.Hydra.BigBattle.html#n:75"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#p:76"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#n:75"><span class="id" title="variable">n</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#p:76"><span class="id" title="variable">p</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minoration" class="idref" href="#minoration"><span class="id" title="lemma">minoration</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> 95 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 95) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.BigBattle.html#N"><span class="id" title="definition">N</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+Hence, the length of the battle is greater or equal than 
+  exp2 (exp2 95 * 95), a number whose representation in base 10 has 
+  at least 10 ^ 30 digits ! 
+<div class="paragraph"> </div>
+
+ from OCAML
+
+<div class="paragraph"> </div>
+
+let exp2 x = 2.0 ** x;;
+
+<div class="paragraph"> </div>
+
+let _ = exp2 95.0 *. 95.0;;
+
+<div class="paragraph"> </div>
+
+float = 3.76333771942755604e+30
+
+<div class="paragraph"> </div>
+
+N &gt;= 2 ** 3.7e+30
+
+<div class="paragraph"> </div>
+
+log10 N &gt;= 3.7e+30 * log10 2 &gt;= 1.0e+30 
+
+<div class="paragraph"> </div>
+
+N &gt;= 10 ** (10 ** 30)
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html b/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html
@@ -0,0 +1,133 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Epsilon0_Needed_Free</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Epsilon0_Needed_Free</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ Pierre Castéran, LaBRI and University of Bordeaux  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#"><span class="id" title="library">Epsilon0_Needed_Generic</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="module">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="module">Epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="module">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="module">Paths</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="module">Relation_Operators</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.O2H.html#"><span class="id" title="module">O2H</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We generalize the results of libraries Omega_Small and Omega2_Small:
+   We prove that no ordinal strictly less than epsilon0 can be used as a variant
+   for proving the termination of all hydra battles.
+We use the so-called "Ketonen-Solovay machinery" for building
+a proof that shares the same structure as for the libraries above, but is much 
+longer 
+ 
+<div class="paragraph"> </div>
+
+<a id="lab257"></a><h2 class="section">Bounded variants</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility_Proof" class="idref" href="#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="mu:1" class="idref" href="#mu:1"><span class="id" title="binder">mu</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hmu:2" class="idref" href="#Hmu:2"><span class="id" title="binder">Hmu</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#mu:1"><span class="id" title="variable">mu</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="m:3" class="idref" href="#m:3"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Var:4" class="idref" href="#Var:4"><span class="id" title="binder">Var</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#m:3"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hy:5" class="idref" href="#Hy:5"><span class="id" title="binder">Hy</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Var:4"><span class="id" title="variable">Var</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#mu:1"><span class="id" title="variable">mu</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.big_h" class="idref" href="#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> := <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h"><span class="id" title="definition">big_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.small_h" class="idref" href="#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> := <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#small_h"><span class="id" title="definition">small_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">nf_m</span> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_ge" class="idref" href="#m_ge"><span class="id" title="lemma">m_ge</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab258"></a><h2 class="section">Proof of the inequality m small_h t1&lt; m big_h</h2>
+
+   
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_variant_LT" class="idref" href="#m_variant_LT"><span class="id" title="lemma">m_variant_LT</span></a> : <span class="id" title="keyword">∀</span> <a id="h:6" class="idref" href="#h:6"><span class="id" title="binder">h</span></a> <a id="h':7" class="idref" href="#h':7"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#h:6"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#h':7"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#h':7"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#h:6"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span>  <a id="big_to_small" class="idref" href="#big_to_small"><span class="id" title="lemma">big_to_small</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_lt" class="idref" href="#m_lt"><span class="id" title="lemma">m_lt</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="self_lt_free" class="idref" href="#self_lt_free"><span class="id" title="lemma">self_lt_free</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> .<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Impossibility_free" class="idref" href="#Impossibility_free"><span class="id" title="lemma">Impossibility_free</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_free"><span class="id" title="lemma">Impossibility_free</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html b/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html
new file mode 100644
index 00000000..26d37db4
--- /dev/null
+++ b/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html
@@ -0,0 +1,98 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Epsilon0_Needed_Generic</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Epsilon0_Needed_Generic</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ Pierre Castéran, LaBRI and University of Bordeaux  
+<div class="paragraph"> </div>
+
+ Technical definitions and lemmas about variants bounded by some ordinal 
+    les than <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> 
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a> <a class="idref" href="hydras.Hydra.O2H.html#"><span class="id" title="library">O2H</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Bounded" class="idref" href="#Bounded"><span class="id" title="section">Bounded</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="B:1" class="idref" href="#B:1"><span class="id" title="binder">B</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="mu:2" class="idref" href="#mu:2"><span class="id" title="binder">mu</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hmu:3" class="idref" href="#Hmu:3"><span class="id" title="binder">Hmu</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu:2"><span class="id" title="variable">mu</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="m:4" class="idref" href="#m:4"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Var:5" class="idref" href="#Var:5"><span class="id" title="binder">Var</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#B:1"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#m:4"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hy:6" class="idref" href="#Hy:6"><span class="id" title="binder">Hy</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Var:5"><span class="id" title="variable">Var</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu:2"><span class="id" title="variable">mu</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Bounded.m_decrease" class="idref" href="#Bounded.m_decrease"><span class="id" title="variable">m_decrease</span></a> : <span class="id" title="keyword">∀</span>  <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> <a id="h:8" class="idref" href="#h:8"><span class="id" title="binder">h</span></a> <a id="h':9" class="idref" href="#h':9"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#h:8"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#h':9"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#h':9"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#h:8"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span>  <a id="nf_m" class="idref" href="#nf_m"><span class="id" title="lemma">nf_m</span></a> : <span class="id" title="keyword">∀</span> <a id="h:11" class="idref" href="#h:11"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#h:11"><span class="id" title="variable">h</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Rem0</span> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="mu_positive" class="idref" href="#mu_positive"><span class="id" title="lemma">mu_positive</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_ge_0" class="idref" href="#m_ge_0"><span class="id" title="lemma">m_ge_0</span></a> <a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#alpha:12"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="big_h" class="idref" href="#big_h"><span class="id" title="definition">big_h</span></a> := <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.mu"><span class="id" title="variable">mu</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="beta_h" class="idref" href="#beta_h"><span class="id" title="definition">beta_h</span></a> := <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h"><span class="id" title="definition">big_h</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="small_h" class="idref" href="#small_h"><span class="id" title="definition">small_h</span></a> := <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#beta_h"><span class="id" title="definition">beta_h</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="mu_beta_h" class="idref" href="#mu_beta_h"><span class="id" title="lemma">mu_beta_h</span></a> : <a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#beta_h"><span class="id" title="definition">beta_h</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Corollary</span> <a id="m_ge_generic" class="idref" href="#m_ge_generic"><span class="id" title="lemma">m_ge_generic</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h"><span class="id" title="definition">big_h</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#small_h"><span class="id" title="definition">small_h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded"><span class="id" title="section">Bounded</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Epsilon0_Needed_Std</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Epsilon0_Needed_Std</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Pierre Castéran, LaBRI and University of Bordeaux 
+
+<div class="paragraph"> </div>
+
+ We prove that the impossibility result of <a href="./hydras.Hydra.Epsilon0_Needed_Free.html"> Epsilon0_Needed_Free</a> still holds for "standard" battles 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#"><span class="id" title="library">Epsilon0_Needed_Generic</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="module">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="module">Epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="module">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="module">Paths</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="module">Relation_Operators</span></a> <a class="idref" href="hydras.Hydra.O2H.html#"><span class="id" title="module">O2H</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="module">Hydra_Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility_Proof" class="idref" href="#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="mu:1" class="idref" href="#mu:1"><span class="id" title="binder">mu</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="m:2" class="idref" href="#m:2"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Var:3" class="idref" href="#Var:3"><span class="id" title="binder">Var</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#m:2"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hy:4" class="idref" href="#Hy:4"><span class="id" title="binder">Hy</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant"><span class="id" title="class">BoundedVariant</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Var:3"><span class="id" title="variable">Var</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#mu:1"><span class="id" title="variable">mu</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.big_h" class="idref" href="#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> := <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h"><span class="id" title="definition">big_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.small_h" class="idref" href="#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> := <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#small_h"><span class="id" title="definition">small_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_ge" class="idref" href="#m_ge"><span class="id" title="lemma">m_ge</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>. 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab264"></a><h2 class="section">Proof of the inequality m small_h t1&lt; m big_h</h2>
+
+<div class="paragraph"> </div>
+
+  The measure is strictly decreasing along any round 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lvar" class="idref" href="#Lvar"><span class="id" title="lemma">Lvar</span></a> : <span class="id" title="keyword">∀</span> <a id="h:5" class="idref" href="#h:5"><span class="id" title="binder">h</span></a> <a id="h0:6" class="idref" href="#h0:6"><span class="id" title="binder">h0</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:5"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:5"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h0:6"><span class="id" title="variable">h0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h0:6"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:5"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Application to standard battles 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_decrease" class="idref" href="#m_decrease"><span class="id" title="lemma">m_decrease</span></a> : <span class="id" title="keyword">∀</span> <a id="j:8" class="idref" href="#j:8"><span class="id" title="binder">j</span></a> <a id="h0:9" class="idref" href="#h0:9"><span class="id" title="binder">h0</span></a> <a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a> <a id="h:11" class="idref" href="#h:11"><span class="id" title="binder">h</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:11"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#i:10"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:11"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#j:8"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h0:9"><span class="id" title="variable">h0</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h0:9"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#h:11"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Conversion of ordinal inequalities into  standard battles  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LT_to_standard_battle" class="idref" href="#LT_to_standard_battle"><span class="id" title="lemma">LT_to_standard_battle</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:23" class="idref" href="#alpha:23"><span class="id" title="binder">alpha</span></a> <a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#beta:24"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#alpha:23"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a> <a id="i:26" class="idref" href="#i:26"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#n:25"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#alpha:23"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#i:26"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#beta:24"><span class="id" title="variable">beta</span></a>). 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem3" class="idref" href="#Rem3"><span class="id" title="lemma">Rem3</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#beta_h"><span class="id" title="definition">beta_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem4" class="idref" href="#Rem4"><span class="id" title="lemma">Rem4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> <a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#n:27"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a>) <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#i:28"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Generic.html#beta_h"><span class="id" title="definition">beta_h</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>)).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Hmu" class="idref" href="#Hmu"><span class="id" title="lemma">Hmu</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.mu"><span class="id" title="variable">mu</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_lt" class="idref" href="#m_lt"><span class="id" title="lemma">m_lt</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+End of the proof 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="self_lt_standard" class="idref" href="#self_lt_standard"><span class="id" title="lemma">self_lt_standard</span></a> : <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Impossibility_std" class="idref" href="#Impossibility_std"><span class="id" title="lemma">Impossibility_std</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_std"><span class="id" title="lemma">Impossibility_std</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Hydra_Definitions.html b/theories/html/hydras.Hydra.Hydra_Definitions.html
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+++ b/theories/html/hydras.Hydra.Hydra_Definitions.html
@@ -0,0 +1,858 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Definitions</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Definitions</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Casteran, LaBRI, University of Bordeaux 
+<div class="paragraph"> </div>
+
+<a id="lab228"></a><h1 class="section">Data types</h1>
+
+
+<div class="paragraph"> </div>
+
+ An hydra is a finitely branched tree. We use the auxiliary type <span class="inlinecode"><span class="id" title="var">Hydrae</span></span>  for
+ representing finite sequences  of hydras.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Hydra" class="idref" href="#Hydra"><span class="id" title="inductive">Hydra</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+| <a id="node" class="idref" href="#node"><span class="id" title="constructor">node</span></a> :  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae:2"><span class="id" title="inductive">Hydrae</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra:1"><span class="id" title="inductive">Hydra</span></a><br/>
+<span class="id" title="keyword">with</span> <a id="Hydrae" class="idref" href="#Hydrae"><span class="id" title="inductive">Hydrae</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+| <a id="hnil" class="idref" href="#hnil"><span class="id" title="constructor">hnil</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae:2"><span class="id" title="inductive">Hydrae</span></a><br/>
+| <a id="hcons" class="idref" href="#hcons"><span class="id" title="constructor">hcons</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra:1"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae:2"><span class="id" title="inductive">Hydrae</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae:2"><span class="id" title="inductive">Hydrae</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Alternative representation (discarded) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt" class="idref" href="#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Alt.Hydra" class="idref" href="#Alt.Hydra"><span class="id" title="inductive">Hydra</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Alt.hnode" class="idref" href="#Alt.hnode"><span class="id" title="constructor">hnode</span></a> (<a id="daughters:7" class="idref" href="#daughters:7"><span class="id" title="binder">daughters</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra:5"><span class="id" title="inductive">Hydra</span></a>).<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+  Mutual induction scheme for types Hydra and Hydrae 
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="Hydra_rect2" class="idref" href="#Hydra_rect2"><span class="id" title="definition">Hydra_rect2</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Type</span><br/>
+<span class="id" title="keyword">with</span>   <a id="Hydrae_rect2" class="idref" href="#Hydrae_rect2"><span class="id" title="definition">Hydrae_rect2</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra_rect2"><span class="id" title="definition">Hydra_rect2</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae_rect2"><span class="id" title="definition">Hydrae_rect2</span></a> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+All elements of s satisfy P 
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="h_forall" class="idref" href="#h_forall"><span class="id" title="definition">h_forall</span></a> (<a id="P:8" class="idref" href="#P:8"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="s:9" class="idref" href="#s:9"><span class="id" title="binder">s</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:9"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">s'</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#P:8"><span class="id" title="variable">P</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h_forall:10"><span class="id" title="definition">h_forall</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#P:8"><span class="id" title="variable">P</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="h_eq_dec" class="idref" href="#h_eq_dec"><span class="id" title="lemma">h_eq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="h:12" class="idref" href="#h:12"><span class="id" title="binder">h</span></a> <a id="h':13" class="idref" href="#h':13"><span class="id" title="binder">h'</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:12"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':13"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:12"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':13"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><br/>
+<span class="id" title="keyword">with</span> <a id="hs_eq_dec" class="idref" href="#hs_eq_dec"><span class="id" title="lemma">hs_eq_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="l:14" class="idref" href="#l:14"><span class="id" title="binder">l</span></a> <a id="l':15" class="idref" href="#l':15"><span class="id" title="binder">l'</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:14"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l':15"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:14"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l':15"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For list usage 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="hs2l" class="idref" href="#hs2l"><span class="id" title="definition">hs2l</span></a> (<a id="s:16" class="idref" href="#s:16"><span class="id" title="binder">s</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:16"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a><br/>
+| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">s'</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hs2l:17"><span class="id" title="definition">hs2l</span></a> <span class="id" title="var">s'</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="l2hs" class="idref" href="#l2hs"><span class="id" title="definition">l2hs</span></a> (<a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>): <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:19"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a><br/>
+| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l2hs:20"><span class="id" title="definition">l2hs</span></a> <span class="id" title="var">l'</span>)<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hs2lK" class="idref" href="#hs2lK"><span class="id" title="lemma">hs2lK</span></a> (<a id="s:22" class="idref" href="#s:22"><span class="id" title="binder">s</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l2hs"><span class="id" title="definition">l2hs</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hs2l"><span class="id" title="definition">hs2l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:22"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:22"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="l2hsK" class="idref" href="#l2hsK"><span class="id" title="lemma">l2hsK</span></a> (<a id="l:23" class="idref" href="#l:23"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hs2l"><span class="id" title="definition">hs2l</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l2hs"><span class="id" title="definition">l2hs</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:23"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:23"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+  Number of nodes (a.k.a. size)
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="hsize" class="idref" href="#hsize"><span class="id" title="definition">hsize</span></a> (<a id="h:24" class="idref" href="#h:24"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:24"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#lhsize:27"><span class="id" title="definition">lhsize</span></a> <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">with</span> <a id="lhsize" class="idref" href="#lhsize"><span class="id" title="definition">lhsize</span></a> <a id="l:25" class="idref" href="#l:25"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:25"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">hs</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize:26"><span class="id" title="definition">hsize</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#lhsize:27"><span class="id" title="definition">lhsize</span></a> <span class="id" title="var">hs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab229"></a><h3 class="section">height (length of longest branch)</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="height" class="idref" href="#height"><span class="id" title="definition">height</span></a>  (<a id="h:30" class="idref" href="#h:30"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:30"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#lheight:33"><span class="id" title="definition">lheight</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">with</span> <a id="lheight" class="idref" href="#lheight"><span class="id" title="definition">lheight</span></a> <a id="l:31" class="idref" href="#l:31"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:31"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">hs</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height:32"><span class="id" title="definition">height</span></a> <span class="id" title="var">h</span>)) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#lheight:33"><span class="id" title="definition">lheight</span></a> <span class="id" title="var">hs</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab230"></a><h2 class="section">Abbreviations</h2>
+
+ 
+<div class="paragraph"> </div>
+
+<a id="lab231"></a><h3 class="section">Heads : A head is just a node without daughters</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab232"></a><h3 class="section">Hydra with 0, 1, 2 or 3 daughters</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="head" class="idref" href="#head"><span class="id" title="abbreviation">head</span></a> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="hyd1" class="idref" href="#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <span class="id" title="var">h</span> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)).<br/>
+<span class="id" title="keyword">Notation</span> <a id="hyd2" class="idref" href="#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <span class="id" title="var">h</span> <span class="id" title="var">h'</span> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h'</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>))).<br/>
+<span class="id" title="keyword">Notation</span> <a id="hyd3" class="idref" href="#hyd3"><span class="id" title="abbreviation">hyd3</span></a> <span class="id" title="var">h</span> <span class="id" title="var">h'</span> <span class="id" title="var">h''</span> := (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h'</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h''</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)))).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab233"></a><h3 class="section">Adds n copies of the same hydra h at the right of  s</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="hcons_mult" class="idref" href="#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a id="h:36" class="idref" href="#h:36"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>)(<a id="n:37" class="idref" href="#n:37"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="s:38" class="idref" href="#s:38"><span class="id" title="binder">s</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>):<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:37"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:38"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:36"><span class="id" title="variable">h</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult:39"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:36"><span class="id" title="variable">h</span></a> <span class="id" title="var">p</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:38"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab234"></a><h3 class="section">Hydra with n equal daughters</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="hyd_mult" class="idref" href="#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a id="h:41" class="idref" href="#h:41"><span class="id" title="binder">h</span></a> <a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:41"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:42"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab235"></a><h2 class="section">Managing sequences of hydras</h2>
+
+ 
+<div class="paragraph"> </div>
+
+<a id="lab236"></a><h3 class="section">Appending</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="hy_app" class="idref" href="#hy_app"><span class="id" title="definition">hy_app</span></a> (<a id="s1:43" class="idref" href="#s1:43"><span class="id" title="binder">s1</span></a> <a id="s2:44" class="idref" href="#s2:44"><span class="id" title="binder">s2</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s1:43"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s2:44"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h1</span> <span class="id" title="var">s1'</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h1</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app:45"><span class="id" title="definition">hy_app</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s2:44"><span class="id" title="variable">s2</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab237"></a><h3 class="section">Adds a  head to  the right of <span class="inlinecode"><span class="id" title="var">s</span></span></h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="add_head_r" class="idref" href="#add_head_r"><span class="id" title="definition">add_head_r</span></a> (<a id="s:47" class="idref" href="#s:47"><span class="id" title="binder">s</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:47"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">s'</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r:48"><span class="id" title="definition">add_head_r</span></a> <span class="id" title="var">s'</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab238"></a><h3 class="section">Adds i   heads to the right of <span class="inlinecode"><span class="id" title="var">s</span></span></h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="add_head_r_plus" class="idref" href="#add_head_r_plus"><span class="id" title="definition">add_head_r_plus</span></a> (<a id="s:50" class="idref" href="#s:50"><span class="id" title="binder">s</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) (<a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:51"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:50"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r"><span class="id" title="definition">add_head_r</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus:52"><span class="id" title="definition">add_head_r_plus</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:50"><span class="id" title="variable">s</span></a> <span class="id" title="var">j</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab239"></a><h3 class="section">adds <span class="inlinecode"><span class="id" title="var">i</span></span> heads to  the root of <span class="inlinecode"><span class="id" title="var">h</span></span> (to the right of its daughters)</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="add_r" class="idref" href="#add_r"><span class="id" title="definition">add_r</span></a> <a id="h:54" class="idref" href="#h:54"><span class="id" title="binder">h</span></a> <a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:54"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus"><span class="id" title="definition">add_head_r_plus</span></a> <span class="id" title="var">s</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:55"><span class="id" title="variable">i</span></a>) <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab240"></a><h1 class="section">Hydra Battles</h1>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab241"></a><h2 class="section">Relation associated with a single round.</h2>
+
+
+<div class="paragraph"> </div>
+
+  We represent the rules of Hydra Battles through a binary relation:
+  <span class="inlinecode"><span class="id" title="var">round_n</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> on the <span class="inlinecode"><span class="id" title="var">Hydra</span></span> type. 
+The proposition  <span class="inlinecode"><span class="id" title="var">round_n</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>  <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds if <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">h</span></span> by a single round of a battle, and <span class="inlinecode"><span class="id" title="var">n</span></span> is the <i>expected</i> replication  factor (irrelevant if the chopped head is at distance 1 from the foot).
+
+<div class="paragraph"> </div>
+
+The proposition <span class="inlinecode"><span class="id" title="var">round_n</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds 
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> if <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">h</span></span> by chopping off one head of height 1. This situation is described by the proposition <span class="inlinecode"><span class="id" title="var">R1</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> Or <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">h</span></span> through a beheading at distance  from the foot, with creation of <span class="inlinecode"><span class="id" title="var">n</span></span> copies of some dub-hydra.
+
+</li>
+</ul>
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+Before giving the definition in Coq of <span class="inlinecode"><span class="id" title="var">round_n</span></span>, we need to define several  auxiliary relations.
+
+<div class="paragraph"> </div>
+
+<a id="lab242"></a><h3 class="section">S0</h3>
+
+
+<div class="paragraph"> </div>
+
+ The proposition <span class="inlinecode"><span class="id" title="var">S0</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">s'</span></span> holds if the sequence <span class="inlinecode"><span class="id" title="var">s'</span></span> is obtained by removing one head from <span class="inlinecode"><span class="id" title="var">s</span></span>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="S0" class="idref" href="#S0"><span class="id" title="inductive">S0</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+| <a id="S0_first" class="idref" href="#S0_first"><span class="id" title="constructor">S0_first</span></a> : <span class="id" title="keyword">∀</span> <a id="s:59" class="idref" href="#s:59"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0:57"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:59"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:59"><span class="id" title="variable">s</span></a><br/>
+| <a id="S0_rest" class="idref" href="#S0_rest"><span class="id" title="constructor">S0_rest</span></a> : <span class="id" title="keyword">∀</span>  <a id="h:60" class="idref" href="#h:60"><span class="id" title="binder">h</span></a> <a id="s:61" class="idref" href="#s:61"><span class="id" title="binder">s</span></a> <a id="s':62" class="idref" href="#s':62"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0:57"><span class="id" title="inductive">S0</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:61"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':62"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0:57"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:60"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:61"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:60"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':62"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab243"></a><h3 class="section">R1</h3>
+
+
+<div class="paragraph"> </div>
+
+    The proposition <span class="inlinecode"><span class="id" title="var">R1</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds if <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained by removing one head from <span class="inlinecode"><span class="id" title="var">h</span></span> at distance 1 from  the foot 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="R1" class="idref" href="#R1"><span class="id" title="inductive">R1</span></a>  :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> :=<br/>
+| <a id="R1_intro" class="idref" href="#R1_intro"><span class="id" title="constructor">R1_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="s:65" class="idref" href="#s:65"><span class="id" title="binder">s</span></a> <a id="s':66" class="idref" href="#s':66"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:65"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':66"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1:63"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:65"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':66"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab244"></a><h3 class="section">S1</h3>
+
+
+<div class="paragraph"> </div>
+
+  <span class="inlinecode"><span class="id" title="var">S1</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">s'</span></span> holds if <span class="inlinecode"><span class="id" title="var">s'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">s</span></span> by replacing some  member <span class="inlinecode"><span class="id" title="var">h</span></span> of <span class="inlinecode"><span class="id" title="var">s</span></span> by  copies of <span class="inlinecode"><span class="id" title="var">h</span></span>,  where <span class="inlinecode"><span class="id" title="var">R1</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds.
+  Thus, <span class="inlinecode"><span class="id" title="var">n</span></span> is the number of new replicas of <span class="inlinecode"><span class="id" title="var">h'</span></span>. 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="S1" class="idref" href="#S1"><span class="id" title="inductive">S1</span></a> (<a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>  :=<br/>
+| <a id="S1_first" class="idref" href="#S1_first"><span class="id" title="constructor">S1_first</span></a> : <span class="id" title="keyword">∀</span> <a id="s:70" class="idref" href="#s:70"><span class="id" title="binder">s</span></a> <a id="h:71" class="idref" href="#h:71"><span class="id" title="binder">h</span></a> <a id="h':72" class="idref" href="#h':72"><span class="id" title="binder">h'</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:71"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':72"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1:68"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:67"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:71"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:70"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':72"><span class="id" title="variable">h'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:67"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:70"><span class="id" title="variable">s</span></a>)<br/>
+| <a id="S1_next" class="idref" href="#S1_next"><span class="id" title="constructor">S1_next</span></a> : <span class="id" title="keyword">∀</span> <a id="h:73" class="idref" href="#h:73"><span class="id" title="binder">h</span></a> <a id="s:74" class="idref" href="#s:74"><span class="id" title="binder">s</span></a> <a id="s':75" class="idref" href="#s':75"><span class="id" title="binder">s'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1:68"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':75"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1:68"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:67"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:73"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:74"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:73"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':75"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab245"></a><h3 class="section">R2</h3>
+
+
+<div class="paragraph"> </div>
+
+The proposition <span class="inlinecode"><span class="id" title="var">R2</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds if some sub-hydra <span class="inlinecode"><span class="id" title="var">h0</span></span> of <span class="inlinecode"><span class="id" title="var">h</span></span> has been replaced by <span class="inlinecode"><span class="id" title="var">h'0</span></span>,
+  where <span class="inlinecode"><span class="id" title="var">R1</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">h0</span></span> <span class="inlinecode"><span class="id" title="var">h'0</span></span>. The <span class="inlinecode"><span class="id" title="var">S2</span></span> relation is just a helper for <span class="inlinecode"><span class="id" title="var">R2</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="R2" class="idref" href="#R2"><span class="id" title="inductive">R2</span></a> (<a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>  :=<br/>
+| <a id="R2_intro" class="idref" href="#R2_intro"><span class="id" title="constructor">R2_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="s:81" class="idref" href="#s:81"><span class="id" title="binder">s</span></a> <a id="s':82" class="idref" href="#s':82"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:81"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':82"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2:77"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:81"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':82"><span class="id" title="variable">s'</span></a>)<br/>
+| <a id="R2_intro_2" class="idref" href="#R2_intro_2"><span class="id" title="constructor">R2_intro_2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:83" class="idref" href="#s:83"><span class="id" title="binder">s</span></a> <a id="s':84" class="idref" href="#s':84"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2:78"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:83"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':84"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2:77"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:83"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s':84"><span class="id" title="variable">s'</span></a>)<br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="S2" class="idref" href="#S2"><span class="id" title="inductive">S2</span></a> (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a id="S2_first" class="idref" href="#S2_first"><span class="id" title="constructor">S2_first</span></a> : <span class="id" title="keyword">∀</span> <a id="h:85" class="idref" href="#h:85"><span class="id" title="binder">h</span></a> <a id="h':86" class="idref" href="#h':86"><span class="id" title="binder">h'</span></a> <a id="s:87" class="idref" href="#s:87"><span class="id" title="binder">s</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2:77"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a>   <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:85"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':86"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2:78"><span class="id" title="inductive">S2</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:85"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:87"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':86"><span class="id" title="variable">h'</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#s:87"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  <a id="S2_next" class="idref" href="#S2_next"><span class="id" title="constructor">S2_next</span></a>  : <span class="id" title="keyword">∀</span> <a id="h:88" class="idref" href="#h:88"><span class="id" title="binder">h</span></a>   <a id="r:89" class="idref" href="#r:89"><span class="id" title="binder">r</span></a> <a id="r':90" class="idref" href="#r':90"><span class="id" title="binder">r'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2:78"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#r:89"><span class="id" title="variable">r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#r':90"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2:78"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:88"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#r:89"><span class="id" title="variable">r</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:88"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#r':90"><span class="id" title="variable">r'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab246"></a><h3 class="section"><span class="inlinecode"><span class="id" title="var">round_n</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+   Let <span class="inlinecode"><span class="id" title="var">n</span></span> be an <i>expected</i> replication number. The relation <span class="inlinecode"><span class="id" title="var">round_n</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span>
+   holds:
+<ul class="doclist">
+<li> if  <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">h</span></span> by removing one head of height 1 (and the factor <span class="inlinecode"><span class="id" title="var">n</span></span> is irrelevant)
+
+</li>
+<li> or  <span class="inlinecode"><span class="id" title="var">h'</span></span> is obtained from <span class="inlinecode"><span class="id" title="var">h</span></span> by removing one head of height greater or equal than 2, and this beheading was made with a relocation factor <span class="inlinecode"><span class="id" title="var">n</span></span>.
+
+</li>
+</ul>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="round_n" class="idref" href="#round_n"><span class="id" title="definition">round_n</span></a> <a id="n:91" class="idref" href="#n:91"><span class="id" title="binder">n</span></a> <a id="h:92" class="idref" href="#h:92"><span class="id" title="binder">h</span></a> <a id="h':93" class="idref" href="#h':93"><span class="id" title="binder">h'</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:92"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':93"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:91"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:92"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':93"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Transition system associated with battles 
+<div class="paragraph"> </div>
+
+<a id="lab247"></a><h3 class="section">Binary relation associated with a battle round</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="round" class="idref" href="#round"><span class="id" title="definition">round</span></a> <a id="h:94" class="idref" href="#h:94"><span class="id" title="binder">h</span></a> <a id="h':95" class="idref" href="#h':95"><span class="id" title="binder">h'</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:96" class="idref" href="#n:96"><span class="id" title="binder">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:96"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:94"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':95"><span class="id" title="variable">h'</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'-1-&gt;'_x" class="idref" href="#:::x_'-1-&gt;'_x"><span class="id" title="notation">&quot;</span></a>-1-&gt;" := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab248"></a><h3 class="section">transitive closures of round</h3>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="round_plus" class="idref" href="#round_plus"><span class="id" title="definition">round_plus</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_trans_1n"><span class="id" title="inductive">clos_trans_1n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="round_star" class="idref" href="#round_star"><span class="id" title="definition">round_star</span></a> <a id="h:97" class="idref" href="#h:97"><span class="id" title="binder">h</span></a> <a id="h':98" class="idref" href="#h':98"><span class="id" title="binder">h'</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:97"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':98"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_plus"><span class="id" title="definition">round_plus</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:97"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':98"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="4b2603e3c2ae6b73edb417c5b09dc498" class="idref" href="#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">&quot;</span></a>-+-&gt;" := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_plus"><span class="id" title="definition">round_plus</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+<span class="id" title="keyword">Infix</span> <a id="01d24958106a67451f21f2873a331e59" class="idref" href="#01d24958106a67451f21f2873a331e59"><span class="id" title="notation">&quot;</span></a>-*-&gt;" := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_star"><span class="id" title="definition">round_star</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab249"></a><h2 class="section">Experimental tactics for interactive battles</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab250"></a><h3 class="section">removes the i-th daughter (should be a head)</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">chop_off</span> <span class="id" title="var">i</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> |- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0_first"><span class="id" title="constructor">S0_first</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ?<span class="id" title="var">j</span> ⇒ <span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0_rest"><span class="id" title="constructor">S0_rest</span></a>; <span class="id" title="var">chop_off</span> <span class="id" title="var">j</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  |- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> 0; <span class="id" title="tactic">left</span>; <span class="id" title="tactic">split</span>; <span class="id" title="var">chop_off</span> <span class="id" title="var">i</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|  |- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span> ; <span class="id" title="tactic">split</span>; <span class="id" title="var">chop_off</span> <span class="id" title="var">i</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Calls the <span class="inlinecode"><span class="id" title="var">R2</span></span> relation 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">h_search_n</span> <span class="id" title="var">n</span>  :=  <span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒  <span class="id" title="tactic">∃</span> <span class="id" title="var">n</span>; <span class="id" title="var">eright</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">h_search</span>   :=  <span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| |- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> ?<span class="id" title="var">p</span> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒ <span class="id" title="var">eright</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">s2_nth</span> <span class="id" title="var">n</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| 0 ⇒ <span class="id" title="var">eleft</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ?<span class="id" title="var">p</span> ⇒ <span class="id" title="var">eright</span> ; <span class="id" title="var">s2_nth</span> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Moves to the <span class="inlinecode"><span class="id" title="var">i</span></span>-th daugther 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">r2_up</span> <span class="id" title="var">i</span> := <span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒ <span class="id" title="var">eright</span>; <span class="id" title="var">s2_nth</span> <span class="id" title="var">i</span> <br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">S1_nth</span> <span class="id" title="var">i</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0 ⇒ <span class="id" title="var">eleft</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ?<span class="id" title="var">j</span> ⇒ <span class="id" title="var">eright</span> ; <span class="id" title="var">S1_nth</span> <span class="id" title="var">j</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+chops off a head which is the <span class="inlinecode"><span class="id" title="var">j</span></span>-th daughter of the 
+    <span class="inlinecode"><span class="id" title="var">i</span></span>-th dauchter
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">r2_d2</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> ?<span class="id" title="var">n</span> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒ <span class="id" title="var">eleft</span>; <span class="id" title="var">S1_nth</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">split</span>; <span class="id" title="var">chop_off</span> <span class="id" title="var">j</span><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+End of the battle 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">stop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_star"><span class="id" title="definition">round_star</span></a> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒ <span class="id" title="tactic">left</span>; <span class="id" title="tactic">reflexivity</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Still fighting 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="round_star_intro" class="idref" href="#round_star_intro"><span class="id" title="lemma">round_star_intro</span></a> <a id="h:99" class="idref" href="#h:99"><span class="id" title="binder">h</span></a> <a id="h'':100" class="idref" href="#h'':100"><span class="id" title="binder">h''</span></a> : <span class="id" title="keyword">∀</span> <a id="h':101" class="idref" href="#h':101"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:99"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x"><span class="id" title="notation">-1-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':101"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':101"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#01d24958106a67451f21f2873a331e59"><span class="id" title="notation">-*-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h'':100"><span class="id" title="variable">h''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:99"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#01d24958106a67451f21f2873a331e59"><span class="id" title="notation">-*-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h'':100"><span class="id" title="variable">h''</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">forward</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|- <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_star"><span class="id" title="definition">round_star</span></a> ?<span class="id" title="var">h</span> ?<span class="id" title="var">h'</span> ⇒ <span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_star_intro"><span class="id" title="lemma">round_star_intro</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab251"></a><h2 class="section">Traces</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="trace_to" class="idref" href="#trace_to"><span class="id" title="inductive">trace_to</span></a>  <a id="dest:102" class="idref" href="#dest:102"><span class="id" title="binder">dest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="trace_to1" class="idref" href="#trace_to1"><span class="id" title="constructor">trace_to1</span></a> : <span class="id" title="keyword">∀</span> <a id="n:105" class="idref" href="#n:105"><span class="id" title="binder">n</span></a> <a id="h:106" class="idref" href="#h:106"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:105"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:106"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#dest:102"><span class="id" title="variable">dest</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to:103"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#dest:102"><span class="id" title="variable">dest</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:105"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:106"><span class="id" title="variable">h</span></a><br/>
+| <a id="trace_toS" class="idref" href="#trace_toS"><span class="id" title="constructor">trace_toS</span></a> : <span class="id" title="keyword">∀</span> <a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a> <a id="t:108" class="idref" href="#t:108"><span class="id" title="binder">t</span></a> <a id="h:109" class="idref" href="#h:109"><span class="id" title="binder">h</span></a> <a id="h':110" class="idref" href="#h':110"><span class="id" title="binder">h'</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:107"><span class="id" title="variable">n</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:109"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':110"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to:103"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#dest:102"><span class="id" title="variable">dest</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#t:108"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':110"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to:103"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#dest:102"><span class="id" title="variable">dest</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:107"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#t:108"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:109"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="trace" class="idref" href="#trace"><span class="id" title="definition">trace</span></a> <a id="t:111" class="idref" href="#t:111"><span class="id" title="binder">t</span></a> <a id="h:112" class="idref" href="#h:112"><span class="id" title="binder">h</span></a> <a id="h':113" class="idref" href="#h':113"><span class="id" title="binder">h'</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':113"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#t:111"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:112"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Let  be an hydra and  and expected replication factor. The next step of 
+  the current battle may be specified by the following type 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="round_spec" class="idref" href="#round_spec"><span class="id" title="definition">round_spec</span></a> <a id="h:114" class="idref" href="#h:114"><span class="id" title="binder">h</span></a> <a id="n:115" class="idref" href="#n:115"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="h':116" class="idref" href="#h':116"><span class="id" title="binder">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#n:115"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:114"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':116"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:114"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab252"></a><h2 class="section">Classes of battles</h2>
+
+
+<div class="paragraph"> </div>
+
+In an hydra battle, the interaction between the two players  may depend on the time (number of rounds) elapsed since the beginning of the fight.
+
+<div class="paragraph"> </div>
+
+Let us formalize this dependence through a relation linking the number of the current round, and the hydra before and after the considered round.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="round_t" class="idref" href="#round_t"><span class="id" title="definition">round_t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Battle" class="idref" href="#Battle"><span class="id" title="record">Battle</span></a> :=<br/>
+&nbsp;&nbsp;{ <a id="battle_rel" class="idref" href="#battle_rel"><span class="id" title="projection">battle_rel</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_t"><span class="id" title="definition">round_t</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="battle_ok" class="idref" href="#battle_ok"><span class="id" title="projection">battle_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="i:119" class="idref" href="#i:119"><span class="id" title="binder">i</span></a> <a id="h:120" class="idref" href="#h:120"><span class="id" title="binder">h</span></a> <a id="h':121" class="idref" href="#h':121"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel:118"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:119"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:120"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':121"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:120"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':121"><span class="id" title="variable">h'</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+In the current state of this development, we will consider two instances of class <span class="inlinecode"><span class="id" title="var">Battle</span></span>:
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> In <span class="inlinecode"><span class="id" title="var">free</span></span>, we hydra may chose any replication factor at any time.
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> In <span class="inlinecode"><span class="id" title="var">standard</span></span> the replication factor at round number <span class="inlinecode"><span class="id" title="var">i</span></span> is just <span class="inlinecode"><span class="id" title="var">i</span></span>
+
+</li>
+</ul>
+ 
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span>, <span class="id" title="tactic">refine</span> ] <span class="id" title="keyword">Instance</span> <a id="free" class="idref" href="#free"><span class="id" title="instance">free</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a> <br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Build_Battle"><span class="id" title="constructor">Build_Battle</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span>  <a id="h:123" class="idref" href="#h:123"><span class="id" title="binder">h</span></a> <a id="h':124" class="idref" href="#h':124"><span class="id" title="binder">h'</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:123"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':124"><span class="id" title="variable">h'</span></a>) <span class="id" title="var">_</span>.<br/>
+ 
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span>, <span class="id" title="tactic">refine</span>] <span class="id" title="keyword">Instance</span> <a id="standard" class="idref" href="#standard"><span class="id" title="instance">standard</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Build_Battle"><span class="id" title="constructor">Build_Battle</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a>  <span class="id" title="var">_</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ The following relation allows us to consider sequences of rounds in a given  class  of battles 
+
+<div class="paragraph"> </div>
+
+ The proposition <span class="inlinecode"><span class="id" title="var">rounds</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode"><span class="id" title="var">h</span></span> <span class="inlinecode"><span class="id" title="var">j</span></span> <span class="inlinecode"><span class="id" title="var">h'</span></span> holds if there is a battle of class <span class="inlinecode"><span class="id" title="var">B</span></span> that
+  starts with hydra <span class="inlinecode"><span class="id" title="var">h</span></span> at round <span class="inlinecode"><span class="id" title="var">i</span></span> and ends with hydra <span class="inlinecode"><span class="id" title="var">h'</span></span> at round <span class="inlinecode"><span class="id" title="var">j</span></span>
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="rounds" class="idref" href="#rounds"><span class="id" title="inductive">rounds</span></a> (<a id="B:125" class="idref" href="#B:125"><span class="id" title="binder">B</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="rounds_1" class="idref" href="#rounds_1"><span class="id" title="constructor">rounds_1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:128" class="idref" href="#i:128"><span class="id" title="binder">i</span></a> <a id="h:129" class="idref" href="#h:129"><span class="id" title="binder">h</span></a>  <a id="h':130" class="idref" href="#h':130"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a>   <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:125"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:128"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:129"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':130"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds:126"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:125"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:128"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:129"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:128"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':130"><span class="id" title="variable">h'</span></a><br/>
+| <a id="rounds_n" class="idref" href="#rounds_n"><span class="id" title="constructor">rounds_n</span></a> : <span class="id" title="keyword">∀</span> <a id="i:131" class="idref" href="#i:131"><span class="id" title="binder">i</span></a> <a id="h:132" class="idref" href="#h:132"><span class="id" title="binder">h</span></a>  <a id="j:133" class="idref" href="#j:133"><span class="id" title="binder">j</span></a> <a id="h':134" class="idref" href="#h':134"><span class="id" title="binder">h'</span></a> <a id="h'':135" class="idref" href="#h'':135"><span class="id" title="binder">h''</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:125"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:131"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:132"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h'':135"><span class="id" title="variable">h''</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds:126"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:125"><span class="id" title="variable">B</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:131"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h'':135"><span class="id" title="variable">h''</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#j:133"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':134"><span class="id" title="variable">h'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds:126"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:125"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:131"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:132"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#j:133"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':134"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+number of steps leading to the hydra's death 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="battle_length" class="idref" href="#battle_length"><span class="id" title="definition">battle_length</span></a> <a id="B:136" class="idref" href="#B:136"><span class="id" title="binder">B</span></a> <a id="k:137" class="idref" href="#k:137"><span class="id" title="binder">k</span></a> <a id="h:138" class="idref" href="#h:138"><span class="id" title="binder">h</span></a> <a id="l:139" class="idref" href="#l:139"><span class="id" title="binder">l</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:136"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#k:137"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:138"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#k:137"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#l:139"><span class="id" title="variable">l</span></a>)%<span class="id" title="var">nat</span>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use rounds")]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="battle" class="idref" href="#battle"><span class="id" title="abbreviation">battle</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab253"></a><h3 class="section">Uniform termination</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Termination" class="idref" href="#Termination"><span class="id" title="definition">Termination</span></a> :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#transp"><span class="id" title="definition">transp</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="B_termination" class="idref" href="#B_termination"><span class="id" title="definition">B_termination</span></a> (<a id="B:140" class="idref" href="#B:140"><span class="id" title="binder">B</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<span class="id" title="keyword">fun</span> <a id="h':141" class="idref" href="#h':141"><span class="id" title="binder">h'</span></a> <a id="h:142" class="idref" href="#h:142"><span class="id" title="binder">h</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:143" class="idref" href="#i:143"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:140"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:143"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:142"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':141"><span class="id" title="variable">h'</span></a> ).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab254"></a><h3 class="section">Variants for proving termination</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Hvariant" class="idref" href="#Hvariant"><span class="id" title="record">Hvariant</span></a> {<a id="A:144" class="idref" href="#A:144"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="Lt:145" class="idref" href="#Lt:145"><span class="id" title="binder">Lt</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#A:144"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Wf:146" class="idref" href="#Wf:146"><span class="id" title="binder">Wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Lt:145"><span class="id" title="variable">Lt</span></a>)(<a id="B:147" class="idref" href="#B:147"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="m:148" class="idref" href="#m:148"><span class="id" title="binder">m</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#A:144"><span class="id" title="variable">A</span></a>): <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;{<a id="variant_decr" class="idref" href="#variant_decr"><span class="id" title="projection">variant_decr</span></a>: <span class="id" title="keyword">∀</span> <a id="i:150" class="idref" href="#i:150"><span class="id" title="binder">i</span></a> <a id="h:151" class="idref" href="#h:151"><span class="id" title="binder">h</span></a> <a id="h':152" class="idref" href="#h':152"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:151"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:147"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:150"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:151"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':152"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Lt:145"><span class="id" title="variable">Lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#m:148"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':152"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#m:148"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:151"><span class="id" title="variable">h</span></a>)}.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Variant bounded by some ordinal alpha &lt; epsilon0 <a id="lab255"></a><h2 class="section">Strictly Bounded variants</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="BoundedVariant" class="idref" href="#BoundedVariant"><span class="id" title="record">BoundedVariant</span></a> {<a id="A:154" class="idref" href="#A:154"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="Lt:155" class="idref" href="#Lt:155"><span class="id" title="binder">Lt</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#A:154"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="Wf:156" class="idref" href="#Wf:156"><span class="id" title="binder">Wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Lt:155"><span class="id" title="variable">Lt</span></a>}{<a id="B:157" class="idref" href="#B:157"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="m:158" class="idref" href="#m:158"><span class="id" title="binder">m</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#A:154"><span class="id" title="variable">A</span></a>} (<a id="Var:159" class="idref" href="#Var:159"><span class="id" title="binder">Var</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Wf:156"><span class="id" title="variable">Wf</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:157"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#m:158"><span class="id" title="variable">m</span></a>)(<a id="mu:160" class="idref" href="#mu:160"><span class="id" title="binder">mu</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#A:154"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;{ <a id="m_bounded" class="idref" href="#m_bounded"><span class="id" title="projection">m_bounded</span></a>: <span class="id" title="keyword">∀</span> <a id="h:162" class="idref" href="#h:162"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Lt:155"><span class="id" title="variable">Lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#m:158"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:162"><span class="id" title="variable">h</span></a> ) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#mu:160"><span class="id" title="variable">mu</span></a> }.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab256"></a><h3 class="section">Liveness</h3>
+
+
+<div class="paragraph"> </div>
+
+ For every reachable configuration <span class="inlinecode">(<span class="id" title="var">i</span>,<span class="id" title="var">h</span>)</span>, either <span class="inlinecode"><span class="id" title="var">h</span></span> is a head, or 
+ there exists a beheading leading to some configuration <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">i</span>,</span> <span class="inlinecode"><span class="id" title="var">h'</span>)</span>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Alive" class="idref" href="#Alive"><span class="id" title="definition">Alive</span></a> (<a id="B:164" class="idref" href="#B:164"><span class="id" title="binder">B</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:165" class="idref" href="#i:165"><span class="id" title="binder">i</span></a> <a id="h:166" class="idref" href="#h:166"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:166"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="h':167" class="idref" href="#h':167"><span class="id" title="binder">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B:164"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#i:165"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h:166"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#h':167"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+++ b/theories/html/hydras.Hydra.Hydra_Examples.html
@@ -0,0 +1,168 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Examples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Examples</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra_rect2"><span class="id" title="definition">Hydra_rect2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Examples" class="idref" href="#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.ex1" class="idref" href="#Examples.ex1"><span class="id" title="definition">ex1</span></a> : <span class="id" title="keyword">∀</span> <a id="h:1" class="idref" href="#h:1"><span class="id" title="binder">h</span></a> <a id="h':2" class="idref" href="#h':2"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd3"><span class="id" title="abbreviation">hyd3</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:1"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h':2"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:1"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h':2"><span class="id" title="variable">h'</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy" class="idref" href="#Examples.Hy"><span class="id" title="definition">Hy</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd3"><span class="id" title="abbreviation">hyd3</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy'" class="idref" href="#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.ex4" class="idref" href="#Examples.ex4"><span class="id" title="definition">ex4</span></a>:  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy"><span class="id" title="definition">Hy</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy''" class="idref" href="#Examples.Hy''"><span class="id" title="definition">Hy''</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>) 5)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy'H''" class="idref" href="#Examples.Hy'H''"><span class="id" title="definition">Hy'H''</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.R2_example" class="idref" href="#Examples.R2_example"><span class="id" title="definition">R2_example</span></a>:  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> 4 <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a>. </div>
+
+<div class="doc">
+move to 2nd sub-hydra (0-based indices) </div>
+<div class="code">
+ </div>
+
+<div class="doc">
+move to first sub-hydra </div>
+<div class="code">
+ </div>
+
+<div class="doc">
+we're at distance 2 from the to-be-chopped-off head 
+        let's go to the first daughter, 
+        then chop-off the leftmost head </div>
+<div class="code">
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Exx" class="idref" href="#Examples.Exx"><span class="id" title="definition">Exx</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><a id="h':3" class="idref" href="#h':3"><span class="id" title="binder">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h':3"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Ex5" class="idref" href="#Examples.Ex5"><span class="id" title="definition">Ex5</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><a id="h':4" class="idref" href="#h':4"><span class="id" title="binder">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy"><span class="id" title="definition">Hy</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#01d24958106a67451f21f2873a331e59"><span class="id" title="notation">-*-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h':4"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy'''" class="idref" href="#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>) 5)) 3 <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.hsize_bigger" class="idref" href="#Examples.hsize_bigger"><span class="id" title="definition">hsize_bigger</span></a> : (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize"><span class="id" title="definition">hsize</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize"><span class="id" title="definition">hsize</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a>)%<span class="id" title="var">nat</span>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.height_not_strictly_decreasing" class="idref" href="#Examples.height_not_strictly_decreasing"><span class="id" title="definition">height_not_strictly_decreasing</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy_1" class="idref" href="#Examples.Hy_1"><span class="id" title="definition">Hy_1</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy"><span class="id" title="definition">Hy</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a>.    
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy_2" class="idref" href="#Examples.Hy_2"><span class="id" title="definition">Hy_2</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> 4 <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.ex_2" class="idref" href="#Examples.ex_2"><span class="id" title="definition">ex_2</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><a id="Hy'':5" class="idref" href="#Hy'':5"><span class="id" title="binder">Hy''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> 4 <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'"><span class="id" title="definition">Hy'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Hy'':5"><span class="id" title="variable">Hy''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy_3" class="idref" href="#Examples.Hy_3"><span class="id" title="definition">Hy_3</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> 2 <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy''_Hy'''" class="idref" href="#Examples.Hy''_Hy'''"><span class="id" title="definition">Hy''_Hy'''</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''"><span class="id" title="definition">Hy''</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x"><span class="id" title="notation">-1-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Examples.Hy_Hy'''" class="idref" href="#Examples.Hy_Hy'''"><span class="id" title="definition">Hy_Hy'''</span></a> :  <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy"><span class="id" title="definition">Hy</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''"><span class="id" title="definition">Hy'''</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Examples"><span class="id" title="module">Examples</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra_ind"><span class="id" title="definition">Hydra_ind</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Bad" class="idref" href="#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span>  <a id="Bad.height_lt_size" class="idref" href="#Bad.height_lt_size"><span class="id" title="lemma">height_lt_size</span></a> (<a id="h:6" class="idref" href="#h:6"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:6"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize"><span class="id" title="definition">hsize</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:6"><span class="id" title="variable">h</span></a>.  <span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+
+<br/>
+ <span class="id" title="keyword">Lemma</span>  <a id="height_lt_size" class="idref" href="#height_lt_size"><span class="id" title="lemma">height_lt_size</span></a> (<a id="h:7" class="idref" href="#h:7"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) :  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:7"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize"><span class="id" title="definition">hsize</span></a> <a class="idref" href="hydras.Hydra.Hydra_Examples.html#h:7"><span class="id" title="variable">h</span></a>.<br/>
+  </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Hydra_Extraction.html b/theories/html/hydras.Hydra.Hydra_Extraction.html
new file mode 100644
index 00000000..23efd715
--- /dev/null
+++ b/theories/html/hydras.Hydra.Hydra_Extraction.html
@@ -0,0 +1,37 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Extraction</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Extraction</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Extraction</span> <span class="id" title="var">Language</span> <span class="id" title="var">OCaml</span>.<br/>
+<span class="id" title="keyword">Recursive Extraction</span> <span class="id" title="var">classic_battle</span> <span class="id" title="var">next_round</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Hydra_Lemmas.html b/theories/html/hydras.Hydra.Hydra_Lemmas.html
new file mode 100644
index 00000000..d3198a4e
--- /dev/null
+++ b/theories/html/hydras.Hydra.Hydra_Lemmas.html
@@ -0,0 +1,331 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Lemmas</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Lemmas</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+  Pierre Casteran 
+      LaBRI, Université de Bordeaux
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Operators_Properties.html#"><span class="id" title="library">Operators_Properties</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">More_Arith</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a>  <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="library">Hydra_Definitions</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab259"></a><h2 class="section">Properties of the  <span class="inlinecode"><span class="id" title="var">hydra</span></span> data type</h2>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="add_r_0" class="idref" href="#add_r_0"><span class="id" title="lemma">add_r_0</span></a> : <span class="id" title="keyword">∀</span> <a id="h:1" class="idref" href="#h:1"><span class="id" title="binder">h</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:1"><span class="id" title="variable">h</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:1"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hy_app_assoc" class="idref" href="#hy_app_assoc"><span class="id" title="lemma">hy_app_assoc</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:2" class="idref" href="#s1:2"><span class="id" title="binder">s1</span></a> <a id="s2:3" class="idref" href="#s2:3"><span class="id" title="binder">s2</span></a> <a id="s3:4" class="idref" href="#s3:4"><span class="id" title="binder">s3</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:2"><span class="id" title="variable">s1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:3"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:4"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:2"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:3"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:4"><span class="id" title="variable">s3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hcons_mult_app" class="idref" href="#hcons_mult_app"><span class="id" title="lemma">hcons_mult_app</span></a> : <span class="id" title="keyword">∀</span> (<a id="h:5" class="idref" href="#h:5"><span class="id" title="binder">h</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) (<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="s:7" class="idref" href="#s:7"><span class="id" title="binder">s</span></a> <a id="s':8" class="idref" href="#s':8"><span class="id" title="binder">s'</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:5"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:6"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:7"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':8"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:5"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:7"><span class="id" title="variable">s</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':8"><span class="id" title="variable">s'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hcons_mult_comm" class="idref" href="#hcons_mult_comm"><span class="id" title="lemma">hcons_mult_comm</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:9" class="idref" href="#i:9"><span class="id" title="binder">i</span></a> <a id="h:10" class="idref" href="#h:10"><span class="id" title="binder">h</span></a> <a id="s:11" class="idref" href="#s:11"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:10"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:9"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:10"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:11"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:10"><span class="id" title="variable">h</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:10"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:9"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:11"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_iff" class="idref" href="#R1_iff"><span class="id" title="lemma">R1_iff</span></a> <a id="s:12" class="idref" href="#s:12"><span class="id" title="binder">s</span></a> <a id="s':13" class="idref" href="#s':13"><span class="id" title="binder">s'</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:12"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':13"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:12"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':13"><span class="id" title="variable">s'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R2_iff" class="idref" href="#R2_iff"><span class="id" title="lemma">R2_iff</span></a> <a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a> <a id="s:15" class="idref" href="#s:15"><span class="id" title="binder">s</span></a> <a id="s':16" class="idref" href="#s':16"><span class="id" title="binder">s'</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:14"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:15"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':16"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:14"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:15"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':16"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:14"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:15"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':16"><span class="id" title="variable">s'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S2_iff" class="idref" href="#S2_iff"><span class="id" title="lemma">S2_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="h:18" class="idref" href="#h:18"><span class="id" title="binder">h</span></a> <a id="h':19" class="idref" href="#h':19"><span class="id" title="binder">h'</span></a> <a id="s:20" class="idref" href="#s:20"><span class="id" title="binder">s</span></a> <a id="s':21" class="idref" href="#s':21"><span class="id" title="binder">s'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:17"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:18"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:20"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':19"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':21"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:18"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':19"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:20"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':21"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:18"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':19"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:20"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':21"><span class="id" title="variable">s'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_app" class="idref" href="#S0_app"><span class="id" title="lemma">S0_app</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:22" class="idref" href="#s1:22"><span class="id" title="binder">s1</span></a> <a id="s2:23" class="idref" href="#s2:23"><span class="id" title="binder">s2</span></a> <a id="s3:24" class="idref" href="#s3:24"><span class="id" title="binder">s3</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:23"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:24"><span class="id" title="variable">s3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:22"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:23"><span class="id" title="variable">s2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:22"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:24"><span class="id" title="variable">s3</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_app" class="idref" href="#R1_app"><span class="id" title="lemma">R1_app</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:25" class="idref" href="#s1:25"><span class="id" title="binder">s1</span></a> <a id="s2:26" class="idref" href="#s2:26"><span class="id" title="binder">s2</span></a> <a id="s3:27" class="idref" href="#s3:27"><span class="id" title="binder">s3</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:26"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:27"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:25"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:26"><span class="id" title="variable">s2</span></a>)) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:25"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:27"><span class="id" title="variable">s3</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S1_app" class="idref" href="#S1_app"><span class="id" title="lemma">S1_app</span></a> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:29" class="idref" href="#s1:29"><span class="id" title="binder">s1</span></a> <a id="s2:30" class="idref" href="#s2:30"><span class="id" title="binder">s2</span></a> <a id="s3:31" class="idref" href="#s3:31"><span class="id" title="binder">s3</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:30"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:31"><span class="id" title="variable">s3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:28"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:29"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:30"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:29"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s3:31"><span class="id" title="variable">s3</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Induction Schemes for R2 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Scheme</span> <a id="R2_ind2" class="idref" href="#R2_ind2"><span class="id" title="definition">R2_ind2</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="S2_ind2" class="idref" href="#S2_ind2"><span class="id" title="definition">S2_ind2</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
+<span class="id" title="keyword">Arguments</span>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#R2_ind2"><span class="id" title="definition">R2_ind2</span></a> <span class="id" title="var">n</span> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+<span class="id" title="keyword">Arguments</span>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#S2_ind2"><span class="id" title="definition">S2_ind2</span></a> <span class="id" title="var">n</span> <span class="id" title="var">P</span> <span class="id" title="var">P0</span> : <span class="id" title="tactic">rename</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S2_app" class="idref" href="#S2_app"><span class="id" title="lemma">S2_app</span></a> : <span class="id" title="keyword">∀</span>  <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a>  <a id="s1:33" class="idref" href="#s1:33"><span class="id" title="binder">s1</span></a> <a id="s2:34" class="idref" href="#s2:34"><span class="id" title="binder">s2</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:32"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:33"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:34"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:35" class="idref" href="#l:35"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:32"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:35"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:33"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:35"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:34"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R2_app" class="idref" href="#R2_app"><span class="id" title="lemma">R2_app</span></a> <a id="n:46" class="idref" href="#n:46"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span>  <a id="s1:47" class="idref" href="#s1:47"><span class="id" title="binder">s1</span></a> <a id="s2:48" class="idref" href="#s2:48"><span class="id" title="binder">s2</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:46"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:47"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:48"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:49" class="idref" href="#l:49"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:46"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:49"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s1:47"><span class="id" title="variable">s1</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:49"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s2:48"><span class="id" title="variable">s2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hcons_mult_S0" class="idref" href="#hcons_mult_S0"><span class="id" title="lemma">hcons_mult_S0</span></a> : <span class="id" title="keyword">∀</span> <a id="h:50" class="idref" href="#h:50"><span class="id" title="binder">h</span></a>  <a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a> <a id="s:52" class="idref" href="#s:52"><span class="id" title="binder">s</span></a> <a id="s':53" class="idref" href="#s':53"><span class="id" title="binder">s'</span></a> , <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:52"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':53"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:50"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:52"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:50"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':53"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hcons_mult_S1" class="idref" href="#hcons_mult_S1"><span class="id" title="lemma">hcons_mult_S1</span></a> <a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span> <a id="h:55" class="idref" href="#h:55"><span class="id" title="binder">h</span></a> <a id="i:56" class="idref" href="#i:56"><span class="id" title="binder">i</span></a>  <a id="s:57" class="idref" href="#s:57"><span class="id" title="binder">s</span></a> <a id="s':58" class="idref" href="#s':58"><span class="id" title="binder">s'</span></a> , <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:54"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:57"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':58"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:54"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:55"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:56"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:57"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:55"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:56"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':58"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hcons_mult_S2" class="idref" href="#hcons_mult_S2"><span class="id" title="lemma">hcons_mult_S2</span></a> <a id="n:59" class="idref" href="#n:59"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span> <a id="h:60" class="idref" href="#h:60"><span class="id" title="binder">h</span></a>  <a id="i:61" class="idref" href="#i:61"><span class="id" title="binder">i</span></a> <a id="s:62" class="idref" href="#s:62"><span class="id" title="binder">s</span></a> <a id="s':63" class="idref" href="#s':63"><span class="id" title="binder">s'</span></a> , <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:59"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:62"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':63"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S2"><span class="id" title="inductive">S2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:59"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:60"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s:62"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:60"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:61"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#s':63"><span class="id" title="variable">s'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab260"></a><h2 class="section">Properties of single rounds</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="head_no_round_n" class="idref" href="#head_no_round_n"><span class="id" title="lemma">head_no_round_n</span></a> : <span class="id" title="keyword">∀</span> <a id="i:64" class="idref" href="#i:64"><span class="id" title="binder">i</span></a> <a id="h:65" class="idref" href="#h:65"><span class="id" title="binder">h</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:64"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:65"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="head_no_round" class="idref" href="#head_no_round"><span class="id" title="lemma">head_no_round</span></a> : <span class="id" title="keyword">∀</span> <a id="h:66" class="idref" href="#h:66"><span class="id" title="binder">h</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round"><span class="id" title="definition">round</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:66"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab261"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">battle</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rounds_trans" class="idref" href="#rounds_trans"><span class="id" title="lemma">rounds_trans</span></a> {<a id="B:67" class="idref" href="#B:67"><span class="id" title="binder">B</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>} :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:68" class="idref" href="#i:68"><span class="id" title="binder">i</span></a> <a id="h:69" class="idref" href="#h:69"><span class="id" title="binder">h</span></a> <a id="j:70" class="idref" href="#j:70"><span class="id" title="binder">j</span></a> <a id="h':71" class="idref" href="#h':71"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#B:67"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:68"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:69"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:70"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':71"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:72" class="idref" href="#k:72"><span class="id" title="binder">k</span></a> <a id="h0:73" class="idref" href="#h0:73"><span class="id" title="binder">h0</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#B:67"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#k:72"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h0:73"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:68"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:69"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#B:67"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#k:72"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h0:73"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:70"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':71"><span class="id" title="variable">h'</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab262"></a><h2 class="section">Properties of standard and free battle classes</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_incl_free" class="idref" href="#standard_incl_free"><span class="id" title="lemma">standard_incl_free</span></a> : <span class="id" title="keyword">∀</span> <a id="i:74" class="idref" href="#i:74"><span class="id" title="binder">i</span></a> <a id="h:75" class="idref" href="#h:75"><span class="id" title="binder">h</span></a> <a id="h':76" class="idref" href="#h':76"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a>   <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:74"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:75"><span class="id" title="variable">h</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':76"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:74"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:75"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':76"><span class="id" title="variable">h'</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="standard_battle_head" class="idref" href="#standard_battle_head"><span class="id" title="lemma">standard_battle_head</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:77" class="idref" href="#i:77"><span class="id" title="binder">i</span></a> <a id="h:78" class="idref" href="#h:78"><span class="id" title="binder">h</span></a> <a id="j:79" class="idref" href="#j:79"><span class="id" title="binder">j</span></a> <a id="h':80" class="idref" href="#h':80"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:77"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:78"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:79"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':80"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:78"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab263"></a><h2 class="section">Generic properties of round_plus</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="round_plus_trans" class="idref" href="#round_plus_trans"><span class="id" title="lemma">round_plus_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="h:81" class="idref" href="#h:81"><span class="id" title="binder">h</span></a> <a id="h':82" class="idref" href="#h':82"><span class="id" title="binder">h'</span></a> <a id="h'':83" class="idref" href="#h'':83"><span class="id" title="binder">h''</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:81"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':82"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':82"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h'':83"><span class="id" title="variable">h''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:81"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h'':83"><span class="id" title="variable">h''</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_remove_r" class="idref" href="#S0_remove_r"><span class="id" title="lemma">S0_remove_r</span></a> : <span class="id" title="keyword">∀</span> <a id="l:84" class="idref" href="#l:84"><span class="id" title="binder">l</span></a> , <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r"><span class="id" title="definition">add_head_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:84"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:84"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_remove_r_i" class="idref" href="#S0_remove_r_i"><span class="id" title="lemma">S0_remove_r_i</span></a> : <span class="id" title="keyword">∀</span> <a id="i:85" class="idref" href="#i:85"><span class="id" title="binder">i</span></a> <a id="l:86" class="idref" href="#l:86"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus"><span class="id" title="definition">add_head_r_plus</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:86"><span class="id" title="variable">l</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:85"><span class="id" title="variable">i</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus"><span class="id" title="definition">add_head_r_plus</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#l:86"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:85"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_remove_r_i" class="idref" href="#R1_remove_r_i"><span class="id" title="lemma">R1_remove_r_i</span></a> : <span class="id" title="keyword">∀</span> <a id="i:87" class="idref" href="#i:87"><span class="id" title="binder">i</span></a> <a id="h:88" class="idref" href="#h:88"><span class="id" title="binder">h</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:88"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:87"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:88"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:87"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="round_n_remove_h" class="idref" href="#round_n_remove_h"><span class="id" title="lemma">round_n_remove_h</span></a> : <span class="id" title="keyword">∀</span> <a id="i:89" class="idref" href="#i:89"><span class="id" title="binder">i</span></a> <a id="j:90" class="idref" href="#j:90"><span class="id" title="binder">j</span></a> <a id="h:91" class="idref" href="#h:91"><span class="id" title="binder">h</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:90"><span class="id" title="variable">j</span></a>  (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:91"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:89"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:91"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:89"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="remove_heads_r" class="idref" href="#remove_heads_r"><span class="id" title="lemma">remove_heads_r</span></a> : <span class="id" title="keyword">∀</span> <a id="i:92" class="idref" href="#i:92"><span class="id" title="binder">i</span></a> <a id="h:93" class="idref" href="#h:93"><span class="id" title="binder">h</span></a> <a id="j:94" class="idref" href="#j:94"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:94"><span class="id" title="variable">j</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:93"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:92"><span class="id" title="variable">i</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:92"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:94"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:93"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="remove_heads_r_free" class="idref" href="#remove_heads_r_free"><span class="id" title="lemma">remove_heads_r_free</span></a>: <span class="id" title="keyword">∀</span> <a id="i:95" class="idref" href="#i:95"><span class="id" title="binder">i</span></a> <a id="h:96" class="idref" href="#h:96"><span class="id" title="binder">h</span></a> <a id="j:97" class="idref" href="#j:97"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:97"><span class="id" title="variable">j</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#add_r"><span class="id" title="definition">add_r</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:96"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:95"><span class="id" title="variable">i</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:95"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:97"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:96"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="variant_mono_free" class="idref" href="#variant_mono_free"><span class="id" title="lemma">variant_mono_free</span></a>  {<a id="A:98" class="idref" href="#A:98"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="Lt:99" class="idref" href="#Lt:99"><span class="id" title="binder">Lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#A:98"><span class="id" title="variable">A</span></a>}{<a id="Tr:100" class="idref" href="#Tr:100"><span class="id" title="binder">Tr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:99"><span class="id" title="variable">Lt</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="Wf:101" class="idref" href="#Wf:101"><span class="id" title="binder">Wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:99"><span class="id" title="variable">Lt</span></a>} <a id="m:102" class="idref" href="#m:102"><span class="id" title="binder">m</span></a> {<a id="V:103" class="idref" href="#V:103"><span class="id" title="binder">V</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Wf:101"><span class="id" title="variable">Wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:102"><span class="id" title="variable">m</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:104" class="idref" href="#i:104"><span class="id" title="binder">i</span></a> <a id="h:105" class="idref" href="#h:105"><span class="id" title="binder">h</span></a> <a id="j:106" class="idref" href="#j:106"><span class="id" title="binder">j</span></a> <a id="h':107" class="idref" href="#h':107"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:104"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:105"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:106"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':107"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:99"><span class="id" title="variable">Lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:102"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':107"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:102"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:105"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m_strict_mono" class="idref" href="#m_strict_mono"><span class="id" title="lemma">m_strict_mono</span></a>  {<a id="A:108" class="idref" href="#A:108"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="Lt:109" class="idref" href="#Lt:109"><span class="id" title="binder">Lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#A:108"><span class="id" title="variable">A</span></a>}{<a id="St:110" class="idref" href="#St:110"><span class="id" title="binder">St</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:109"><span class="id" title="variable">Lt</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="Wf:111" class="idref" href="#Wf:111"><span class="id" title="binder">Wf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:109"><span class="id" title="variable">Lt</span></a>} <a id="m:112" class="idref" href="#m:112"><span class="id" title="binder">m</span></a> (<a id="V:113" class="idref" href="#V:113"><span class="id" title="binder">V</span></a>: @<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Wf:111"><span class="id" title="variable">Wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:112"><span class="id" title="variable">m</span></a>){<a id="h:114" class="idref" href="#h:114"><span class="id" title="binder">h</span></a> <a id="h':115" class="idref" href="#h':115"><span class="id" title="binder">h'</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:114"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':115"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Lt:109"><span class="id" title="variable">Lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:112"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':115"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:112"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:114"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+companion lemmas for R1 and R2 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="round_n_inv" class="idref" href="#round_n_inv"><span class="id" title="lemma">round_n_inv</span></a> <a id="i:116" class="idref" href="#i:116"><span class="id" title="binder">i</span></a> <a id="h:117" class="idref" href="#h:117"><span class="id" title="binder">h</span></a> <a id="h':118" class="idref" href="#h':118"><span class="id" title="binder">h'</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:116"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:117"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':118"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:117"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':118"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:116"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:117"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':118"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rounds_free_equiv1" class="idref" href="#rounds_free_equiv1"><span class="id" title="lemma">rounds_free_equiv1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:119" class="idref" href="#i:119"><span class="id" title="binder">i</span></a> <a id="j:120" class="idref" href="#j:120"><span class="id" title="binder">j</span></a> <a id="h:121" class="idref" href="#h:121"><span class="id" title="binder">h</span></a> <a id="h':122" class="idref" href="#h':122"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:119"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:121"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:120"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':122"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:121"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':122"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rounds_free_equiv2" class="idref" href="#rounds_free_equiv2"><span class="id" title="lemma">rounds_free_equiv2</span></a> : <span class="id" title="keyword">∀</span> <a id="h:123" class="idref" href="#h:123"><span class="id" title="binder">h</span></a> <a id="h':124" class="idref" href="#h':124"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:123"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':124"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:125" class="idref" href="#i:125"><span class="id" title="binder">i</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="j:126" class="idref" href="#j:126"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#i:125"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:123"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#j:126"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':124"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Termination_strong" class="idref" href="#Termination_strong"><span class="id" title="lemma">Termination_strong</span></a> (<a id="B:127" class="idref" href="#B:127"><span class="id" title="binder">B</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Battle"><span class="id" title="class">Battle</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Termination"><span class="id" title="definition">Termination</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#B_termination"><span class="id" title="definition">B_termination</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#B:127"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Hvariant_Termination" class="idref" href="#Hvariant_Termination"><span class="id" title="lemma">Hvariant_Termination</span></a> {<a id="A:128" class="idref" href="#A:128"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} {<a id="lt:129" class="idref" href="#lt:129"><span class="id" title="binder">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#A:128"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hwf:130" class="idref" href="#Hwf:130"><span class="id" title="binder">Hwf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#lt:129"><span class="id" title="variable">lt</span></a>) (<a id="m:131" class="idref" href="#m:131"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#A:128"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;@<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#Hwf:130"><span class="id" title="variable">Hwf</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#m:131"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Termination"><span class="id" title="definition">Termination</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If the hydra is not reduced to a head, there exists at 
+    least one head that  Hercules can chop off  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="next_round_dec" class="idref" href="#next_round_dec"><span class="id" title="definition">next_round_dec</span></a> <a id="n:136" class="idref" href="#n:136"><span class="id" title="binder">n</span></a> (<a id="h:137" class="idref" href="#h:137"><span class="id" title="binder">h</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:137"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="h':138" class="idref" href="#h':138"><span class="id" title="binder">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:137"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':138"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:136"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:137"><span class="id" title="variable">h</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h':138"><span class="id" title="variable">h'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>. </div>
+
+<div class="doc">
+chose  a lowest head 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="next_round" class="idref" href="#next_round"><span class="id" title="definition">next_round</span></a>  <a id="n:143" class="idref" href="#n:143"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="h:144" class="idref" href="#h:144"><span class="id" title="binder">h</span></a> , <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_spec"><span class="id" title="definition">round_spec</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:144"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:143"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="next_step" class="idref" href="#next_step"><span class="id" title="definition">next_step</span></a> (<a id="f:147" class="idref" href="#f:147"><span class="id" title="binder">f</span></a> : <span class="id" title="keyword">∀</span> <a id="n:145" class="idref" href="#n:145"><span class="id" title="binder">n</span></a> <a id="h:146" class="idref" href="#h:146"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_spec"><span class="id" title="definition">round_spec</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:146"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:145"><span class="id" title="variable">n</span></a>) <a id="n:148" class="idref" href="#n:148"><span class="id" title="binder">n</span></a> <a id="h:149" class="idref" href="#h:149"><span class="id" title="binder">h</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#f:147"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:148"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:149"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inright"><span class="id" title="constructor">inright</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#inleft"><span class="id" title="constructor">inleft</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> <span class="id" title="var">h'</span> <span class="id" title="var">_</span>) ⇒  <span class="id" title="var">h'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="classic_battle" class="idref" href="#classic_battle"><span class="id" title="definition">classic_battle</span></a> <a id="f:150" class="idref" href="#f:150"><span class="id" title="binder">f</span></a> <a id="t:151" class="idref" href="#t:151"><span class="id" title="binder">t</span></a> <a id="h:152" class="idref" href="#h:152"><span class="id" title="binder">h</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="go:159" class="idref" href="#go:159"><span class="id" title="binder">go</span></a> <a id="n:153" class="idref" href="#n:153"><span class="id" title="binder">n</span></a> <a id="t:154" class="idref" href="#t:154"><span class="id" title="binder">t</span></a> <a id="h:155" class="idref" href="#h:155"><span class="id" title="binder">h</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:155"><span class="id" title="variable">h</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#t:154"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:155"><span class="id" title="variable">h</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|   <span class="id" title="var">_</span>, 0 ⇒ <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:155"><span class="id" title="variable">h</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">t'</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#go:156"><span class="id" title="variable">go</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:153"><span class="id" title="variable">n</span></a>) <span class="id" title="var">t'</span> (<a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#next_step"><span class="id" title="definition">next_step</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#f:150"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#n:153"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:155"><span class="id" title="variable">h</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#go:159"><span class="id" title="variable">go</span></a> 1 <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#t:151"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#h:152"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,173 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Termination</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Termination</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab218"></a><h2 class="section">Proof of termination of all hydra battles</h2>
+
+<div class="paragraph"> </div>
+
+ Pierre Casteran, Univ. Bordeaux, LaBRI, UMR 5800 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#"><span class="id" title="library">Hessenberg</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="module">ON_Generic</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab219"></a><h3 class="section">Converting any hydra into an ordinal less than</h3>
+
+  epsilon0   
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="m" class="idref" href="#m"><span class="id" title="definition">m</span></a> (<a id="h:1" class="idref" href="#h:1"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:1"><span class="id" title="variable">h</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <span class="id" title="var">hs</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms:4"><span class="id" title="definition">ms</span></a> <span class="id" title="var">hs</span><br/>
+<span class="id" title="keyword">end</span> <br/>
+<span class="id" title="keyword">with</span> <a id="ms" class="idref" href="#ms"><span class="id" title="definition">ms</span></a> (<a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a>) :  <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:2"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h</span> <span class="id" title="var">s'</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m:3"><span class="id" title="definition">m</span></a> <span class="id" title="var">h</span>) <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms:4"><span class="id" title="definition">ms</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ms_eqn2" class="idref" href="#ms_eqn2"><span class="id" title="lemma">ms_eqn2</span></a> :  <span class="id" title="keyword">∀</span> <a id="h:7" class="idref" href="#h:7"><span class="id" title="binder">h</span></a> <a id="s:8" class="idref" href="#s:8"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:7"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:8"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:7"><span class="id" title="variable">h</span></a>) <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:8"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="o_finite_mult_S_rw" class="idref" href="#o_finite_mult_S_rw"><span class="id" title="lemma">o_finite_mult_S_rw</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> <a id="a:10" class="idref" href="#a:10"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:9"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Termination.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#a:10"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ The functions <span class="inlinecode"><span class="id" title="var">m</span></span> and <span class="inlinecode"><span class="id" title="var">ms</span></span> return well formed ordinals (less than epsilon0)
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="m_nf" class="idref" href="#m_nf"><span class="id" title="lemma">m_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="h:11" class="idref" href="#h:11"><span class="id" title="binder">h</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:11"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ms_nf" class="idref" href="#ms_nf"><span class="id" title="lemma">ms_nf</span></a> : <span class="id" title="keyword">∀</span> <a id="s:14" class="idref" href="#s:14"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:14"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">m_nf</span> <span class="id" title="var">nf_phi0</span> <span class="id" title="var">ms_nf</span> : <span class="id" title="var">T1</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ms_eqn3" class="idref" href="#ms_eqn3"><span class="id" title="lemma">ms_eqn3</span></a> :  <span class="id" title="keyword">∀</span> <a id="h:15" class="idref" href="#h:15"><span class="id" title="binder">h</span></a> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> <a id="s:17" class="idref" href="#s:17"><span class="id" title="binder">s</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:15"><span class="id" title="variable">h</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:17"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hessenberg.html#o_finite_mult"><span class="id" title="definition">o_finite_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:16"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:15"><span class="id" title="variable">h</span></a>)) <a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:17"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab220"></a><h2 class="section">Monotonicity lemmas for S0, R1, etc .</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_decr_0" class="idref" href="#S0_decr_0"><span class="id" title="lemma">S0_decr_0</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:18" class="idref" href="#s:18"><span class="id" title="binder">s</span></a> <a id="s':19" class="idref" href="#s':19"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:18"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':19"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':19"><span class="id" title="variable">s'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:18"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_decr" class="idref" href="#S0_decr"><span class="id" title="lemma">S0_decr</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:20" class="idref" href="#s:20"><span class="id" title="binder">s</span></a> <a id="s':21" class="idref" href="#s':21"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:20"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':21"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':21"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:20"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_decr_0" class="idref" href="#R1_decr_0"><span class="id" title="lemma">R1_decr_0</span></a> : <span class="id" title="keyword">∀</span> <a id="h:22" class="idref" href="#h:22"><span class="id" title="binder">h</span></a> <a id="h':23" class="idref" href="#h':23"><span class="id" title="binder">h'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:22"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':23"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':23"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:22"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_decr" class="idref" href="#R1_decr"><span class="id" title="lemma">R1_decr</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="h:24" class="idref" href="#h:24"><span class="id" title="binder">h</span></a> <a id="h':25" class="idref" href="#h':25"><span class="id" title="binder">h'</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:24"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':25"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':25"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:24"><span class="id" title="variable">h</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S1_decr_0" class="idref" href="#S1_decr_0"><span class="id" title="lemma">S1_decr_0</span></a> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:27" class="idref" href="#s:27"><span class="id" title="binder">s</span></a> <a id="s':28" class="idref" href="#s':28"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:27"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':28"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':28"><span class="id" title="variable">s'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:27"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S1_decr" class="idref" href="#S1_decr"><span class="id" title="lemma">S1_decr</span></a> <a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:30" class="idref" href="#s:30"><span class="id" title="binder">s</span></a> <a id="s':31" class="idref" href="#s':31"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:29"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:30"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':31"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s':31"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:30"><span class="id" title="variable">s</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="m_ms" class="idref" href="#m_ms"><span class="id" title="lemma">m_ms</span></a> : <span class="id" title="keyword">∀</span> <a id="s:32" class="idref" href="#s:32"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:32"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#ms"><span class="id" title="definition">ms</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#s:32"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R2_decr_0" class="idref" href="#R2_decr_0"><span class="id" title="lemma">R2_decr_0</span></a> <a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span> <a id="h:34" class="idref" href="#h:34"><span class="id" title="binder">h</span></a> <a id="h':35" class="idref" href="#h':35"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:33"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:34"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':35"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':35"><span class="id" title="variable">h'</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:34"><span class="id" title="variable">h</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R2_decr" class="idref" href="#R2_decr"><span class="id" title="lemma">R2_decr</span></a> <a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a> : <span class="id" title="keyword">∀</span> <a id="h:43" class="idref" href="#h:43"><span class="id" title="binder">h</span></a> <a id="h':44" class="idref" href="#h':44"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#n:42"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:43"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':44"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':44"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:43"><span class="id" title="variable">h</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="round_decr" class="idref" href="#round_decr"><span class="id" title="lemma">round_decr</span></a> : <span class="id" title="keyword">∀</span> <a id="h:45" class="idref" href="#h:45"><span class="id" title="binder">h</span></a> <a id="h':46" class="idref" href="#h':46"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:45"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x"><span class="id" title="notation">-1-&gt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':46"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h':46"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#h:45"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span>, <span class="id" title="var">program</span> ] <span class="id" title="keyword">Instance</span> <a id="var" class="idref" href="#var"><span class="id" title="instance">var</span></a> (<a id="h:47" class="idref" href="#h:47"><span class="id" title="binder">h</span></a>:<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>:= @<a class="idref" href="hydras.Epsilon0.E0.html#mkord"><span class="id" title="constructor">mkord</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a> <span class="id" title="var">h</span>) <span class="id" title="var">_</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="HVariant_0" class="idref" href="#HVariant_0"><span class="id" title="instance">HVariant_0</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">m</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="HVariant" class="idref" href="#HVariant"><span class="id" title="instance">HVariant</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#Epsilon0"><span class="id" title="instance">Epsilon0</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#var"><span class="id" title="instance">var</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="every_battle_terminates" class="idref" href="#every_battle_terminates"><span class="id" title="lemma">every_battle_terminates</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Termination"><span class="id" title="definition">Termination</span></a>.<br/>
+ </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.Hydra_Theorems.html b/theories/html/hydras.Hydra.Hydra_Theorems.html
new file mode 100644
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+++ b/theories/html/hydras.Hydra.Hydra_Theorems.html
@@ -0,0 +1,240 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Hydra_Theorems</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Hydra_Theorems</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a>  <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#"><span class="id" title="library">Epsilon0_Needed_Free</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Epsilon0_Needed_Std.html#"><span class="id" title="library">Epsilon0_Needed_Std</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Termination.html#"><span class="id" title="library">Hydra_Termination</span></a> <a class="idref" href="hydras.Epsilon0.L_alpha.html#"><span class="id" title="library">L_alpha</span></a> <a class="idref" href="hydras.Hydra.Battle_length.html#"><span class="id" title="library">Battle_length</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#"><span class="id" title="library">Ack</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="module">E0</span></a> <a class="idref" href="hydras.Epsilon0.Large_Sets.html#"><span class="id" title="module">Large_Sets</span></a> <a class="idref" href="hydras.Epsilon0.Hprime.html#"><span class="id" title="module">Hprime</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="module">Paths</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="module">MoreLists</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#"><span class="id" title="module">O2H</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="module">Hydra_Definitions</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="module">Iterates</span></a>.<br/>
+<span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="module">Hydra_Definitions</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab221"></a><h2 class="section">Liveness</h2>
+
+
+<div class="paragraph"> </div>
+
+  If the hydra is not reduced to one  head, then Hercules can chop off 
+  some head
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Alive_free" class="idref" href="#Alive_free"><span class="id" title="lemma">Alive_free</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Alive"><span class="id" title="definition">Alive</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a>. 
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Alive_standard" class="idref" href="#Alive_standard"><span class="id" title="lemma">Alive_standard</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Alive"><span class="id" title="definition">Alive</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a>. 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab222"></a><h2 class="section">Termination of free battles</h2>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Variant_LT_free_0" class="idref" href="#Variant_LT_free_0"><span class="id" title="lemma">Variant_LT_free_0</span></a> :  @<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a>  <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">Hydra_Termination.m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Variant_lt_free" class="idref" href="#Variant_lt_free"><span class="id" title="lemma">Variant_lt_free</span></a>:  @<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#var"><span class="id" title="instance">Hydra_Termination.var</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Variant_LT_standard" class="idref" href="#Variant_LT_standard"><span class="id" title="lemma">Variant_LT_standard</span></a> : @<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.T1.html#T1_wf"><span class="id" title="lemma">T1_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#m"><span class="id" title="definition">Hydra_Termination.m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Variant_lt_standard" class="idref" href="#Variant_lt_standard"><span class="id" title="lemma">Variant_lt_standard</span></a> : @<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.Epsilon0.E0.html#E0lt_wf"><span class="id" title="lemma">E0lt_wf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Hydra_Termination.html#var"><span class="id" title="instance">Hydra_Termination.var</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab223"></a><h2 class="section">Impossibility theorems</h2>
+
+
+<div class="paragraph"> </div>
+
+  Termination of free battles cannot be proven with a variant from hydras into a segment  with  
+
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+  Impossiblity to define a variant bounded by some ordinal less than 
+     <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_free"><span class="id" title="lemma">Impossibility_free</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">About</span>  <a class="idref" href="hydras.Hydra.Battle_length.html#battle_length_std"><span class="id" title="lemma">battle_length_std</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="battle_length_std_Hardy" class="idref" href="#battle_length_std_Hardy"><span class="id" title="lemma">battle_length_std_Hardy</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Theorems.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0zero"><span class="id" title="instance">E0zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:2" class="idref" href="#k:2"><span class="id" title="binder">k</span></a> , 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:2"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l:3" class="idref" href="#l:3"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Hprime.html#H'_"><span class="id" title="definition">H'_</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:2"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:2"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#l:3"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_length"><span class="id" title="definition">battle_length</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:2"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#alpha:1"><span class="id" title="variable">alpha</span></a>)) <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#l:3"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab224"></a><h2 class="section">Battle length is not PR</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#"><span class="id" title="library">F_alpha</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#"><span class="id" title="library">AckNotPR</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#"><span class="id" title="library">PrimRecExamples</span></a> <a class="idref" href="hydras.Epsilon0.F_omega.html#"><span class="id" title="library">F_omega</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="module">E0</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="battle_length_notPR" class="idref" href="#battle_length_notPR"><span class="id" title="section">battle_length_notPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="H:5" class="idref" href="#H:5"><span class="id" title="binder">H</span></a>: <span class="id" title="keyword">∀</span> <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.Hydra.Battle_length.html#l_std"><span class="id" title="definition">l_std</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#alpha:4"><span class="id" title="variable">alpha</span></a>)).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A counter example 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="battle_length_notPR.alpha" class="idref" href="#battle_length_notPR.alpha"><span class="id" title="variable">alpha</span></a> := <a class="idref" href="hydras.Epsilon0.E0.html#E0_phi0"><span class="id" title="instance">E0_phi0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="battle_length_notPR.h" class="idref" href="#battle_length_notPR.h"><span class="id" title="variable">h</span></a> := <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+let us get rid of the substraction ... 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="battle_length_notPR.m" class="idref" href="#battle_length_notPR.m"><span class="id" title="variable">m</span></a> <a id="k:6" class="idref" href="#k:6"><span class="id" title="binder">k</span></a> := <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:6"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_eqn" class="idref" href="#m_eqn"><span class="id" title="lemma">m_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="k:7" class="idref" href="#k:7"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:7"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.Hydra.Battle_length.html#l_std"><span class="id" title="definition">l_std</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:7"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:7"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span>. 
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="mIsPR" class="idref" href="#mIsPR"><span class="id" title="instance">mIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_ge_F_omega" class="idref" href="#m_ge_F_omega"><span class="id" title="lemma">m_ge_F_omega</span></a> <a id="k:10" class="idref" href="#k:10"><span class="id" title="binder">k</span></a>: <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:10"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#k:10"><span class="id" title="variable">k</span></a>). 
+<br/>
+</div>
+
+<div class="doc">
+We compare <span class="inlinecode"><span class="id" title="var">m</span></span> with the Ackermann function 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_ge_Ack" class="idref" href="#m_ge_Ack"><span class="id" title="lemma">m_ge_Ack</span></a>:  <span class="id" title="keyword">∀</span> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:11"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:11"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:11"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:11"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_dominates_Ack_from_3" class="idref" href="#m_dominates_Ack_from_3"><span class="id" title="lemma">m_dominates_Ack_from_3</span></a> : <span class="id" title="keyword">∀</span> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a>, 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:12"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="m_dominates_Ack" class="idref" href="#m_dominates_Ack"><span class="id" title="lemma">m_dominates_Ack</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> (<span class="id" title="keyword">fun</span> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:13"><span class="id" title="variable">n</span></a>)) (<span class="id" title="keyword">fun</span> <a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack.Ack</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:14"><span class="id" title="variable">n</span></a>).  
+<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SmNotPR" class="idref" href="#SmNotPR"><span class="id" title="lemma">SmNotPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#n:15"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="LNotPR" class="idref" href="#LNotPR"><span class="id" title="lemma">LNotPR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR"><span class="id" title="section">battle_length_notPR</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Battle_length.html#l_std_ok"><span class="id" title="lemma">l_std_ok</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Hydra.Hydra_Theorems.html#LNotPR"><span class="id" title="lemma">LNotPR</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Epsilon0.L_alpha.html#L_"><span class="id" title="definition">L_</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Hydra.KP_example.html b/theories/html/hydras.Hydra.KP_example.html
new file mode 100644
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+++ b/theories/html/hydras.Hydra.KP_example.html
@@ -0,0 +1,152 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.KP_example</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.KP_example</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Casteran, Univ. Bordeaux and LaBRI  
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="library">Hydra_Definitions</span></a> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The hydra from page 286 of <span class="inlinecode"><span class="id" title="var">KP</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="KP" class="idref" href="#KP"><span class="id" title="section">KP</span></a>.<br/>
+</div>
+
+<div class="doc">
+initial state 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="h0" class="idref" href="#h0"><span class="id" title="definition">h0</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 3)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a id="F1" class="idref" href="#F1"><span class="id" title="lemma">F1</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hsize"><span class="id" title="definition">hsize</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#h0"><span class="id" title="definition">h0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 14.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a id="F2" class="idref" href="#F2"><span class="id" title="lemma">F2</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#h0"><span class="id" title="definition">h0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 4.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ after stage 1 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="h1" class="idref" href="#h1"><span class="id" title="definition">h1</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd3"><span class="id" title="abbreviation">hyd3</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+After stage 2 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="hyd4" class="idref" href="#hyd4"><span class="id" title="abbreviation">hyd4</span></a> <span class="id" title="var">h1</span> <span class="id" title="var">h2</span> <span class="id" title="var">h3</span> <span class="id" title="var">h4</span> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h1</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h2</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h3</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <span class="id" title="var">h4</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span>  <a id="KP.h'" class="idref" href="#KP.h'"><span class="id" title="variable">h'</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="h2" class="idref" href="#h2"><span class="id" title="definition">h2</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.KP_example.html#hyd4"><span class="id" title="abbreviation">hyd4</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd3"><span class="id" title="abbreviation">hyd3</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> 2)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a>  <a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+After stage 3 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="h3" class="idref" href="#h3"><span class="id" title="definition">h3</span></a> := <a class="idref" href="hydras.Hydra.KP_example.html#hyd4"><span class="id" title="abbreviation">hyd4</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>) 4<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>))))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#KP.h'"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a id="h0_h1" class="idref" href="#h0_h1"><span class="id" title="lemma">h0_h1</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> 1 <a class="idref" href="hydras.Hydra.KP_example.html#h0"><span class="id" title="definition">h0</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#h1"><span class="id" title="definition">h1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a id="h1_h2" class="idref" href="#h1_h2"><span class="id" title="lemma">h1_h2</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> 2 <a class="idref" href="hydras.Hydra.KP_example.html#h1"><span class="id" title="definition">h1</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#h2"><span class="id" title="definition">h2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a id="h2_h3" class="idref" href="#h2_h3"><span class="id" title="lemma">h2_h3</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> 3 <a class="idref" href="hydras.Hydra.KP_example.html#h2"><span class="id" title="definition">h2</span></a> <a class="idref" href="hydras.Hydra.KP_example.html#h3"><span class="id" title="definition">h3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="battle_example" class="idref" href="#battle_example"><span class="id" title="lemma">battle_example</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a> 1 <a class="idref" href="hydras.Hydra.KP_example.html#h0"><span class="id" title="definition">h0</span></a> 4 <a class="idref" href="hydras.Hydra.KP_example.html#h3"><span class="id" title="definition">h3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+= 28 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.KP_example.html#KP"><span class="id" title="section">KP</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<title>hydras.Hydra.O2H</title>
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+
+<body>
+
+<div id="page">
+
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+
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+
+<h1 class="libtitle">Library hydras.Hydra.O2H</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Injection from ordinals (less than epsilon0) in CNF into hydras.
+
+<div class="paragraph"> </div>
+
+ Pierre Castéran, LaBRI and Univ. Bordeaux 
+
+<div class="paragraph"> </div>
+
+We define a function <span class="inlinecode"><span class="id" title="keyword">iota</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T1</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">Hydra</span></span> such that if <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span>, then there exists a battle from <span class="inlinecode"><span class="id" title="keyword">iota</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span> to <span class="inlinecode"><span class="id" title="keyword">iota</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>.
+
+<div class="paragraph"> </div>
+
+Note that <span class="inlinecode"><span class="id" title="keyword">iota</span></span> is not a bijection, but is sufficient for proving 
+impossibility lemmas or minoration of battle lengths (see Hydra_Theorems).
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="library">Canon</span></a> <a class="idref" href="hydras.Epsilon0.Paths.html#"><span class="id" title="library">Paths</span></a>  .<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#"><span class="id" title="module">Hydra_Definitions</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us transform any ordinal term into an hydra 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="iota" class="idref" href="#iota"><span class="id" title="definition">iota</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.O2H.html#a:1"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">c</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota:3"><span class="id" title="definition">iota</span></a> <span class="id" title="var">c</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) (<a class="idref" href="hydras.Hydra.O2H.html#iotas:4"><span class="id" title="definition">iotas</span></a> <span class="id" title="var">b</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span> <br/>
+<span class="id" title="keyword">with</span> <a id="iotas" class="idref" href="#iotas"><span class="id" title="definition">iotas</span></a> (<a id="a:2" class="idref" href="#a:2"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Hydra.O2H.html#a:2"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">a0</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>  ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota:3"><span class="id" title="definition">iota</span></a> <span class="id" title="var">a0</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) (<a class="idref" href="hydras.Hydra.O2H.html#iotas:4"><span class="id" title="definition">iotas</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ We now prove  a lot of technical lemmas that relate Hydras and 
+ordinals. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_iotas" class="idref" href="#iota_iotas"><span class="id" title="lemma">iota_iotas</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iotas"><span class="id" title="definition">iotas</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:7"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:7"><span class="id" title="variable">alpha</span></a> .<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iotas_succ" class="idref" href="#iotas_succ"><span class="id" title="lemma">iotas_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iotas"><span class="id" title="definition">iotas</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:8"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iotas"><span class="id" title="definition">iotas</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:8"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hy_app_l_nil" class="idref" href="#hy_app_l_nil"><span class="id" title="lemma">hy_app_l_nil</span></a> : <span class="id" title="keyword">∀</span> <a id="l:9" class="idref" href="#l:9"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <a class="idref" href="hydras.Hydra.O2H.html#l:9"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.O2H.html#l:9"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_succ_R1" class="idref" href="#iota_succ_R1"><span class="id" title="lemma">iota_succ_R1</span></a> : <span class="id" title="keyword">∀</span>  <a id="o:10" class="idref" href="#o:10"><span class="id" title="binder">o</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:10"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:10"><span class="id" title="variable">o</span></a>)) (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:10"><span class="id" title="variable">o</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_succ_round_n" class="idref" href="#iota_succ_round_n"><span class="id" title="lemma">iota_succ_round_n</span></a> : <span class="id" title="keyword">∀</span>  <a id="i:11" class="idref" href="#i:11"><span class="id" title="binder">i</span></a> <a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:11"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:12"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:12"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_succ_round" class="idref" href="#iota_succ_round"><span class="id" title="lemma">iota_succ_round</span></a> : <span class="id" title="keyword">∀</span>  <a id="o:13" class="idref" href="#o:13"><span class="id" title="binder">o</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:13"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:13"><span class="id" title="variable">o</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x"><span class="id" title="notation">-1-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#o:13"><span class="id" title="variable">o</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_rw1" class="idref" href="#iota_rw1"><span class="id" title="lemma">iota_rw1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:15"><span class="id" title="variable">alpha</span></a> 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:14"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:15"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:14"><span class="id" title="variable">i</span></a>))).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_rw2" class="idref" href="#iota_rw2"><span class="id" title="lemma">iota_rw2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:16" class="idref" href="#i:16"><span class="id" title="binder">i</span></a> <a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="alpha:18" class="idref" href="#alpha:18"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:18"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:17"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:16"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:18"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:17"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:18"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:16"><span class="id" title="variable">i</span></a>))) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_rw3" class="idref" href="#iota_rw3"><span class="id" title="lemma">iota_rw3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:19" class="idref" href="#i:19"><span class="id" title="binder">i</span></a>  <a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:20"><span class="id" title="variable">alpha</span></a>) 0 <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:19"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:20"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:19"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_rw4" class="idref" href="#iota_rw4"><span class="id" title="lemma">iota_rw4</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:21" class="idref" href="#i:21"><span class="id" title="binder">i</span></a> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>  <a id="alpha:23" class="idref" href="#alpha:23"><span class="id" title="binder">alpha</span></a> ,  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:23"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:23"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:22"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:21"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:23"><span class="id" title="variable">alpha</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:22"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:23"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:21"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_tail" class="idref" href="#iota_tail"><span class="id" title="lemma">iota_tail</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:24" class="idref" href="#i:24"><span class="id" title="binder">i</span></a>   <a id="alpha:25" class="idref" href="#alpha:25"><span class="id" title="binder">alpha</span></a> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> <a id="beta:27" class="idref" href="#beta:27"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:27"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#beta:27"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:27"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:24"><span class="id" title="variable">i</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:25"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:26"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Hydra.O2H.html#iotas"><span class="id" title="definition">iotas</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:27"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:24"><span class="id" title="variable">i</span></a>)))).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_hcons" class="idref" href="#R1_hcons"><span class="id" title="lemma">R1_hcons</span></a> : <span class="id" title="keyword">∀</span> <a id="h:28" class="idref" href="#h:28"><span class="id" title="binder">h</span></a> <a id="s1:29" class="idref" href="#s1:29"><span class="id" title="binder">s1</span></a> <a id="s2:30" class="idref" href="#s2:30"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s1:29"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s2:30"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:28"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s1:29"><span class="id" title="variable">s1</span></a>)) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:28"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s2:30"><span class="id" title="variable">s2</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_hcons_mult" class="idref" href="#R1_hcons_mult"><span class="id" title="lemma">R1_hcons_mult</span></a>  : <span class="id" title="keyword">∀</span> <a id="n:31" class="idref" href="#n:31"><span class="id" title="binder">n</span></a> <a id="h:32" class="idref" href="#h:32"><span class="id" title="binder">h</span></a> <a id="s1:33" class="idref" href="#s1:33"><span class="id" title="binder">s1</span></a> <a id="s2:34" class="idref" href="#s2:34"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s1:33"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s2:34"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#h:32"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s1:33"><span class="id" title="variable">s1</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#h:32"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:31"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s2:34"><span class="id" title="variable">s2</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_R2" class="idref" href="#R1_R2"><span class="id" title="lemma">R1_R2</span></a> : <span class="id" title="keyword">∀</span> <a id="h:35" class="idref" href="#h:35"><span class="id" title="binder">h</span></a> <a id="h':36" class="idref" href="#h':36"><span class="id" title="binder">h'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:35"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h':36"><span class="id" title="variable">h'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R2"><span class="id" title="inductive">R2</span></a> 0 (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:35"><span class="id" title="variable">h</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#h':36"><span class="id" title="variable">h'</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="mem_head" class="idref" href="#mem_head"><span class="id" title="inductive">mem_head</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydrae"><span class="id" title="inductive">Hydrae</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="hh_1" class="idref" href="#hh_1"><span class="id" title="constructor">hh_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:39" class="idref" href="#s:39"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.O2H.html#mem_head:37"><span class="id" title="inductive">mem_head</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:39"><span class="id" title="variable">s</span></a>)<br/>
+| <a id="hh_2" class="idref" href="#hh_2"><span class="id" title="constructor">hh_2</span></a> : <span class="id" title="keyword">∀</span> <a id="h:40" class="idref" href="#h:40"><span class="id" title="binder">h</span></a> <a id="s:41" class="idref" href="#s:41"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.O2H.html#mem_head:37"><span class="id" title="inductive">mem_head</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:41"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#mem_head:37"><span class="id" title="inductive">mem_head</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:40"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:41"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S0_mem_head" class="idref" href="#S0_mem_head"><span class="id" title="lemma">S0_mem_head</span></a> : <span class="id" title="keyword">∀</span> <a id="s:42" class="idref" href="#s:42"><span class="id" title="binder">s</span></a> <a id="s':43" class="idref" href="#s':43"><span class="id" title="binder">s'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#S0"><span class="id" title="inductive">S0</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:42"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s':43"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#mem_head"><span class="id" title="inductive">mem_head</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:42"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mem_head_mult_inv" class="idref" href="#mem_head_mult_inv"><span class="id" title="lemma">mem_head_mult_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="n:44" class="idref" href="#n:44"><span class="id" title="binder">n</span></a> <a id="h:45" class="idref" href="#h:45"><span class="id" title="binder">h</span></a> <a id="s:46" class="idref" href="#s:46"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Hydra.O2H.html#mem_head"><span class="id" title="inductive">mem_head</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:45"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:44"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:46"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#h:45"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Hydra.O2H.html#mem_head"><span class="id" title="inductive">mem_head</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:46"><span class="id" title="variable">s</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="R1_mem_head" class="idref" href="#R1_mem_head"><span class="id" title="lemma">R1_mem_head</span></a> : <span class="id" title="keyword">∀</span> <a id="l:47" class="idref" href="#l:47"><span class="id" title="binder">l</span></a> <a id="l':48" class="idref" href="#l':48"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#l:47"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <a class="idref" href="hydras.Hydra.O2H.html#l':48"><span class="id" title="variable">l'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#mem_head"><span class="id" title="inductive">mem_head</span></a> <a class="idref" href="hydras.Hydra.O2H.html#l:47"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_no_head" class="idref" href="#limit_no_head"><span class="id" title="lemma">limit_no_head</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.O2H.html#mem_head"><span class="id" title="inductive">mem_head</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iotas"><span class="id" title="definition">iotas</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:49"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_no_R1" class="idref" href="#limit_no_R1"><span class="id" title="lemma">limit_no_R1</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:50" class="idref" href="#alpha:50"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:50"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:50"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="h':51" class="idref" href="#h':51"><span class="id" title="binder">h'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#R1"><span class="id" title="inductive">R1</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:50"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Hydra.O2H.html#h':51"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_of_succ" class="idref" href="#iota_of_succ"><span class="id" title="lemma">iota_of_succ</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:52" class="idref" href="#beta:52"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:52"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:52"><span class="id" title="variable">beta</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> <span class="id" title="var">s</span> ⇒ <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hy_app"><span class="id" title="definition">hy_app</span></a> <span class="id" title="var">s</span> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons"><span class="id" title="constructor">hcons</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_iota_i" class="idref" href="#canonS_iota_i"><span class="id" title="lemma">canonS_iota_i</span></a> : <span class="id" title="keyword">∀</span> <a id="i:83" class="idref" href="#i:83"><span class="id" title="binder">i</span></a> <a id="alpha:84" class="idref" href="#alpha:84"><span class="id" title="binder">alpha</span></a> , <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:84"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#alpha:84"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:83"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:84"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:84"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:83"><span class="id" title="variable">i</span></a>))).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="canonS_iota" class="idref" href="#canonS_iota"><span class="id" title="lemma">canonS_iota</span></a> <a id="i:85" class="idref" href="#i:85"><span class="id" title="binder">i</span></a> <a id="a:86" class="idref" href="#a:86"><span class="id" title="binder">a</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x"><span class="id" title="notation">-1-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:86"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:85"><span class="id" title="variable">i</span></a>)). 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="canonS_rel_rounds" class="idref" href="#canonS_rel_rounds"><span class="id" title="lemma">canonS_rel_rounds</span></a> : <span class="id" title="keyword">∀</span> <a id="n:87" class="idref" href="#n:87"><span class="id" title="binder">n</span></a> <a id="alpha:88" class="idref" href="#alpha:88"><span class="id" title="binder">alpha</span></a> <a id="beta:89" class="idref" href="#beta:89"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:88"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:89"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#alpha:88"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#beta:89"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:88"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#n:87"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:88"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:89"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="trace_to_round_plus" class="idref" href="#trace_to_round_plus"><span class="id" title="lemma">trace_to_round_plus</span></a> <a id="h':90" class="idref" href="#h':90"><span class="id" title="binder">h'</span></a> : <span class="id" title="keyword">∀</span> <a id="t:91" class="idref" href="#t:91"><span class="id" title="binder">t</span></a> <a id="h:92" class="idref" href="#h:92"><span class="id" title="binder">h</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h':90"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Hydra.O2H.html#t:91"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:92"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_plus"><span class="id" title="definition">round_plus</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:92"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h':90"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.MoreLists.html#"><span class="id" title="module">MoreLists</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="trace_to_std" class="idref" href="#trace_to_std"><span class="id" title="lemma">trace_to_std</span></a> <a id="i:97" class="idref" href="#i:97"><span class="id" title="binder">i</span></a> <a id="h:98" class="idref" href="#h:98"><span class="id" title="binder">h</span></a> <a id="j:99" class="idref" href="#j:99"><span class="id" title="binder">j</span></a> <a id="h':100" class="idref" href="#h':100"><span class="id" title="binder">h'</span></a>: (<a class="idref" href="hydras.Hydra.O2H.html#i:97"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.O2H.html#j:99"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h':100"><span class="id" title="variable">h'</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Hydra.O2H.html#i:97"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.O2H.html#j:99"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Hydra.O2H.html#h:98"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#rounds"><span class="id" title="inductive">rounds</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#standard"><span class="id" title="instance">standard</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#i:97"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.O2H.html#h:98"><span class="id" title="variable">h</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.O2H.html#j:99"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Hydra.O2H.html#h':100"><span class="id" title="variable">h'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="path_toS_trace" class="idref" href="#path_toS_trace"><span class="id" title="lemma">path_toS_trace</span></a> <a id="alpha:101" class="idref" href="#alpha:101"><span class="id" title="binder">alpha</span></a> <a id="s:102" class="idref" href="#s:102"><span class="id" title="binder">s</span></a> <a id="beta:103" class="idref" href="#beta:103"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:103"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:102"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:101"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:101"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#trace_to"><span class="id" title="inductive">trace_to</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:103"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Hydra.O2H.html#s:102"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:101"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="path_toS_round_plus" class="idref" href="#path_toS_round_plus"><span class="id" title="lemma">path_toS_round_plus</span></a> <a id="alpha:104" class="idref" href="#alpha:104"><span class="id" title="binder">alpha</span></a> <a id="s:105" class="idref" href="#s:105"><span class="id" title="binder">s</span></a> <a id="beta:106" class="idref" href="#beta:106"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_toS"><span class="id" title="inductive">path_toS</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:106"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:105"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:106"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="path_to_round_plus" class="idref" href="#path_to_round_plus"><span class="id" title="lemma">path_to_round_plus</span></a> <a id="a:107" class="idref" href="#a:107"><span class="id" title="binder">a</span></a> <a id="s:108" class="idref" href="#s:108"><span class="id" title="binder">s</span></a> <a id="b:109" class="idref" href="#b:109"><span class="id" title="binder">b</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#path_to"><span class="id" title="inductive">path_to</span></a> <a class="idref" href="hydras.Hydra.O2H.html#b:109"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Hydra.O2H.html#s:108"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:107"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:107"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:107"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#b:109"><span class="id" title="variable">b</span></a>. 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="acc_from_to_round_plus" class="idref" href="#acc_from_to_round_plus"><span class="id" title="lemma">acc_from_to_round_plus</span></a> <a id="alpha:110" class="idref" href="#alpha:110"><span class="id" title="binder">alpha</span></a> <a id="beta:111" class="idref" href="#beta:111"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.Paths.html#acc_from"><span class="id" title="definition">acc_from</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:111"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#alpha:110"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#beta:111"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Any strict inequality on <span class="inlinecode"><span class="id" title="var">T1</span></span> can be converted into a (free) battle 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LT_to_round_plus" class="idref" href="#LT_to_round_plus"><span class="id" title="lemma">LT_to_round_plus</span></a> <a id="a:112" class="idref" href="#a:112"><span class="id" title="binder">a</span></a> <a id="b:113" class="idref" href="#b:113"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Hydra.O2H.html#b:113"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:112"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#a:112"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.O2H.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Hydra.O2H.html#b:113"><span class="id" title="variable">b</span></a>. </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Omega2_Small</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Omega2_Small</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Castéran, University of Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#"><span class="id" title="library">ON_Omega2</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="module">ON_Generic</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+There is no measure into omega^2  for proving termination
+of all hydra battles 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility_Proof" class="idref" href="#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us assume there is a variant from <span class="inlinecode"><span class="id" title="var">Hydra</span></span> into <span class="inlinecode"><span class="id" title="var">omega</span>^2</span> 
+  for proving the termination of all hydra battles 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Impossibility_Proof.m" class="idref" href="#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">rep</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#Omega2"><span class="id" title="instance">Omega2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hvar:2" class="idref" href="#Hvar:2"><span class="id" title="binder">Hvar</span></a>: <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#Omega2"><span class="id" title="instance">Omega2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.big_h" class="idref" href="#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd2"><span class="id" title="abbreviation">hyd2</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+To every pair  of natural numbers we associate an hydra 
+        with  branches of length 2 and  branches of length 1 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.iota" class="idref" href="#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> (<a id="p:3" class="idref" href="#p:3"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">ON_Omega2.t</span></a>) := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#node"><span class="id" title="constructor">node</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#p:3"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hcons_mult"><span class="id" title="definition">hcons_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#p:3"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hnil"><span class="id" title="constructor">hnil</span></a>)).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.small_h" class="idref" href="#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> := <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab225"></a><h3 class="section">Proof of the inequality <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">small_h</span></span> <span class="inlinecode"><span class="id" title="var">o</span>&lt;</span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">big_h</span></span></h3>
+
+   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Hydra.Omega2_Small.html#R1"><span class="id" title="inductive">R1</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#S1"><span class="id" title="inductive">S1</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#S2"><span class="id" title="inductive">S2</span></a> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_big_h_not_null" class="idref" href="#m_big_h_not_null"><span class="id" title="lemma">m_big_h_not_null</span></a> : <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="big_to_small" class="idref" href="#big_to_small"><span class="id" title="lemma">big_to_small</span></a> : <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Corollary</span> <a id="m_lt" class="idref" href="#m_lt"><span class="id" title="lemma">m_lt</span></a> : <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab226"></a><h3 class="section">Proof of the inequality <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">big_h</span></span> <span class="inlinecode"><span class="id" title="var">o</span>≤</span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">small_h</span></span></h3>
+
+<div class="paragraph"> </div>
+
+<a id="lab227"></a><h3 class="section">Let us decompose any inequality p o&lt; q into elementary steps</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="step" class="idref" href="#step"><span class="id" title="inductive">step</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="succ_step" class="idref" href="#succ_step"><span class="id" title="constructor">succ_step</span></a> : <span class="id" title="keyword">∀</span> <a id="i:6" class="idref" href="#i:6"><span class="id" title="binder">i</span></a> <a id="j:7" class="idref" href="#j:7"><span class="id" title="binder">j</span></a>,  <a class="idref" href="hydras.Hydra.Omega2_Small.html#step:4"><span class="id" title="inductive">step</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:6"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:7"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:6"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:7"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a id="limit_step" class="idref" href="#limit_step"><span class="id" title="constructor">limit_step</span></a> : <span class="id" title="keyword">∀</span> <a id="i:8" class="idref" href="#i:8"><span class="id" title="binder">i</span></a> <a id="j:9" class="idref" href="#j:9"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Hydra.Omega2_Small.html#step:4"><span class="id" title="inductive">step</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#i:8"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:8"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:9"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="succ_rounds" class="idref" href="#succ_rounds"><span class="id" title="lemma">succ_rounds</span></a> : <span class="id" title="keyword">∀</span> <a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a> <a id="j:11" class="idref" href="#j:11"><span class="id" title="binder">j</span></a>,  <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:10"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:11"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:10"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:11"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="limit_rounds_0" class="idref" href="#limit_rounds_0"><span class="id" title="lemma">limit_rounds_0</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a> <a id="j:13" class="idref" href="#j:13"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#round_n"><span class="id" title="definition">round_n</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:13"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#i:12"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) (<a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:12"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:13"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="limit_rounds" class="idref" href="#limit_rounds"><span class="id" title="lemma">limit_rounds</span></a> : <span class="id" title="keyword">∀</span> <a id="i:14" class="idref" href="#i:14"><span class="id" title="binder">i</span></a> <a id="j:15" class="idref" href="#j:15"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#i:14"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Hydra.Omega2_Small.html#i:14"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#j:15"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Hydra.Omega2_Small.html#step"><span class="id" title="inductive">step</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#clos_trans_1n"><span class="id" title="inductive">clos_trans_1n</span></a> : <span class="id" title="var">hydra</span>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">lex_1</span> <span class="id" title="var">lex_2</span>: <span class="id" title="var">hydra</span>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="step_to_battle" class="idref" href="#step_to_battle"><span class="id" title="lemma">step_to_battle</span></a> : <span class="id" title="keyword">∀</span> <a id="p:16" class="idref" href="#p:16"><span class="id" title="binder">p</span></a> <a id="q:17" class="idref" href="#q:17"><span class="id" title="binder">q</span></a>, <a class="idref" href="hydras.Hydra.Omega2_Small.html#step"><span class="id" title="inductive">step</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#p:16"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#q:17"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#p:16"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498"><span class="id" title="notation">-+-&gt;</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#q:17"><span class="id" title="variable">q</span></a>. 
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">step_to_battle</span> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_ge" class="idref" href="#m_ge"><span class="id" title="lemma">m_ge</span></a> : <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>.  
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Impossible" class="idref" href="#Impossible"><span class="id" title="lemma">Impossible</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,147 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Hydra.Omega_Small</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Hydra.Omega_Small</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Castéran, University of Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Hydra.Hydra_Lemmas.html#"><span class="id" title="library">Hydra_Lemmas</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#"><span class="id" title="library">ON_Omega</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="height_var" class="idref" href="#height_var"><span class="id" title="instance">height_var</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a>.<br/>
+<span class="id" title="keyword">Abort</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="height_bad" class="idref" href="#height_bad"><span class="id" title="lemma">height_bad</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#height"><span class="id" title="definition">height</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+There is no measure into omega for proving termination
+of all hydra battles 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility_Proof" class="idref" href="#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us assume there exists a variant from Hydra  into nat 
+      for proving the termination of all hydra battles 
+      <span class="inlinecode"><span class="id" title="var">Omega</span></span> is an ordinal notation for the least infinite ordinal
+      <span class="inlinecode"><span class="id" title="var">omega</span></span>, whose members are the natural numbers.
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Impossibility_Proof.m" class="idref" href="#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hydra"><span class="id" title="inductive">Hydra</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="Hvar:2" class="idref" href="#Hvar:2"><span class="id" title="binder">Hvar</span></a> : <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#Hvariant"><span class="id" title="class">Hvariant</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.iota" class="idref" href="#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> (<a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd_mult"><span class="id" title="definition">hyd_mult</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#i:3"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.big_h" class="idref" href="#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> := <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> (<a class="idref" href="hydras.Hydra.Hydra_Definitions.html#hyd1"><span class="id" title="abbreviation">hyd1</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#head"><span class="id" title="abbreviation">head</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility_Proof.small_h" class="idref" href="#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> := <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="big_to_small" class="idref" href="#big_to_small"><span class="id" title="lemma">big_to_small</span></a>: <span class="id" title="keyword">∀</span> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#i:4"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>. 
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">big_to_small</span> : <span class="id" title="var">hydra</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_lt" class="idref" href="#m_lt"><span class="id" title="lemma">m_lt</span></a> : <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ In order to find a contradiction, we  prove the inequality
+       m big_h &lt;= m small_h, i.e.  m big_h &lt;= m (iota (m h)) 
+
+<div class="paragraph"> </div>
+
+       For that purpose, we prove the inequality i &lt;= m (iota i) for any i 
+   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="round_S" class="idref" href="#round_S"><span class="id" title="lemma">round_S</span></a>: <span class="id" title="keyword">∀</span> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#battle_rel"><span class="id" title="method">battle_rel</span></a> <a class="idref" href="hydras.Hydra.Hydra_Definitions.html#free"><span class="id" title="instance">free</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#n:6"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#i:5"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.iota"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#i:5"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="m_ge" class="idref" href="#m_ge"><span class="id" title="lemma">m_ge</span></a> : <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h"><span class="id" title="variable">big_h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.small_h"><span class="id" title="variable">small_h</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Contradiction" class="idref" href="#Contradiction"><span class="id" title="lemma">Contradiction</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Hydra.Omega_Small.html#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,420 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.Ack</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.Ack</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Wf.html#"><span class="id" title="library">Program.Wf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.ssrnat_extracts.html#"><span class="id" title="library">ssrnat_extracts</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The famous Ackermann function 
+<div class="paragraph"> </div>
+
+ The following definition fails, because Coq cannot guess a 
+  decreasing argument.
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="var">Fail</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ack" class="idref" href="#Ack"><span class="id" title="definition">Ack</span></a> (<a id="m:1" class="idref" href="#m:1"><span class="id" title="binder">m</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.Ack.html#m:1"><span class="id" title="variable">m</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0, <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:2"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">m</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>, 0 ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:3"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:1"><span class="id" title="variable">m</span></a> 1<br/>
+&nbsp;&nbsp;| <span class="id" title="var">m0</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>, <span class="id" title="var">p</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:3"><span class="id" title="definition">Ack</span></a> <span class="id" title="var">m0</span> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack:3"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:1"><span class="id" title="variable">m</span></a> <span class="id" title="var">p</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition (with inner fixpoint) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt" class="idref" href="#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Alt.Ack" class="idref" href="#Alt.Ack"><span class="id" title="definition">Ack</span></a> (<a id="m:6" class="idref" href="#m:6"><span class="id" title="binder">m</span></a> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.Ack.html#m:6"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#n:7"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">p</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="Ackm:13" class="idref" href="#Ackm:13"><span class="id" title="binder">Ackm</span></a> (<a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.Ack.html#n:10"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:8"><span class="id" title="definition">Ack</span></a> <span class="id" title="var">p</span> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">q</span> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:8"><span class="id" title="definition">Ack</span></a> <span class="id" title="var">p</span> (<a class="idref" href="hydras.MoreAck.Ack.html#Ackm:11"><span class="id" title="variable">Ackm</span></a> <span class="id" title="var">q</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.MoreAck.Ack.html#Ackm:13"><span class="id" title="variable">Ackm</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:7"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition using the <span class="inlinecode"><span class="id" title="var">iterate</span></span> functional:
+<pre>
+   iterate : forall {A : Type}, (A -&gt; A) -&gt; nat -&gt; A -&gt; A
+</pre>
+
+<div class="paragraph"> </div>
+
+Allows to infer monotony properties of <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">m</span>)</span> from <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Ack" class="idref" href="#Ack"><span class="id" title="definition">Ack</span></a> (<a id="m:14" class="idref" href="#m:14"><span class="id" title="binder">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.Ack.html#m:14"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <span class="id" title="keyword">fun</span> <a id="k:17" class="idref" href="#k:17"><span class="id" title="binder">k</span></a> ⇒  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack:15"><span class="id" title="definition">Ack</span></a> <span class="id" title="var">n</span>) <a class="idref" href="hydras.MoreAck.Ack.html#k:17"><span class="id" title="variable">k</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> 1<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Using the lexicographic ordering 
+   (post by Anton Trunov in stackoverflow (May 2018)) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="lex_nat" class="idref" href="#lex_nat"><span class="id" title="definition">lex_nat</span></a> (<a id="ab1:18" class="idref" href="#ab1:18"><span class="id" title="binder">ab1</span></a> <a id="ab2:19" class="idref" href="#ab2:19"><span class="id" title="binder">ab2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.Ack.html#ab1:18"><span class="id" title="variable">ab1</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#ab2:19"><span class="id" title="variable">ab2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="var">b1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">b2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+this is defined in stdlib, but unfortunately it is opaque 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_wf_ind" class="idref" href="#lt_wf_ind"><span class="id" title="lemma">lt_wf_ind</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> (<a id="P:23" class="idref" href="#P:23"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:25" class="idref" href="#m:25"><span class="id" title="binder">m</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#m:25"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:23"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:25"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:23"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:24"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:23"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:22"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+this is defined in stdlib, but unfortunately it is opaque too 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_wf_double_ind" class="idref" href="#lt_wf_double_ind"><span class="id" title="lemma">lt_wf_double_ind</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:26" class="idref" href="#P:26"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> <a id="m:28" class="idref" href="#m:28"><span class="id" title="binder">m</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="p:29" class="idref" href="#p:29"><span class="id" title="binder">p</span></a> (<a id="q:30" class="idref" href="#q:30"><span class="id" title="binder">q</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.MoreAck.Ack.html#p:29"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:29"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#q:30"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="p:31" class="idref" href="#p:31"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#p:31"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:28"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:31"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:28"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> <a id="m:33" class="idref" href="#m:33"><span class="id" title="binder">m</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:32"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:33"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lex_nat_wf" class="idref" href="#lex_nat_wf"><span class="id" title="lemma">lex_nat_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#lex_nat"><span class="id" title="definition">lex_nat</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt2" class="idref" href="#Alt2"><span class="id" title="module">Alt2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Program Fixpoint</span> <a id="Alt2.Ack" class="idref" href="#Alt2.Ack"><span class="id" title="definition">Ack</span></a> (<a id="ab:34" class="idref" href="#ab:34"><span class="id" title="binder">ab</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">wf</span> <a class="idref" href="hydras.MoreAck.Ack.html#lex_nat"><span class="id" title="definition">lex_nat</span></a> <span class="id" title="var">ab</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">ab</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">b</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:35"><span class="id" title="definition">Ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack:35"><span class="id" title="definition">Ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack:35"><span class="id" title="definition">Ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Alt2.test1" class="idref" href="#Alt2.test1"><span class="id" title="definition">test1</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Alt2.Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 4 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Example</span> <a id="Alt2.test2" class="idref" href="#Alt2.test2"><span class="id" title="definition">test2</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Alt2.Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 4<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 125 := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Alt2"><span class="id" title="module">Alt2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+With the Equations plug-in 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Equations</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Lex_nat_wf" class="idref" href="#Lex_nat_wf"><span class="id" title="instance">Lex_nat_wf</span></a> : <span class="id" title="class">WellFounded</span> <a class="idref" href="hydras.MoreAck.Ack.html#lex_nat"><span class="id" title="definition">lex_nat</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt3" class="idref" href="#Alt3"><span class="id" title="module">Alt3</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Equations</span> <a id="Alt3.ack" class="idref" href="#Alt3.ack"><span class="id" title="definition">ack</span></a> (<a id="p:37" class="idref" href="#p:37"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span> <span class="id" title="var">p</span> <a class="idref" href="hydras.MoreAck.Ack.html#lex_nat"><span class="id" title="definition">lex_nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#ack:40"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:39"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#ack:42"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.Ack.html#m:41"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="hydras.MoreAck.Ack.html#ack:38"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#ack:45"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.Ack.html#m:43"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:44"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="hydras.MoreAck.Ack.html#ack:38"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#ack:38"><span class="id" title="definition">ack</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">m</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Alt3"><span class="id" title="module">Alt3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab314"></a><h3 class="section">Exercise</h3>
+
+
+<div class="paragraph"> </div>
+
+   Prove that the four definitions of the Ackermann function 
+   <span class="inlinecode"><span class="id" title="var">Ack</span></span> , <span class="inlinecode"><span class="id" title="var">Alt.Ack</span></span>, <span class="inlinecode"><span class="id" title="var">Alt2.Ack</span></span>,  and <span class="inlinecode"><span class="id" title="var">Alt3.ack</span></span> are extensionnally equal 
+ 
+<div class="paragraph"> </div>
+
+<a id="lab315"></a><h3 class="section">Usual equations</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_0" class="idref" href="#Ack_0"><span class="id" title="lemma">Ack_0</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_S_0" class="idref" href="#Ack_S_0"><span class="id" title="lemma">Ack_S_0</span></a> <a id="m:46" class="idref" href="#m:46"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:46"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:46"><span class="id" title="variable">m</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_S_S" class="idref" href="#Ack_S_S"><span class="id" title="lemma">Ack_S_S</span></a> : <span class="id" title="keyword">∀</span> <a id="m:47" class="idref" href="#m:47"><span class="id" title="binder">m</span></a> <a id="p:48" class="idref" href="#p:48"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:47"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:48"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:47"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:47"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:48"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab316"></a><h3 class="section">First values</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_1_n" class="idref" href="#Ack_1_n"><span class="id" title="lemma">Ack_1_n</span></a> <a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> 1 <a class="idref" href="hydras.MoreAck.Ack.html#n:49"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:49"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_2_n" class="idref" href="#Ack_2_n"><span class="id" title="lemma">Ack_2_n</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> 2  <a class="idref" href="hydras.MoreAck.Ack.html#n:50"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:50"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 3. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_3_n" class="idref" href="#Ack_3_n"><span class="id" title="lemma">Ack_3_n</span></a> <a id="n:51" class="idref" href="#n:51"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> 3 <a class="idref" href="hydras.MoreAck.Ack.html#n:51"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:51"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#331a87f8f0f3c9336601c27f1e458dc3"><span class="id" title="notation">.+3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 3. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_4_n" class="idref" href="#Ack_4_n"><span class="id" title="lemma">Ack_4_n</span></a> <a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> 4 <a class="idref" href="hydras.MoreAck.Ack.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:52"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#331a87f8f0f3c9336601c27f1e458dc3"><span class="id" title="notation">.+3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 3. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab317"></a><h3 class="section">monotony properties</h3>
+
+
+<div class="paragraph"> </div>
+
+  We prove simultaneously 3 properties of <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> by induction on <span class="inlinecode"><span class="id" title="var">m</span></span>:
+<ul class="doclist">
+<li> <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> is strictly monotonous,
+
+</li>
+<li> <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">&gt;</span> <span class="inlinecode"><span class="id" title="var">n</span></span>
+
+</li>
+<li> <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">m</span>)</span> <span class="inlinecode"><span class="id" title="var">n</span></span>
+
+</li>
+</ul>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Ack_Properties" class="idref" href="#Ack_Properties"><span class="id" title="section">Ack_Properties</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Ack_Properties.P" class="idref" href="#Ack_Properties.P"><span class="id" title="variable">P</span></a> (<a id="m:53" class="idref" href="#m:53"><span class="id" title="binder">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:53"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:53"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:53"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:54"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:53"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:54"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="P0" class="idref" href="#P0"><span class="id" title="lemma">P0</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.P"><span class="id" title="variable">P</span></a> 0.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Ack_Properties.Induc_step" class="idref" href="#Ack_Properties.Induc_step"><span class="id" title="section">Induc_step</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Ack_Properties.Induc_step.m" class="idref" href="#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Ack_Properties.Induc_step.Hm" class="idref" href="#Ack_Properties.Induc_step.Hm"><span class="id" title="variable">Hm</span></a> : <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span>  <a id="Rem1" class="idref" href="#Rem1"><span class="id" title="lemma">Rem1</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem2" class="idref" href="#Rem2"><span class="id" title="lemma">Rem2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Ack_m_mono_weak" class="idref" href="#Ack_m_mono_weak"><span class="id" title="lemma">Ack_m_mono_weak</span></a> : <span class="id" title="keyword">∀</span> <a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> <a id="p:58" class="idref" href="#p:58"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:58"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:58"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Rem3" class="idref" href="#Rem3"><span class="id" title="lemma">Rem3</span></a> : <span class="id" title="keyword">∀</span> <a id="n:59" class="idref" href="#n:59"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:59"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:59"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L5" class="idref" href="#L5"><span class="id" title="lemma">L5</span></a>: <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step"><span class="id" title="section">Induc_step</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Ack_properties" class="idref" href="#Ack_properties"><span class="id" title="lemma">Ack_properties</span></a> : <span class="id" title="keyword">∀</span> <a id="m:60" class="idref" href="#m:60"><span class="id" title="binder">m</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:60"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Ack_Properties"><span class="id" title="section">Ack_Properties</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_S_Ack" class="idref" href="#le_S_Ack"><span class="id" title="lemma">le_S_Ack</span></a> <a id="m:61" class="idref" href="#m:61"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#fun_le"><span class="id" title="definition">fun_le</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:61"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_strict_mono" class="idref" href="#Ack_strict_mono"><span class="id" title="lemma">Ack_strict_mono</span></a> <a id="m:62" class="idref" href="#m:62"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:62"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_mono_l" class="idref" href="#Ack_mono_l"><span class="id" title="lemma">Ack_mono_l</span></a> <a id="m:63" class="idref" href="#m:63"><span class="id" title="binder">m</span></a> <a id="n:64" class="idref" href="#n:64"><span class="id" title="binder">n</span></a> :   <a class="idref" href="hydras.MoreAck.Ack.html#m:63"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:64"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="p:65" class="idref" href="#p:65"><span class="id" title="binder">p</span></a>,  <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:63"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:65"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:64"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:65"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_mono_r" class="idref" href="#Ack_mono_r"><span class="id" title="lemma">Ack_mono_r</span></a> <a id="m:66" class="idref" href="#m:66"><span class="id" title="binder">m</span></a>: <span class="id" title="keyword">∀</span> <a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a> <a id="p:68" class="idref" href="#p:68"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:68"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:66"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:66"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:68"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab318"></a><h2 class="section">Bounding nested calls of <span class="inlinecode"><span class="id" title="var">Ack</span></span></h2>
+
+
+<div class="paragraph"> </div>
+
+   The following inequality is applied in the proof that <span class="inlinecode"><span class="id" title="var">Ack</span></span> is not primitive 
+   recursive, allowing to eliminate patterns of the form <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode">(<span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">_</span></span> <span class="inlinecode"><span class="id" title="var">_</span></span> <span class="inlinecode"><span class="id" title="var">_</span>)</span>.
+
+<div class="paragraph"> </div>
+
+<pre>
+ Lemma nested_Ack_bound : 
+    forall k m n, Ack k (Ack m n) &lt;= Ack (2 + max k m) n.
+</pre>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_nested_Ack_bound" class="idref" href="#Proof_of_nested_Ack_bound"><span class="id" title="section">Proof_of_nested_Ack_bound</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R0" class="idref" href="#R0"><span class="id" title="lemma">R0</span></a> (<a id="m:69" class="idref" href="#m:69"><span class="id" title="binder">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <span class="id" title="keyword">∀</span> <a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:69"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.MoreAck.Ack.html#n:70"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <span class="id" title="keyword">∀</span> <a id="m:71" class="idref" href="#m:71"><span class="id" title="binder">m</span></a> <a id="n:72" class="idref" href="#n:72"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:71"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:72"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:71"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.MoreAck.Ack.html#n:72"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> (<a id="m:73" class="idref" href="#m:73"><span class="id" title="binder">m</span></a> <a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:73"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:73"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:74"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:73"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:74"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> (<a id="m:75" class="idref" href="#m:75"><span class="id" title="binder">m</span></a> <a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:75"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:75"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:76"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:75"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:76"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="nested_Ack_bound" class="idref" href="#nested_Ack_bound"><span class="id" title="lemma">nested_Ack_bound</span></a> <a id="k:77" class="idref" href="#k:77"><span class="id" title="binder">k</span></a> <a id="m:78" class="idref" href="#m:78"><span class="id" title="binder">m</span></a> <a id="n:79" class="idref" href="#n:79"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#k:77"><span class="id" title="variable">k</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:78"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:79"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#k:77"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:78"><span class="id" title="variable">m</span></a>) <a class="idref" href="hydras.MoreAck.Ack.html#n:79"><span class="id" title="variable">n</span></a>. 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.Ack.html#Proof_of_nested_Ack_bound"><span class="id" title="section">Proof_of_nested_Ack_bound</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_Sn_plus" class="idref" href="#Ack_Sn_plus"><span class="id" title="lemma">Ack_Sn_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="n:80" class="idref" href="#n:80"><span class="id" title="binder">n</span></a> <a id="p:81" class="idref" href="#p:81"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#n:80"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:81"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:80"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:81"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab319"></a><h2 class="section"><span class="inlinecode"><span class="id" title="var">Ack</span></span> is (partially) strictly monotonous in its first argument</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a> <a id="m:82" class="idref" href="#m:82"><span class="id" title="binder">m</span></a> <a id="n:83" class="idref" href="#n:83"><span class="id" title="binder">n</span></a> :  <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:82"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:83"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:82"><span class="id" title="variable">m</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:83"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ack_strict_mono_l" class="idref" href="#Ack_strict_mono_l"><span class="id" title="lemma">Ack_strict_mono_l</span></a> : <span class="id" title="keyword">∀</span> <a id="n:84" class="idref" href="#n:84"><span class="id" title="binder">n</span></a> <a id="m:85" class="idref" href="#m:85"><span class="id" title="binder">m</span></a> <a id="p:86" class="idref" href="#p:86"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.MoreAck.Ack.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:85"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:86"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#m:85"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#p:86"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>. 
+<br/>
+</div>
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+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.AckNotPR</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab320"></a><h1 class="section">Proof that Ack is not primitive recursive</h1>
+
+
+<div class="paragraph"> </div>
+
+  After <a href="https://planetmath.org/ackermannfunctionisnotprimitiverecursive">
+planetmath page </a>
+and
+<a href="http://www.enseignement.polytechnique.fr/informatique/INF412/uploads/Main/pc-primrec-sujet2014.pdf"> Bruno Salvy's lecture</a>.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#"><span class="id" title="library">Ack</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#"><span class="id" title="library">MoreVectors</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="module">extEqualNat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#VectorNotations"><span class="id" title="module">VectorNotations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="module">Vector</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Uncurried apply :
+ <span class="inlinecode"><span class="id" title="var">v_apply</span></span> <span class="inlinecode"><span class="id" title="var">f</span></span> <span class="inlinecode">(<span class="id" title="var">x1</span>::<span class="id" title="var">x2</span>::</span> <span class="inlinecode">...</span> <span class="inlinecode">::<span class="id" title="var">xn</span>::<span class="id" title="var">nil</span>)</span>  is <span class="inlinecode"><span class="id" title="var">f</span></span> <span class="inlinecode"><span class="id" title="var">x1</span></span> <span class="inlinecode"><span class="id" title="var">x2</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" title="var">xn</span></span> 
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id=":::'v_apply'_x_x" class="idref" href="#:::'v_apply'_x_x"><span class="id" title="notation">&quot;</span></a>'v_apply' f v" := (<a class="idref" href="hydras.Ackermann.primRec.html#evalList"><span class="id" title="definition">evalList</span></a> <span class="id" title="var">_</span> <span class="id" title="var">v</span> <span class="id" title="var">f</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#06a81d089cb210523d05cd9d098003fa"><span class="id" title="notation">[</span></a>4<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#06a81d089cb210523d05cd9d098003fa"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> : <span class="id" title="keyword">∀</span> (<a id="f:1" class="idref" href="#f:1"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) <a id="x:2" class="idref" href="#x:2"><span class="id" title="binder">x</span></a> <a id="y:3" class="idref" href="#y:3"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#x:2"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#y:3"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:2"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:3"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex4" class="idref" href="#Ex4"><span class="id" title="definition">Ex4</span></a> : <span class="id" title="keyword">∀</span> (<a id="f:4" class="idref" href="#f:4"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 4) <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> <a id="z:7" class="idref" href="#z:7"><span class="id" title="binder">z</span></a> <a id="t:8" class="idref" href="#t:8"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:4"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#x:5"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#y:6"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#z:7"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#t:8"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:4"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#z:7"><span class="id" title="variable">z</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#t:8"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab321"></a><h2 class="section">Comparing an n-ary and a binary functions</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="majorized" class="idref" href="#majorized"><span class="id" title="definition">majorized</span></a> {<a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a>} (<a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:9"><span class="id" title="variable">n</span></a>) (<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="q:12" class="idref" href="#q:12"><span class="id" title="binder">q</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="v:13" class="idref" href="#v:13"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:9"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:13"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#q:12"><span class="id" title="variable">q</span></a> (<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:13"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="majorizedPR" class="idref" href="#majorizedPR"><span class="id" title="definition">majorizedPR</span></a> {<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a>} (<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:14"><span class="id" title="variable">n</span></a>) <a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#majorized"><span class="id" title="definition">majorized</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRec"><span class="id" title="definition">evalPrimRec</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:15"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#A:16"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For vectors of functions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="majorizedS" class="idref" href="#majorizedS"><span class="id" title="definition">majorizedS</span></a> {<a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="m:18" class="idref" href="#m:18"><span class="id" title="binder">m</span></a>} (<a id="fs:19" class="idref" href="#fs:19"><span class="id" title="binder">fs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:17"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:18"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2):=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="N:21" class="idref" href="#N:21"><span class="id" title="binder">N</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> (<a id="v:22" class="idref" href="#v:22"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:17"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#map"><span class="id" title="definition">Vector.map</span></a> (<span class="id" title="keyword">fun</span> <a id="f:23" class="idref" href="#f:23"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:23"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:22"><span class="id" title="variable">v</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#fs:19"><span class="id" title="variable">fs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#N:21"><span class="id" title="variable">N</span></a> (<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:22"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="majorizedSPR" class="idref" href="#majorizedSPR"><span class="id" title="definition">majorizedSPR</span></a> {<a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> <a id="m:25" class="idref" href="#m:25"><span class="id" title="binder">m</span></a>} (<a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRecs"><span class="id" title="inductive">PrimRecs</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:25"><span class="id" title="variable">m</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#majorizedS"><span class="id" title="definition">majorizedS</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecs"><span class="id" title="definition">evalPrimRecs</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:26"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="evalList" class="idref" href="#evalList"><span class="id" title="section">evalList</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="evalList_Const" class="idref" href="#evalList_Const"><span class="id" title="lemma">evalList_Const</span></a> : <span class="id" title="keyword">∀</span> <a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> (<a id="v:28" class="idref" href="#v:28"><span class="id" title="binder">v</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:27"><span class="id" title="variable">n</span></a>) <a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalConstFunc"><span class="id" title="definition">evalConstFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:27"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:29"><span class="id" title="variable">x</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:28"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:29"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="proj_le_max" class="idref" href="#proj_le_max"><span class="id" title="lemma">proj_le_max</span></a> : <span class="id" title="keyword">∀</span> <a id="n:30" class="idref" href="#n:30"><span class="id" title="binder">n</span></a>, <span class="id" title="keyword">∀</span> <a id="v:31" class="idref" href="#v:31"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:30"><span class="id" title="variable">n</span></a>, <span class="id" title="keyword">∀</span> <a id="k:32" class="idref" href="#k:32"><span class="id" title="binder">k</span></a> (<a id="H:33" class="idref" href="#H:33"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.MoreAck.AckNotPR.html#k:32"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:30"><span class="id" title="variable">n</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalProjFunc"><span class="id" title="definition">evalProjFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#k:32"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#H:33"><span class="id" title="variable">H</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:31"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:31"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="evalListComp" class="idref" href="#evalListComp"><span class="id" title="lemma">evalListComp</span></a> : <span class="id" title="keyword">∀</span> <a id="n:34" class="idref" href="#n:34"><span class="id" title="binder">n</span></a>  (<a id="v:35" class="idref" href="#v:35"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:34"><span class="id" title="variable">n</span></a>) <a id="m:36" class="idref" href="#m:36"><span class="id" title="binder">m</span></a> (<a id="gs:37" class="idref" href="#gs:37"><span class="id" title="binder">gs</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:34"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:36"><span class="id" title="variable">m</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="h:38" class="idref" href="#h:38"><span class="id" title="binder">h</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:36"><span class="id" title="variable">m</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalComposeFunc"><span class="id" title="definition">evalComposeFunc</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#gs:37"><span class="id" title="variable">gs</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#h:38"><span class="id" title="variable">h</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:35"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#h:38"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <a id="g:39" class="idref" href="#g:39"><span class="id" title="binder">g</span></a> ⇒  <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:39"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:35"><span class="id" title="variable">v</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#gs:37"><span class="id" title="variable">gs</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="evalListCompose2" class="idref" href="#evalListCompose2"><span class="id" title="lemma">evalListCompose2</span></a> : <span class="id" title="keyword">∀</span> <a id="n:40" class="idref" href="#n:40"><span class="id" title="binder">n</span></a>  (<a id="v:41" class="idref" href="#v:41"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:40"><span class="id" title="variable">n</span></a>)  (<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:40"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:43" class="idref" href="#g:43"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:40"><span class="id" title="variable">n</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:43"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:41"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:43"><span class="id" title="variable">g</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalList"><span class="id" title="definition">evalList</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:41"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:42"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:41"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="evalListPrimrec_0" class="idref" href="#evalListPrimrec_0"><span class="id" title="lemma">evalListPrimrec_0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:44" class="idref" href="#n:44"><span class="id" title="binder">n</span></a>  (<a id="v:45" class="idref" href="#v:45"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:44"><span class="id" title="variable">n</span></a>) (<a id="f:46" class="idref" href="#f:46"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:44"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:47" class="idref" href="#g:47"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:44"><span class="id" title="variable">n</span></a>))),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a>  <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:47"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> 0 <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:45"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:45"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="evalListPrimrec_S" class="idref" href="#evalListPrimrec_S"><span class="id" title="lemma">evalListPrimrec_S</span></a> : <span class="id" title="keyword">∀</span> <a id="n:48" class="idref" href="#n:48"><span class="id" title="binder">n</span></a>  (<a id="v:49" class="idref" href="#v:49"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:48"><span class="id" title="variable">n</span></a>) (<a id="f:50" class="idref" href="#f:50"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:48"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:51" class="idref" href="#g:51"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:48"><span class="id" title="variable">n</span></a>))) <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a>  <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:51"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#a:52"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:49"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:51"><span class="id" title="variable">g</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">v_apply</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:48"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:51"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:49"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#v:49"><span class="id" title="variable">v</span></a><a class="idref" href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#evalList"><span class="id" title="section">evalList</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab322"></a><h2 class="section">Every primitive recursive function is majorized by <span class="inlinecode"><span class="id" title="var">Ack</span></span></h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab323"></a><h3 class="section">Base cases</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorSucc" class="idref" href="#majorSucc"><span class="id" title="lemma">majorSucc</span></a> : <a class="idref" href="hydras.MoreAck.AckNotPR.html#majorizedPR"><span class="id" title="definition">majorizedPR</span></a>  <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorZero" class="idref" href="#majorZero"><span class="id" title="lemma">majorZero</span></a> : <a class="idref" href="hydras.MoreAck.AckNotPR.html#majorizedPR"><span class="id" title="definition">majorizedPR</span></a>  <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorProjection" class="idref" href="#majorProjection"><span class="id" title="lemma">majorProjection</span></a> (<a id="n:53" class="idref" href="#n:53"><span class="id" title="binder">n</span></a> <a id="m:54" class="idref" href="#m:54"><span class="id" title="binder">m</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="H:55" class="idref" href="#H:55"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:54"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:53"><span class="id" title="variable">n</span></a>): <a class="idref" href="hydras.MoreAck.AckNotPR.html#majorizedPR"><span class="id" title="definition">majorizedPR</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#m:54"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#H:55"><span class="id" title="variable">H</span></a>) <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab324"></a><h3 class="section">The general case is proved by  induction on x</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorAnyPR" class="idref" href="#majorAnyPR"><span class="id" title="lemma">majorAnyPR</span></a>: <span class="id" title="keyword">∀</span> <a id="n:56" class="idref" href="#n:56"><span class="id" title="binder">n</span></a> (<a id="x:57" class="idref" href="#x:57"><span class="id" title="binder">x</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:56"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.MoreAck.AckNotPR.html#majorizedPR"><span class="id" title="definition">majorizedPR</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:57"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+Let us specialize Lemma <span class="inlinecode"><span class="id" title="var">majorAnyPR</span></span> to unary and binary  functions 
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorPR1" class="idref" href="#majorPR1"><span class="id" title="lemma">majorPR1</span></a>  (<a id="f:74" class="idref" href="#f:74"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 1)(<a id="Hf:75" class="idref" href="#Hf:75"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:74"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a> <span class="id" title="keyword">∀</span> <a id="x:77" class="idref" href="#x:77"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:74"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:77"><span class="id" title="variable">x</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:76"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:77"><span class="id" title="variable">x</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorPR2" class="idref" href="#majorPR2"><span class="id" title="lemma">majorPR2</span></a> (<a id="f:78" class="idref" href="#f:78"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2)(<a id="Hf:79" class="idref" href="#Hf:79"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:78"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="n:80" class="idref" href="#n:80"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">),</span></a> <span class="id" title="keyword">∀</span> <a id="x:81" class="idref" href="#x:81"><span class="id" title="binder">x</span></a> <a id="y:82" class="idref" href="#y:82"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:78"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:81"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:82"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:80"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:81"><span class="id" title="variable">x</span></a>  <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:82"><span class="id" title="variable">y</span></a>).  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="majorPR2_strict" class="idref" href="#majorPR2_strict"><span class="id" title="lemma">majorPR2_strict</span></a> (<a id="f:83" class="idref" href="#f:83"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2)(<a id="Hf:84" class="idref" href="#Hf:84"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:83"><span class="id" title="variable">f</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:86" class="idref" href="#x:86"><span class="id" title="binder">x</span></a> <a id="y:87" class="idref" href="#y:87"><span class="id" title="binder">y</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:87"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#f:83"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:87"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:85"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:87"><span class="id" title="variable">y</span></a>). 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab325"></a><h3 class="section">Now, let us assume that <span class="inlinecode"><span class="id" title="var">Ack</span></span> is PR</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility_Proof" class="idref" href="#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="HAck:88" class="idref" href="#HAck:88"><span class="id" title="binder">HAck</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Ack_not_PR" class="idref" href="#Ack_not_PR"><span class="id" title="lemma">Ack_not_PR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.  
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Impossibility_Proof"><span class="id" title="section">Impossibility_Proof</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab326"></a><h3 class="section">Any function which dominates (diagonalized) <span class="inlinecode"><span class="id" title="var">Ack</span></span> fails to be PR</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="dom_AckNotPR" class="idref" href="#dom_AckNotPR"><span class="id" title="section">dom_AckNotPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="dom_AckNotPR.f" class="idref" href="#dom_AckNotPR.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="dom_AckNotPR.Hf" class="idref" href="#dom_AckNotPR.Hf"><span class="id" title="variable">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR.f"><span class="id" title="variable">f</span></a> (<span class="id" title="keyword">fun</span> <a id="n:90" class="idref" href="#n:90"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:90"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:90"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="dom_AckNotPR" class="idref" href="#dom_AckNotPR"><span class="id" title="lemma">dom_AckNotPR</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR"><span class="id" title="section">dom_AckNotPR</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab327"></a><h2 class="section">Nevertheless, for any <span class="inlinecode"><span class="id" title="var">n</span></span>, <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is primitive recursive.</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AckSn_as_iterate" class="idref" href="#AckSn_as_iterate"><span class="id" title="lemma">AckSn_as_iterate</span></a> (<a id="n:92" class="idref" href="#n:92"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:92"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="k:93" class="idref" href="#k:93"><span class="id" title="binder">k</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:92"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#k:93"><span class="id" title="variable">k</span></a>) 1).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AckSn_as_PRiterate" class="idref" href="#AckSn_as_PRiterate"><span class="id" title="lemma">AckSn_as_PRiterate</span></a> (<a id="n:94" class="idref" href="#n:94"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:94"><span class="id" title="variable">n</span></a>)) (<span class="id" title="keyword">fun</span> <a id="k:95" class="idref" href="#k:95"><span class="id" title="binder">k</span></a> ⇒ <a class="idref" href="hydras.Ackermann.primRec.html#iterate"><span class="id" title="definition">primRec.iterate</span></a> (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:94"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#k:95"><span class="id" title="variable">k</span></a>) 1).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_nat_rec" class="idref" href="#iterate_nat_rec"><span class="id" title="lemma">iterate_nat_rec</span></a> (<a id="g:96" class="idref" href="#g:96"><span class="id" title="binder">g</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="x:98" class="idref" href="#x:98"><span class="id" title="binder">x</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:96"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:98"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#x:98"><span class="id" title="variable">x</span></a> (<span class="id" title="keyword">fun</span> <a id="x:99" class="idref" href="#x:99"><span class="id" title="binder">x</span></a> <a id="y:100" class="idref" href="#y:100"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.MoreAck.AckNotPR.html#g:96"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:100"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:97"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_Ackn_PR" class="idref" href="#Proof_of_Ackn_PR"><span class="id" title="section">Proof_of_Ackn_PR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Ackn_PR.S_step" class="idref" href="#Proof_of_Ackn_PR.S_step"><span class="id" title="section">S_step</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Ackn_PR.S_step.n" class="idref" href="#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="IHn:102" class="idref" href="#IHn:102"><span class="id" title="binder">IHn</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:103" class="idref" href="#a:103"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="y:104" class="idref" href="#y:104"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:104"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#a:103"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="R2" class="idref" href="#R2"><span class="id" title="instance">R2</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="a:105" class="idref" href="#a:105"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="y:106" class="idref" href="#y:106"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#y:106"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#a:105"><span class="id" title="variable">a</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="iSPR_Ack_Sn" class="idref" href="#iSPR_Ack_Sn"><span class="id" title="instance">iSPR_Ack_Sn</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step"><span class="id" title="section">S_step</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Ackn_IsPR" class="idref" href="#Ackn_IsPR"><span class="id" title="instance">Ackn_IsPR</span></a> (<a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 (<a class="idref" href="hydras.MoreAck.Ack.html#Ack"><span class="id" title="definition">Ack</span></a> <a class="idref" href="hydras.MoreAck.AckNotPR.html#n:107"><span class="id" title="variable">n</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">n</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="var">cbn</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Ackermann.primRec.html#succIsPR"><span class="id" title="instance">succIsPR</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#iSPR_Ack_Sn"><span class="id" title="instance">iSPR_Ack_Sn</span></a>; <span class="id" title="tactic">auto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR"><span class="id" title="section">Proof_of_Ackn_PR</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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--- /dev/null
+++ b/theories/html/hydras.MoreAck.BadSubst.html
@@ -0,0 +1,98 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.BadSubst</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.BadSubst</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab333"></a><h2 class="section">Preliminary lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">Ensembles.In</span></a> {<span class="id" title="var">_</span>} .<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> {<span class="id" title="var">_</span>} .<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="BadSubst" class="idref" href="#BadSubst"><span class="id" title="module">BadSubst</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="BadSubst.substF" class="idref" href="#BadSubst.substF"><span class="id" title="definition">substF</span></a> <a id="L:1" class="idref" href="#L:1"><span class="id" title="binder">L</span></a> (<a id="F:2" class="idref" href="#F:2"><span class="id" title="binder">F</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a>) <a id="v:3" class="idref" href="#v:3"><span class="id" title="binder">v</span></a> (<a id="t:4" class="idref" href="#t:4"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#F:2"><span class="id" title="variable">F</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <span class="id" title="var">t1</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> <span class="id" title="var">t2</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a>  <span class="id" title="var">r</span> <span class="id" title="var">s</span> ⇒  <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a> <span class="id" title="var">r</span> (<a class="idref" href="hydras.Ackermann.folProp.html#substTs"><span class="id" title="definition">substTs</span></a> <span class="id" title="var">s</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a>  <span class="id" title="var">G</span> <span class="id" title="var">H</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.MoreAck.BadSubst.html#substF:5"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a> <span class="id" title="var">G</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>) (<a class="idref" href="hydras.MoreAck.BadSubst.html#substF:5"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a> <span class="id" title="var">H</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a>  <span class="id" title="var">G</span> ⇒ <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.MoreAck.BadSubst.html#substF:5"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a> <span class="id" title="var">G</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a>  <span class="id" title="var">w</span> <span class="id" title="var">G</span> ⇒ <span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.eq_dec"><span class="id" title="lemma">Nat.eq_dec</span></a> <span class="id" title="var">w</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#F:2"><span class="id" title="variable">F</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <span class="id" title="var">w</span> (<a class="idref" href="hydras.MoreAck.BadSubst.html#substF:5"><span class="id" title="definition">substF</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#L:1"><span class="id" title="variable">L</span></a> <span class="id" title="var">G</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.MoreAck.BadSubst.html#t:4"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#BadSubst"><span class="id" title="module">BadSubst</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="BadSubstF2" class="idref" href="#BadSubstF2"><span class="id" title="module">BadSubstF2</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#BadSubstF2"><span class="id" title="module">BadSubstF2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#"><span class="id" title="library">FolExamples</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy"><span class="id" title="module">Toy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="BadExample" class="idref" href="#BadExample"><span class="id" title="section">BadExample</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="BadExample.F" class="idref" href="#BadExample.F"><span class="id" title="variable">F</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> 2<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="BadExample.F1" class="idref" href="#BadExample.F1"><span class="id" title="variable">F1</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> :=  (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> 2<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.BadSubst.html#BadExample"><span class="id" title="section">BadExample</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.MoreAck.FolExamples.html b/theories/html/hydras.MoreAck.FolExamples.html
new file mode 100644
index 00000000..a41361a8
--- /dev/null
+++ b/theories/html/hydras.MoreAck.FolExamples.html
@@ -0,0 +1,588 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.FolExamples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.FolExamples</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Use of FOL notations (Experimental)  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>  <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.folLogic.html#"><span class="id" title="library">folLogic</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folLogic2.html#"><span class="id" title="library">folLogic2</span></a> <a class="idref" href="hydras.Ackermann.folLogic3.html#"><span class="id" title="library">folLogic3</span></a> <a class="idref" href="hydras.Ackermann.subAll.html#"><span class="id" title="library">subAll</span></a> <a class="idref" href="hydras.Ackermann.Deduction.html#"><span class="id" title="library">Deduction</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab328"></a><h2 class="section">Preliminary lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">Ensembles.In</span></a> {<span class="id" title="var">_</span>} .<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> {<span class="id" title="var">_</span>} .<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Ackermann.fol.html#Term_Terms_rec_full"><span class="id" title="definition">Term_Terms_rec_full</span></a>.<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Ackermann.fol.html#Formula_rect"><span class="id" title="definition">Formula_rect</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Ackermann.fol.html#Formula_depth_rec"><span class="id" title="definition">Formula_depth_rec</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+depth-order vs structural order 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="depth_rec_demo" class="idref" href="#depth_rec_demo"><span class="id" title="section">depth_rec_demo</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="depth_rec_demo.L" class="idref" href="#depth_rec_demo.L"><span class="id" title="variable">L</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Language"><span class="id" title="record">Language</span></a>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="depth_rec_demo.P" class="idref" href="#depth_rec_demo.P"><span class="id" title="variable">P</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#depth_rec_demo.L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+<span class="id" title="keyword">Variable</span> <a id="depth_rec_demo.a" class="idref" href="#depth_rec_demo.a"><span class="id" title="variable">a</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">fol.Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#depth_rec_demo.L"><span class="id" title="variable">L</span></a>.<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">a</span></a>.<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">P</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#depth_rec_demo"><span class="id" title="section">depth_rec_demo</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="module">Ensembles</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_add1" class="idref" href="#In_add1"><span class="id" title="lemma">In_add1</span></a> {<a id="T:4" class="idref" href="#T:4"><span class="id" title="binder">T</span></a>:<span class="id" title="keyword">Type</span>}(<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.MoreAck.FolExamples.html#T:4"><span class="id" title="variable">T</span></a>)(<a id="S:6" class="idref" href="#S:6"><span class="id" title="binder">S</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#T:4"><span class="id" title="variable">T</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Add</span></a>  <a class="idref" href="hydras.MoreAck.FolExamples.html#S:6"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#a:5"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.MoreAck.FolExamples.html#a:5"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_add2" class="idref" href="#In_add2"><span class="id" title="lemma">In_add2</span></a> {<a id="T:7" class="idref" href="#T:7"><span class="id" title="binder">T</span></a>:<span class="id" title="keyword">Type</span>}(<a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a> <a id="b:9" class="idref" href="#b:9"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.MoreAck.FolExamples.html#T:7"><span class="id" title="variable">T</span></a>)(<a id="S:10" class="idref" href="#S:10"><span class="id" title="binder">S</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#T:7"><span class="id" title="variable">T</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#S:10"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#a:8"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Add</span></a>  <a class="idref" href="hydras.MoreAck.FolExamples.html#S:10"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#b:9"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.MoreAck.FolExamples.html#a:8"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a>:  <span class="id" title="var">core</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">In_add1</span> <span class="id" title="var">In_add2</span> <span class="id" title="var">AXM</span>: <span class="id" title="var">core</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">fol_scope</span></span> allows us to read and write FOL terms and formulas 
+    in a syntax close to Coq's 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> <a id="L:11" class="idref" href="#L:11"><span class="id" title="binder">L</span></a> (<a id="t:12" class="idref" href="#t:12"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:11"><span class="id" title="variable">L</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#t:12"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#t:12"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#t:12"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+A small variation on MP (without appending contexts) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="MPSys" class="idref" href="#MPSys"><span class="id" title="lemma">MPSys</span></a> <a id="L:13" class="idref" href="#L:13"><span class="id" title="binder">L</span></a> (<a id="G:14" class="idref" href="#G:14"><span class="id" title="binder">G</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#System"><span class="id" title="definition">System</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:13"><span class="id" title="variable">L</span></a>) (<a id="A:15" class="idref" href="#A:15"><span class="id" title="binder">A</span></a> <a id="B:16" class="idref" href="#B:16"><span class="id" title="binder">B</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:13"><span class="id" title="variable">L</span></a>) : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:13"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#G:14"><span class="id" title="variable">G</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#A:15"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#B:16"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:13"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#G:14"><span class="id" title="variable">G</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#A:15"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#L:13"><span class="id" title="variable">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#G:14"><span class="id" title="variable">G</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#B:16"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A small language 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Toy" class="idref" href="#Toy"><span class="id" title="module">Toy</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Toy.Rel" class="idref" href="#Toy.Rel"><span class="id" title="inductive">Rel</span></a>: <span class="id" title="keyword">Set</span> := <a id="Toy.A_" class="idref" href="#Toy.A_"><span class="id" title="constructor">A_</span></a> | <a id="Toy.B_" class="idref" href="#Toy.B_"><span class="id" title="constructor">B_</span></a> | <a id="Toy.C_" class="idref" href="#Toy.C_"><span class="id" title="constructor">C_</span></a> | <a id="Toy.P_" class="idref" href="#Toy.P_"><span class="id" title="constructor">P_</span></a> | <a id="Toy.Q_" class="idref" href="#Toy.Q_"><span class="id" title="constructor">Q_</span></a> | <a id="Toy.R_" class="idref" href="#Toy.R_"><span class="id" title="constructor">R_</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Toy.Fun" class="idref" href="#Toy.Fun"><span class="id" title="inductive">Fun</span></a> : <span class="id" title="keyword">Set</span> := <a id="Toy.a_" class="idref" href="#Toy.a_"><span class="id" title="constructor">a_</span></a> | <a id="Toy.b_" class="idref" href="#Toy.b_"><span class="id" title="constructor">b_</span></a> | <a id="Toy.f_" class="idref" href="#Toy.f_"><span class="id" title="constructor">f_</span></a> | <a id="Toy.g_" class="idref" href="#Toy.g_"><span class="id" title="constructor">g_</span></a> | <a id="Toy.h_" class="idref" href="#Toy.h_"><span class="id" title="constructor">h_</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Toy.arityR" class="idref" href="#Toy.arityR"><span class="id" title="definition">arityR</span></a> (<a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Rel"><span class="id" title="inductive">Rel</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#x:21"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P_"><span class="id" title="constructor">P_</span></a> | <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Q_"><span class="id" title="constructor">Q_</span></a> ⇒ 1 | <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R_"><span class="id" title="constructor">R_</span></a> ⇒ 2 | <span class="id" title="var">_</span> ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="Toy.arityF" class="idref" href="#Toy.arityF"><span class="id" title="definition">arityF</span></a> (<a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Fun"><span class="id" title="inductive">Fun</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#x:23"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span>  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f_"><span class="id" title="constructor">f_</span></a> |  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.g_"><span class="id" title="constructor">g_</span></a> ⇒ 1 | <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h_"><span class="id" title="constructor">h_</span></a> ⇒ 2 | <span class="id" title="var">_</span> ⇒ 0  <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="Toy.L" class="idref" href="#Toy.L"><span class="id" title="definition">L</span></a> := <a class="idref" href="hydras.Ackermann.fol.html#language"><span class="id" title="constructor">language</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Rel"><span class="id" title="inductive">Rel</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Fun"><span class="id" title="inductive">Fun</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.arityR"><span class="id" title="definition">arityR</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.arityF"><span class="id" title="definition">arityF</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="Toy.a" class="idref" href="#Toy.a"><span class="id" title="abbreviation">a</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a_"><span class="id" title="constructor">a_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Toy.b" class="idref" href="#Toy.b"><span class="id" title="abbreviation">b</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.b_"><span class="id" title="constructor">b_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Toy.f" class="idref" href="#Toy.f"><span class="id" title="abbreviation">f</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f_"><span class="id" title="constructor">f_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Toy.g" class="idref" href="#Toy.g"><span class="id" title="abbreviation">g</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.g_"><span class="id" title="constructor">g_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+<span class="id" title="keyword">Notation</span> <a id="Toy.h" class="idref" href="#Toy.h"><span class="id" title="abbreviation">h</span></a> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> := (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h_"><span class="id" title="constructor">h_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t1</span> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t2</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.t0" class="idref" href="#Toy.t0"><span class="id" title="definition">t0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> :=  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.t1" class="idref" href="#Toy.t1"><span class="id" title="definition">t1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.t0"><span class="id" title="definition">t0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.t2" class="idref" href="#Toy.t2"><span class="id" title="definition">t2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h"><span class="id" title="abbreviation">h</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.t1"><span class="id" title="definition">t1</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.t0"><span class="id" title="definition">t0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.t3" class="idref" href="#Toy.t3"><span class="id" title="definition">t3</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h"><span class="id" title="abbreviation">h</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)) (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.g"><span class="id" title="abbreviation">g</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1)).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm" class="idref" href="#Toy.Unnamed_thm"><span class="id" title="definition">t0</span></a> <a id="Toy.Unnamed_thm" class="idref" href="#Toy.Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm" class="idref" href="#Toy.Unnamed_thm"><span class="id" title="definition">a</span></a>.  <br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm0" class="idref" href="#Toy.Unnamed_thm0"><span class="id" title="definition">t1</span></a> <a id="Toy.Unnamed_thm0" class="idref" href="#Toy.Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm0" class="idref" href="#Toy.Unnamed_thm0"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm0" class="idref" href="#Toy.Unnamed_thm0"><span class="id" title="definition">a</span></a>.  <br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">t2</span></a> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">h</span></a> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">a</span></a><a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">)</span></a> <a id="Toy.Unnamed_thm1" class="idref" href="#Toy.Unnamed_thm1"><span class="id" title="definition">a</span></a>.  <br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Abreviations for the toy language L 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.A" class="idref" href="#Toy.A"><span class="id" title="abbreviation">A</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A_"><span class="id" title="constructor">A_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.B" class="idref" href="#Toy.B"><span class="id" title="abbreviation">B</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B_"><span class="id" title="constructor">B_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.C" class="idref" href="#Toy.C"><span class="id" title="abbreviation">C</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C_"><span class="id" title="constructor">C_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.P" class="idref" href="#Toy.P"><span class="id" title="abbreviation">P</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P_"><span class="id" title="constructor">P_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.Q" class="idref" href="#Toy.Q"><span class="id" title="abbreviation">Q</span></a> <span class="id" title="var">t</span> := (<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Q_"><span class="id" title="constructor">Q_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Toy.R" class="idref" href="#Toy.R"><span class="id" title="abbreviation">R</span></a> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> := (@<a class="idref" href="hydras.Ackermann.fol.html#atomic"><span class="id" title="constructor">atomic</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R_"><span class="id" title="constructor">R_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t1</span> (<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> <span class="id" title="var">t2</span> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F1" class="idref" href="#Toy.F1"><span class="id" title="definition">F1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> :=  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.b"><span class="id" title="abbreviation">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F2" class="idref" href="#Toy.F2"><span class="id" title="definition">F2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 1 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#impH"><span class="id" title="constructor">impH</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1)) (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F3" class="idref" href="#Toy.F3"><span class="id" title="definition">F3</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F4" class="idref" href="#Toy.F4"><span class="id" title="definition">F4</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 0 (<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F5" class="idref" href="#Toy.F5"><span class="id" title="definition">F5</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.F6" class="idref" href="#Toy.F6"><span class="id" title="definition">F6</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a>:= (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.F1"><span class="id" title="definition">F1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.F2"><span class="id" title="definition">F2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.F3"><span class="id" title="definition">F3</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The following computation expands some derived connectives and 
+    quantifiers. Within <span class="inlinecode"><span class="id" title="var">fol_scope</span></span>, we print them with a 
+    similar syntax (with primed symbols) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Toy.PrimedSymbols" class="idref" href="#Toy.PrimedSymbols"><span class="id" title="section">PrimedSymbols</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">F3</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">∧</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">F1</span></a><a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">)%</span></a><a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">fol</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">(~(~</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">¬</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">F3</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">→</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">¬</span></a> <a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">F1</span></a><a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">))%</span></a><a id="Toy.Unnamed_thm2" class="idref" href="#Toy.Unnamed_thm2"><span class="id" title="definition">fol</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.F6"><span class="id" title="definition">F6</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Unset</span> <span class="id" title="keyword">Printing</span> <span class="id" title="var">Notations</span>.<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.F6"><span class="id" title="definition">F6</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.PrimedSymbols"><span class="id" title="section">PrimedSymbols</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">forallH</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">1</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">equal</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">var</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">1)</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">a</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">)</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">≠</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">forallH</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">0</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">equal</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">var</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">0)</span></a> <a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">a</span></a><a id="Toy.Unnamed_thm3" class="idref" href="#Toy.Unnamed_thm3"><span class="id" title="definition">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+a 
+</div>
+<div class="code">
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">apply</span></a> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">a_</span></a> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">Tnil</span></a> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm4" class="idref" href="#Toy.Unnamed_thm4"><span class="id" title="definition">a</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+f a 
+</div>
+<div class="code">
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">apply</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">f_</span></a>  <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">Tcons</span></a>  <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">apply</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">a_</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">Tnil</span></a><a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">)</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">Tnil</span></a><a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">)</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm5" class="idref" href="#Toy.Unnamed_thm5"><span class="id" title="definition">a</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+f (f v1) 
+</div>
+<div class="code">
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">apply</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">f_</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">Tcons</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">apply</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">f_</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">Tcons</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">var</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">1)</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">Tnil</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">))</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">Tnil</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">)</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">var</span></a>  <a id="Toy.Unnamed_thm6" class="idref" href="#Toy.Unnamed_thm6"><span class="id" title="definition">1)))</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Toy.Ldec" class="idref" href="#Toy.Ldec"><span class="id" title="definition">Ldec</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#language_decidable"><span class="id" title="definition">language_decidable</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Formula_eqdec is opaque ! 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">lt_depth</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">L</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#0</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#1</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">∨</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">exH</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">2,</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#1</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#2)%</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">fol</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">(</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#0</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#1</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">∧</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">exH</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">2,</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#1</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">=</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">f</span></a> <a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">v</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">#2)%</span></a><a id="Toy.Unnamed_thm7" class="idref" href="#Toy.Unnamed_thm7"><span class="id" title="definition">fol</span></a>. 
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Toy.Unnamed_thm8" class="idref" href="#Toy.Unnamed_thm8"><span class="id" title="definition">lt_depth</span></a> <a id="Toy.Unnamed_thm8" class="idref" href="#Toy.Unnamed_thm8"><span class="id" title="definition">_</span></a> <a id="Toy.Unnamed_thm8" class="idref" href="#Toy.Unnamed_thm8"><span class="id" title="definition">F1</span></a> <a id="Toy.Unnamed_thm8" class="idref" href="#Toy.Unnamed_thm8"><span class="id" title="definition">F2</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.subAll.html#subAllFormula"><span class="id" title="definition">subAllFormula</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Toy.OnSubstF" class="idref" href="#Toy.OnSubstF"><span class="id" title="section">OnSubstF</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Toy.OnSubstF.F" class="idref" href="#Toy.OnSubstF.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> 2<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.OnSubstF"><span class="id" title="section">OnSubstF</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx1" class="idref" href="#Toy.PrfEx1"><span class="id" title="definition">PrfEx1</span></a>: <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a>  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">[</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a>)%<span class="id" title="var">fol</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">]</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.PrfEx2" class="idref" href="#Toy.PrfEx2"><span class="id" title="lemma">PrfEx2</span></a>:  <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">]</span></a>%<span class="id" title="var">fol</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.PrfEx2"><span class="id" title="lemma">PrfEx2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.MP'" class="idref" href="#Toy.MP'"><span class="id" title="lemma">MP'</span></a> <a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> <a id="g:31" class="idref" href="#g:31"><span class="id" title="binder">g</span></a> <a id="H1:32" class="idref" href="#H1:32"><span class="id" title="binder">H1</span></a> <a id="H2:33" class="idref" href="#H2:33"><span class="id" title="binder">H2</span></a> <a id="H:34" class="idref" href="#H:34"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.MoreAck.FolExamples.html#H:34"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#H1:32"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#H2:33"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#H1:32"><span class="id" title="variable">H1</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#g:31"><span class="id" title="variable">g</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#H2:33"><span class="id" title="variable">H2</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#H:34"><span class="id" title="variable">H</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#g:31"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Cuts the current list of hypotheses as (G++?H), then applies MP 
+</div>
+<div class="code">
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">cutMP</span> <span class="id" title="var">G</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;|- <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> ?<span class="id" title="var">L</span> ?<span class="id" title="var">H</span>  ?<span class="id" title="var">F</span> ⇒  <span class="id" title="tactic">eapply</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.MP'"><span class="id" title="lemma">MP'</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">H1</span> := <span class="id" title="var">G</span>); <br/>
+[<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">reflexivity</span> | <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Ackermann.folProof.html#AXM"><span class="id" title="constructor">AXM</span></a> | <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Ackermann.folProof.html#AXM"><span class="id" title="constructor">AXM</span></a> ] <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx2'" class="idref" href="#Toy.PrfEx2'"><span class="id" title="definition">PrfEx2'</span></a>:  <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">]</span></a>%<span class="id" title="var">fol</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a>.  <span class="id" title="keyword">Section</span> <a id="Toy.ProofOfEx3" class="idref" href="#Toy.ProofOfEx3"><span class="id" title="section">ProofOfEx3</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>]  <span class="id" title="var">Arguments</span> <a class="idref" href="hydras.Ackermann.folProof.html#MP"><span class="id" title="constructor">MP</span></a> {<span class="id" title="var">L</span> <span class="id" title="var">Hyp1</span> <span class="id" title="var">Hyp2</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx3" class="idref" href="#Toy.PrfEx3"><span class="id" title="definition">PrfEx3</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a>)%<span class="id" title="var">fol</span>.  
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.PrfEx3"><span class="id" title="definition">PrfEx3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.ProofOfEx3"><span class="id" title="section">ProofOfEx3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Rule of contradiction 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx4" class="idref" href="#Toy.PrfEx4"><span class="id" title="definition">PrfEx4</span></a> (<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a> <a id="B:36" class="idref" href="#B:36"><span class="id" title="binder">B</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a>): <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#B:36"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#B:36"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#A:35"><span class="id" title="variable">A</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.PrfEx4"><span class="id" title="definition">PrfEx4</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab329"></a><h2 class="section">Universal quantifier</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx5" class="idref" href="#Toy.PrfEx5"><span class="id" title="definition">PrfEx5</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1 2<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 2<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.a"><span class="id" title="abbreviation">a</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx6" class="idref" href="#Toy.PrfEx6"><span class="id" title="definition">PrfEx6</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfContrex7" class="idref" href="#Toy.PrfContrex7"><span class="id" title="definition">PrfContrex7</span></a> : <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 1<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.  
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx8" class="idref" href="#Toy.PrfEx8"><span class="id" title="definition">PrfEx8</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Q"><span class="id" title="abbreviation">Q</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Q"><span class="id" title="abbreviation">Q</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.eq_refl" class="idref" href="#Toy.eq_refl"><span class="id" title="lemma">eq_refl</span></a> (<a id="t:37" class="idref" href="#t:37"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a>): <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#t:37"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#t:37"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Ackermann.folProof.html#EQ4"><span class="id" title="constructor">EQ4</span></a>.<br/>
+<span class="id" title="keyword">Check</span> @<a class="idref" href="hydras.Ackermann.folProof.html#EQ4"><span class="id" title="constructor">EQ4</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R_"><span class="id" title="constructor">R_</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx9" class="idref" href="#Toy.PrfEx9"><span class="id" title="definition">PrfEx9</span></a>: <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx10" class="idref" href="#Toy.PrfEx10"><span class="id" title="definition">PrfEx10</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>3 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>3 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfContrex9" class="idref" href="#Toy.PrfContrex9"><span class="id" title="definition">PrfContrex9</span></a>: <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.PrfEx11" class="idref" href="#Toy.PrfEx11"><span class="id" title="definition">PrfEx11</span></a>: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ae9a5e1034e143b218b09d8e454472bd"><span class="id" title="notation">[]</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>3 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h"><span class="id" title="abbreviation">h</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.h"><span class="id" title="abbreviation">h</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>3 <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.  
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.ded1" class="idref" href="#Toy.ded1"><span class="id" title="lemma">ded1</span></a>:  <span class="id" title="keyword">∀</span> (<a id="A:38" class="idref" href="#A:38"><span class="id" title="binder">A</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <span class="id" title="var">_</span>) <a class="idref" href="hydras.MoreAck.FolExamples.html#A:38"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="pf:39" class="idref" href="#pf:39"><span class="id" title="binder">pf</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.MoreAck.FolExamples.html#A:38"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.ded2" class="idref" href="#Toy.ded2"><span class="id" title="lemma">ded2</span></a>:  <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a>)%<span class="id" title="var">fol</span>.<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a>.<br/>
+<span class="id" title="keyword">Search</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Add</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.ded3" class="idref" href="#Toy.ded3"><span class="id" title="lemma">ded3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="pf:40" class="idref" href="#pf:40"><span class="id" title="binder">pf</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#Prf"><span class="id" title="inductive">Prf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">Ensembles.In</span></a> {<span class="id" title="var">_</span>} .<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="var">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Ensembles.Add</span></a> {<span class="id" title="var">_</span>} .<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="module">Ensembles</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.Ackermann.fol.html#mem"><span class="id" title="definition">mem</span></a>: <span class="id" title="var">sets</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">In_add1</span> <span class="id" title="var">In_add2</span>: <span class="id" title="var">sets</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Toy.Adds" class="idref" href="#Toy.Adds"><span class="id" title="definition">Adds</span></a> {<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="X:42" class="idref" href="#X:42"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#A:41"><span class="id" title="variable">A</span></a>)(<a id="l:43" class="idref" href="#l:43"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#A:41"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#l:43"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.MoreAck.FolExamples.html#X:42"><span class="id" title="variable">X</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Add</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Adds:44"><span class="id" title="definition">Adds</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#X:42"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#l:43"><span class="id" title="variable">l</span></a>) <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Toy.SysPrfEx2" class="idref" href="#Toy.SysPrfEx2"><span class="id" title="definition">SysPrfEx2</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:46" class="idref" href="#x:46"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#x:46"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#e76c6291366b599375c28eba0aae94bb"><span class="id" title="notation">]</span></a>%<span class="id" title="var">fol</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.C"><span class="id" title="abbreviation">C</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (?<span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> ?<span class="id" title="var">B</span>)%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.fol.html#notH"><span class="id" title="constructor">notH</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> (<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> ?<span class="id" title="var">L</span> ?<span class="id" title="var">T</span> (?<span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> ?<span class="id" title="var">B</span>)%<span class="id" title="var">fol</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> ?<span class="id" title="var">L</span> ?<span class="id" title="var">T</span> ?<span class="id" title="var">B</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> (<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">fol</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (?<span class="id" title="var">a</span> <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">b</span> )%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.folProp.html#substF"><span class="id" title="definition">substF</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> ?<span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">fol</span> (<a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">allH</span></a> ?<span class="id" title="var">v</span><a class="idref" href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Toy.PeirceProof" class="idref" href="#Toy.PeirceProof"><span class="id" title="section">PeirceProof</span></a>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Add"><span class="id" title="definition">Add</span></a> {<span class="id" title="var">U</span>}.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> {<span class="id" title="var">U</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Toy.Peirce" class="idref" href="#Toy.Peirce"><span class="id" title="definition">Peirce</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">((</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.A"><span class="id" title="abbreviation">A</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Toy.peirce" class="idref" href="#Toy.peirce"><span class="id" title="lemma">peirce</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a>  <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Peirce"><span class="id" title="definition">Peirce</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.PeirceProof"><span class="id" title="section">PeirceProof</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Toy.Drinkers_theorem" class="idref" href="#Toy.Drinkers_theorem"><span class="id" title="section">Drinkers_theorem</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Toy.D0" class="idref" href="#Toy.D0"><span class="id" title="lemma">D0</span></a> : <span class="id" title="keyword">∀</span> <a id="i:47" class="idref" href="#i:47"><span class="id" title="binder">i</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <span class="id" title="var">_</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;( <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#i:47"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#i:47"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#i:47"><span class="id" title="variable">i</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#i:47"><span class="id" title="variable">i</span></a>)<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Toy.D01" class="idref" href="#Toy.D01"><span class="id" title="lemma">D01</span></a> <a id="T:48" class="idref" href="#T:48"><span class="id" title="binder">T</span></a> <a id="i:49" class="idref" href="#i:49"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#T:48"><span class="id" title="variable">T</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#i:49"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#i:49"><span class="id" title="variable">i</span></a>)) <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#i:49"><span class="id" title="variable">i</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#i:49"><span class="id" title="variable">i</span></a>)<a class="idref" href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448"><span class="id" title="notation">)</span></a><a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Toy.Drinkers_theorem.f" class="idref" href="#Toy.Drinkers_theorem.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">exH</span></a> 0<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0) <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 1 (<a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.P"><span class="id" title="abbreviation">P</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1))<a class="idref" href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Toy.drinkers_thm" class="idref" href="#Toy.drinkers_thm"><span class="id" title="lemma">drinkers_thm</span></a> : <a class="idref" href="hydras.Ackermann.folProof.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.L"><span class="id" title="definition">L</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a> <span class="id" title="var">_</span>) <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Drinkers_theorem.f"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy.Drinkers_theorem"><span class="id" title="section">Drinkers_theorem</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#Toy"><span class="id" title="module">Toy</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+v1 + 0 
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="t1_0" class="idref" href="#t1_0"><span class="id" title="definition">t1_0</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> := <br/>
+&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Plus_"><span class="id" title="constructor">Plus_</span></a> <br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>) <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> )).<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.FolExamples.html#t1_0"><span class="id" title="definition">t1_0</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+forall v0, v0 = 0 \/ exists v1,  v0 = S v1 
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="f1" class="idref" href="#f1"><span class="id" title="definition">f1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> ))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="f2" class="idref" href="#f2"><span class="id" title="definition">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> )))).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="f3" class="idref" href="#f3"><span class="id" title="definition">f3</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0) (<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Zero_"><span class="id" title="constructor">Zero_</span></a> <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Succ_</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) <a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a>))))).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.MoreAck.Iterate_compat.html b/theories/html/hydras.MoreAck.Iterate_compat.html
new file mode 100644
index 00000000..dccaeffd
--- /dev/null
+++ b/theories/html/hydras.MoreAck.Iterate_compat.html
@@ -0,0 +1,54 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.Iterate_compat</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.Iterate_compat</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Compatibility between primRec.iterate and Iterates.iterate 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_compat" class="idref" href="#iterate_compat"><span class="id" title="lemma">iterate_compat</span></a> (<a id="f:1" class="idref" href="#f:1"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> <a id="x:3" class="idref" href="#x:3"><span class="id" title="binder">x</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">Iterates.iterate</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#x:3"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#iterate"><span class="id" title="definition">primRec.iterate</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#f:1"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#x:3"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_extEqual" class="idref" href="#iterate_extEqual"><span class="id" title="lemma">iterate_extEqual</span></a> (<a id="f:4" class="idref" href="#f:4"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">Iterates.iterate</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#f:4"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#n:5"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Ackermann.primRec.html#iterate"><span class="id" title="definition">primRec.iterate</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#f:4"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.Iterate_compat.html#n:5"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.MoreAck.LNN_Examples.html b/theories/html/hydras.MoreAck.LNN_Examples.html
new file mode 100644
index 00000000..3232e619
--- /dev/null
+++ b/theories/html/hydras.MoreAck.LNN_Examples.html
@@ -0,0 +1,170 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.LNN_Examples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.LNN_Examples</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Experimental 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">Lists.List</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#"><span class="id" title="library">fol</span></a> <a class="idref" href="hydras.Ackermann.folProp.html#"><span class="id" title="library">folProp</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#"><span class="id" title="library">Languages</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="library">LNN</span></a> <a class="idref" href="hydras.Ackermann.folProof.html#"><span class="id" title="library">folProof</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations"><span class="id" title="module">FolNotations</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="bare_syntax" class="idref" href="#bare_syntax"><span class="id" title="section">bare_syntax</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="f0" class="idref" href="#f0"><span class="id" title="definition">f0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#orH"><span class="id" title="definition">orH</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a>  0) <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#equal"><span class="id" title="constructor">equal</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#apply"><span class="id" title="constructor">apply</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.Languages.html#Succ_"><span class="id" title="constructor">Languages.Succ_</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Functions"><span class="id" title="projection">Functions</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#Tcons"><span class="id" title="constructor">Tcons</span></a>  (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) (@<a class="idref" href="hydras.Ackermann.fol.html#Tnil"><span class="id" title="constructor">Tnil</span></a> <span class="id" title="var">_</span>)))))).<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#"><span class="id" title="module">LNN</span></a>.<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#f0"><span class="id" title="definition">f0</span></a>.<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>.<br/>
+<span class="id" title="keyword">Set Printing All</span>.<br/>
+<span class="id" title="keyword">Unset Printing All</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#f0"><span class="id" title="definition">f0</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#bare_syntax"><span class="id" title="section">bare_syntax</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span>  <a class="idref" href="hydras.MoreAck.LNN_Examples.html#f0"><span class="id" title="definition">f0</span></a>.<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">f0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">allH</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">0,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">v</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">#0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">Zero</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∨</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">exH</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">1,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">v</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">#0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">Succ</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">v</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">#1)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">fol</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">Plus</span>.<br/>
+<span class="id" title="keyword">Locate</span> "_ + _".<br/>
+&nbsp;<span class="id" title="keyword">Example</span> <a id="t1_0" class="idref" href="#t1_0"><span class="id" title="definition">t1_0</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <span class="id" title="var">_</span> := <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1))%<span class="id" title="var">fol</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>.<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#t1_0"><span class="id" title="definition">t1_0</span></a>.<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#t1_0"><span class="id" title="definition">t1_0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Examples" class="idref" href="#Examples"><span class="id" title="section">Examples</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Examples.t1" class="idref" href="#Examples.t1"><span class="id" title="variable">t1</span></a>: <a class="idref" href="hydras.Ackermann.fol.html#Term"><span class="id" title="inductive">Term</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> := <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#var"><span class="id" title="constructor">var</span></a> 1) <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+forall v0, v0 = 0 \/ exists v1,  v0 = S v1 
+</div>
+<div class="code">
+<span class="id" title="keyword">Let</span> <a id="Examples.f1" class="idref" href="#Examples.f1"><span class="id" title="variable">f1</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#forallH"><span class="id" title="constructor">forallH</span></a> 0 <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1))))%<span class="id" title="var">fol</span>.<br/>
+<span class="id" title="keyword">Print</span> <span class="id" title="var">f1</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.Ackermann.fol.html#Relations"><span class="id" title="projection">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Examples.f2" class="idref" href="#Examples.f2"><span class="id" title="variable">f2</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 2 (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.LT"><span class="id" title="definition">LT</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2) <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 4 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2) (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)))%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Examples.f2'" class="idref" href="#Examples.f2'"><span class="id" title="variable">f2'</span></a> : <a class="idref" href="hydras.Ackermann.fol.html#Formula"><span class="id" title="inductive">Formula</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 2 (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2 <a class="idref" href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 4 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2) (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)))%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Examples.f3" class="idref" href="#Examples.f3"><span class="id" title="variable">f3</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a> <a class="idref" href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Ackermann.fol.html#existH"><span class="id" title="definition">existH</span></a> 1 (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Succ"><span class="id" title="definition">Succ</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)))%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="Examples.f4" class="idref" href="#Examples.f4"><span class="id" title="variable">f4</span></a> := (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.LNN.html#natToTerm"><span class="id" title="definition">natToTerm</span></a> 2<a class="idref" href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <span class="id" title="var">f4</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Plus"><span class="id" title="definition">Plus</span></a> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>  <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.Zero"><span class="id" title="definition">Zero</span></a>)%<span class="id" title="var">fol</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Locate</span> <span class="id" title="var">LT</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.LNN_Examples.html#Examples"><span class="id" title="section">Examples</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.folProof.html#GEN"><span class="id" title="constructor">GEN</span></a> <a class="idref" href="hydras.Ackermann.Languages.html#LNN"><span class="id" title="definition">LNN</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0)%<span class="id" title="var">fol</span> 1.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Prf</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">LNN</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">forallH</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">0</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">v</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">#0</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">v</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">#0))%</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">fol</span></a>  <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Prf</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">LNN</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">nil</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Zero</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">Zero</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">fol</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,481 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.PrimRecExamples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.PrimRecExamples</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Unicode.Utf8.html#"><span class="id" title="library">Utf8</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#VectorNotations"><span class="id" title="module">VectorNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="module">extEqualNat</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations"><span class="id" title="module">PRNotations</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#leBool"><span class="id" title="definition">leBool</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#plus"><span class="id" title="abbreviation">plus</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> 42: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<span class="id" title="keyword">fun</span> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> <a id="p:2" class="idref" href="#p:2"><span class="id" title="binder">p</span></a> <a id="q:3" class="idref" href="#q:3"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#p:2"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#q:3"><span class="id" title="variable">q</span></a>): <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 3.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="extEqual_ex1" class="idref" href="#extEqual_ex1"><span class="id" title="definition">extEqual_ex1</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.mul"><span class="id" title="definition">Nat.mul</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2<a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">fun</span> <a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a> <a id="y:5" class="idref" href="#y:5"><span class="id" title="binder">y</span></a> ⇒  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:5"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:4"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:4"><span class="id" title="variable">x</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab330"></a><h2 class="section">Examples of terms of type <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> and their interpretation</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> <a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:6"><span class="id" title="variable">a</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex3" class="idref" href="#Ex3"><span class="id" title="definition">Ex3</span></a> <a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> <a id="b:8" class="idref" href="#b:8"><span class="id" title="binder">b</span></a> <a id="c:9" class="idref" href="#c:9"><span class="id" title="binder">c</span></a> <a id="d:10" class="idref" href="#d:10"><span class="id" title="binder">d</span></a> <a id="e:11" class="idref" href="#e:11"><span class="id" title="binder">e</span></a> <a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a>: <span class="id" title="keyword">∀</span> (<a id="H:13" class="idref" href="#H:13"><span class="id" title="binder">H</span></a>: 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> 6),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#projFunc"><span class="id" title="constructor">projFunc</span></a> 6 2 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#H:13"><span class="id" title="variable">H</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#b:8"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#c:9"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#d:10"><span class="id" title="variable">d</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#e:11"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:12"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#d:10"><span class="id" title="variable">d</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex4" class="idref" href="#Ex4"><span class="id" title="definition">Ex4</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 4 (<span class="id" title="keyword">fun</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a> <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a> <a id="c:16" class="idref" href="#c:16"><span class="id" title="binder">c</span></a> <a id="d:17" class="idref" href="#d:17"><span class="id" title="binder">d</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ 0)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PRnil"><span class="id" title="constructor">PRnil</span></a> <span class="id" title="var">_</span>))).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Composition" class="idref" href="#Composition"><span class="id" title="section">Composition</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Composition.x" class="idref" href="#Composition.x"><span class="id" title="variable">x</span></a> <a id="Composition.y" class="idref" href="#Composition.y"><span class="id" title="variable">y</span></a> <a id="Composition.z" class="idref" href="#Composition.z"><span class="id" title="variable">z</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 2) (<a id="Composition.t" class="idref" href="#Composition.t"><span class="id" title="variable">t</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 3).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.u" class="idref" href="#Composition.u"><span class="id" title="variable">u</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 2 := <a class="idref" href="hydras.Ackermann.primRec.html#composeFunc"><span class="id" title="constructor">composeFunc</span></a> 2 3 <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.x"><span class="id" title="variable">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.y"><span class="id" title="variable">y</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.z"><span class="id" title="variable">z</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">]</span></a>%<span class="id" title="var">pr</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.t"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.f" class="idref" href="#Composition.f"><span class="id" title="variable">f</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.g" class="idref" href="#Composition.g"><span class="id" title="variable">g</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.y"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.h" class="idref" href="#Composition.h"><span class="id" title="variable">h</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.z"><span class="id" title="variable">z</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.k" class="idref" href="#Composition.k"><span class="id" title="variable">k</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.t"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> <a id="p:23" class="idref" href="#p:23"><span class="id" title="binder">p</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">PReval</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">u</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">k</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">h</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Composition.v" class="idref" href="#Composition.v"><span class="id" title="variable">v</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.t"><span class="id" title="variable">t</span></a>  <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">[</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi1_2"><span class="id" title="abbreviation">pi1_2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi1_2"><span class="id" title="abbreviation">pi1_2</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi2_2"><span class="id" title="abbreviation">pi2_2</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">]</span></a>)%<span class="id" title="var">pr</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition.v"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Composition"><span class="id" title="section">Composition</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Primitive_recursion" class="idref" href="#Primitive_recursion"><span class="id" title="section">Primitive_recursion</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Primitive_recursion.x" class="idref" href="#Primitive_recursion.x"><span class="id" title="variable">x</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 2)(<a id="Primitive_recursion.y" class="idref" href="#Primitive_recursion.y"><span class="id" title="variable">y</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 4).<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Primitive_recursion.z" class="idref" href="#Primitive_recursion.z"><span class="id" title="variable">z</span></a> := (<a class="idref" href="hydras.Ackermann.primRec.html#primRecFunc"><span class="id" title="constructor">primRecFunc</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.x"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.y"><span class="id" title="variable">y</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Primitive_recursion.g" class="idref" href="#Primitive_recursion.g"><span class="id" title="variable">g</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Primitive_recursion.h" class="idref" href="#Primitive_recursion.h"><span class="id" title="variable">h</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.y"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="Primitive_recursion.f" class="idref" href="#Primitive_recursion.f"><span class="id" title="variable">f</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.z"><span class="id" title="variable">z</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">∀</span></a> <a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">0</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">a</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">b</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">a</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">b</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">∀</span></a> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> <a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> <a id="b:30" class="idref" href="#b:30"><span class="id" title="binder">b</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">n</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">.+1</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">a</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">b</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">h</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">a</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">b</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">a</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion"><span class="id" title="section">Primitive_recursion</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="compose2Examples" class="idref" href="#compose2Examples"><span class="id" title="section">compose2Examples</span></a>.<br/>
+<span class="id" title="keyword">Variables</span> (<a id="compose2Examples.f" class="idref" href="#compose2Examples.f"><span class="id" title="variable">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 1) (<a id="compose2Examples.g" class="idref" href="#compose2Examples.g"><span class="id" title="variable">g</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="compose2Examples.h" class="idref" href="#compose2Examples.h"><span class="id" title="variable">h</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 3)<br/>
+&nbsp;&nbsp;(<a id="compose2Examples.f'" class="idref" href="#compose2Examples.f'"><span class="id" title="variable">f'</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 4) (<a id="compose2Examples.g'" class="idref" href="#compose2Examples.g'"><span class="id" title="variable">g'</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 5).<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.g"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 2 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.g"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.h"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">simpl</span> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.g'"><span class="id" title="variable">g'</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#compose2Examples"><span class="id" title="section">compose2Examples</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MoreExamples" class="idref" href="#MoreExamples"><span class="id" title="module">MoreExamples</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The unary constant function which returns 0 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="MoreExamples.cst0" class="idref" href="#MoreExamples.cst0"><span class="id" title="definition">cst0</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1 := (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PRnil"><span class="id" title="constructor">PRnil</span></a> <span class="id" title="var">_</span>))%<span class="id" title="var">pr</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The unary constant function which returns i 
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="MoreExamples.cst" class="idref" href="#MoreExamples.cst"><span class="id" title="definition">cst</span></a> (<a id="i:36" class="idref" href="#i:36"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1 :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:36"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.cst0"><span class="id" title="definition">cst0</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.PrimRecExamples.html#cst:37"><span class="id" title="definition">cst</span></a> <span class="id" title="var">j</span><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a>)%<span class="id" title="var">pr</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Addition 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="MoreExamples.plus" class="idref" href="#MoreExamples.plus"><span class="id" title="definition">plus</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 2 := <br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRrec"><span class="id" title="abbreviation">PRrec</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi1_1"><span class="id" title="abbreviation">pi1_1</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi2_3"><span class="id" title="abbreviation">pi2_3</span></a><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a>))%<span class="id" title="var">pr</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Multiplication 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="MoreExamples.mult" class="idref" href="#MoreExamples.mult"><span class="id" title="definition">mult</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 2 :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRrec"><span class="id" title="abbreviation">PRrec</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.cst0"><span class="id" title="definition">cst0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi2_3"><span class="id" title="abbreviation">pi2_3</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi3_3"><span class="id" title="abbreviation">pi3_3</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">]</span></a>%<span class="id" title="var">pr</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Factorial function 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="MoreExamples.fact" class="idref" href="#MoreExamples.fact"><span class="id" title="definition">fact</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1 := <br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRrec"><span class="id" title="abbreviation">PRrec</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.mult"><span class="id" title="definition">mult</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi2_2"><span class="id" title="abbreviation">pi2_2</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">;</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#succFunc"><span class="id" title="constructor">succFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi1_2"><span class="id" title="abbreviation">pi1_2</span></a><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a><a class="idref" href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74"><span class="id" title="notation">]</span></a>))%<span class="id" title="var">pr</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples"><span class="id" title="module">MoreExamples</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#MoreExamples"><span class="id" title="module">MoreExamples</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cst0_correct" class="idref" href="#cst0_correct"><span class="id" title="lemma">cst0_correct</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#cst0"><span class="id" title="definition">cst0</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ 0).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cst_correct" class="idref" href="#cst_correct"><span class="id" title="lemma">cst_correct</span></a> (<a id="k:40" class="idref" href="#k:40"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :  <a class="idref" href="hydras.Ackermann.primRec.html#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#cst"><span class="id" title="definition">cst</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#k:40"><span class="id" title="variable">k</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#k:40"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_correct" class="idref" href="#plus_correct"><span class="id" title="lemma">plus_correct</span></a>:  <a class="idref" href="hydras.Ackermann.primRec.html#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.add"><span class="id" title="definition">Nat.add</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="mult_eqn1" class="idref" href="#mult_eqn1"><span class="id" title="lemma">mult_eqn1</span></a> <a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a> <a id="p:42" class="idref" href="#p:42"><span class="id" title="binder">p</span></a>: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#mult"><span class="id" title="definition">mult</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:41"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#p:42"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#plus"><span class="id" title="definition">plus</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#mult"><span class="id" title="definition">mult</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:41"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#p:42"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#p:42"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_correct" class="idref" href="#mult_correct"><span class="id" title="lemma">mult_correct</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#mult"><span class="id" title="definition">mult</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.mul"><span class="id" title="definition">Nat.mul</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fact_correct" class="idref" href="#fact_correct"><span class="id" title="lemma">fact_correct</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PRcorrect"><span class="id" title="definition">PRcorrect</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#fact"><span class="id" title="definition">fact</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Factorial.html#fact"><span class="id" title="definition">Coq.Arith.Factorial.fact</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab331"></a><h2 class="section">Understanding some constructions ...</h2>
+
+
+<div class="paragraph"> </div>
+
+ These lemmas are trivial and theoretically useless, but they may help to 
+   make the construction more   concrete 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="PRcompose1" class="idref" href="#PRcompose1"><span class="id" title="definition">PRcompose1</span></a> (<a id="g:45" class="idref" href="#g:45"><span class="id" title="binder">g</span></a> <a id="f:46" class="idref" href="#f:46"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1) : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1 := <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:45"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:46"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a>%<span class="id" title="var">pr</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">∀</span></a> <a id="f:47" class="idref" href="#f:47"><span class="id" title="binder">f</span></a> <a id="g:48" class="idref" href="#g:48"><span class="id" title="binder">g</span></a> <a id="x:49" class="idref" href="#x:49"><span class="id" title="binder">x</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">evalPrimRec</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">1</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">(</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">PRcompose1</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">f</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">x</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">evalPrimRec</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">1</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">(</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">evalPrimRec</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">1</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">x</span></a><a id="Unnamed_thm2" class="idref" href="#Unnamed_thm2"><span class="id" title="definition">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="compose2_0" class="idref" href="#compose2_0"><span class="id" title="lemma">compose2_0</span></a> (<a id="a:50" class="idref" href="#a:50"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:51" class="idref" href="#g:51"><span class="id" title="binder">g</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:51"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:51"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:50"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="compose2_1" class="idref" href="#compose2_1"><span class="id" title="lemma">compose2_1</span></a> (<a id="f:52" class="idref" href="#f:52"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:53" class="idref" href="#g:53"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="x:54" class="idref" href="#x:54"><span class="id" title="binder">x</span></a><br/>
+&nbsp;&nbsp;: <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:53"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:54"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:53"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:52"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:54"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:54"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="compose2_2" class="idref" href="#compose2_2"><span class="id" title="lemma">compose2_2</span></a>  (<a id="f:55" class="idref" href="#f:55"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="g:56" class="idref" href="#g:56"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 3) <a id="x:57" class="idref" href="#x:57"><span class="id" title="binder">x</span></a> <a id="y:58" class="idref" href="#y:58"><span class="id" title="binder">y</span></a><br/>
+&nbsp;&nbsp;: <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 2 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:56"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:57"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:58"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:56"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:57"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:58"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:57"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:58"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="compose2_3" class="idref" href="#compose2_3"><span class="id" title="lemma">compose2_3</span></a>  (<a id="f:59" class="idref" href="#f:59"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 3) (<a id="g:60" class="idref" href="#g:60"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 4) <a id="x:61" class="idref" href="#x:61"><span class="id" title="binder">x</span></a> <a id="y:62" class="idref" href="#y:62"><span class="id" title="binder">y</span></a> <a id="z:63" class="idref" href="#z:63"><span class="id" title="binder">z</span></a> <br/>
+&nbsp;&nbsp;: <a class="idref" href="hydras.Ackermann.primRec.html#compose2"><span class="id" title="definition">compose2</span></a> 3 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:59"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:60"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:62"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#z:63"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:60"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:59"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:62"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#z:63"><span class="id" title="variable">z</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:62"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#z:63"><span class="id" title="variable">z</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_0_0" class="idref" href="#PrimRec_0_0"><span class="id" title="lemma">PrimRec_0_0</span></a> (<a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="g:65" class="idref" href="#g:65"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:65"><span class="id" title="variable">g</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:64"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_0_S" class="idref" href="#PrimRec_0_S"><span class="id" title="lemma">PrimRec_0_S</span></a> (<a id="a:66" class="idref" href="#a:66"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="g:67" class="idref" href="#g:67"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="i:68" class="idref" href="#i:68"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="phi:69" class="idref" href="#phi:69"><span class="id" title="binder">phi</span></a> :=  <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:67"><span class="id" title="variable">g</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:69"><span class="id" title="variable">phi</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:68"><span class="id" title="variable">i</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:67"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:68"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:69"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:68"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_1_0" class="idref" href="#PrimRec_1_0"><span class="id" title="lemma">PrimRec_1_0</span></a> (<a id="f:70" class="idref" href="#f:70"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="g:71" class="idref" href="#g:71"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:72" class="idref" href="#x:72"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:71"><span class="id" title="variable">g</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:72"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:72"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_1_S" class="idref" href="#PrimRec_1_S"><span class="id" title="lemma">PrimRec_1_S</span></a> (<a id="f:73" class="idref" href="#f:73"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:74" class="idref" href="#g:74"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="i:75" class="idref" href="#i:75"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="phi:76" class="idref" href="#phi:76"><span class="id" title="binder">phi</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 1  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:74"><span class="id" title="variable">g</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span> <span class="id" title="keyword">∀</span> <a id="x:77" class="idref" href="#x:77"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:76"><span class="id" title="variable">phi</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:75"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:74"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:75"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:76"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:75"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:77"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:77"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_2_0" class="idref" href="#PrimRec_2_0"><span class="id" title="lemma">PrimRec_2_0</span></a> (<a id="f:78" class="idref" href="#f:78"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="g:79" class="idref" href="#g:79"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 4) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:80" class="idref" href="#x:80"><span class="id" title="binder">x</span></a> <a id="y:81" class="idref" href="#y:81"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 2 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:78"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:79"><span class="id" title="variable">g</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:81"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:78"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:80"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:81"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="PrimRec_2_S" class="idref" href="#PrimRec_2_S"><span class="id" title="lemma">PrimRec_2_S</span></a>  (<a id="f:82" class="idref" href="#f:82"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2) (<a id="g:83" class="idref" href="#g:83"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 4) (<a id="i:84" class="idref" href="#i:84"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="phi:85" class="idref" href="#phi:85"><span class="id" title="binder">phi</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#evalPrimRecFunc"><span class="id" title="definition">evalPrimRecFunc</span></a> 2  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#f:82"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:83"><span class="id" title="variable">g</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="tactic">in</span>  <span class="id" title="keyword">∀</span> <a id="x:86" class="idref" href="#x:86"><span class="id" title="binder">x</span></a> <a id="y:87" class="idref" href="#y:87"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:85"><span class="id" title="variable">phi</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:84"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:87"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#g:83"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:84"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#phi:85"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#i:84"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:87"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:87"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab332"></a><h1 class="section">First proofs of isPR statements</h1>
+
+
+<div class="paragraph"> </div>
+
+  The module <span class="inlinecode"><span class="id" title="var">Alt</span></span> presents proofs of lemmas already proven in <span class="inlinecode"><span class="id" title="var">primRec.v</span></span>
+  We just hope that such a redundancy will help the reader to get familiar 
+  with the various patterns allowed by that library.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Alt" class="idref" href="#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.zeroIsPR" class="idref" href="#Alt.zeroIsPR"><span class="id" title="instance">zeroIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 0 0.  
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.succIsPR" class="idref" href="#Alt.succIsPR"><span class="id" title="instance">succIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.addIsPR" class="idref" href="#Alt.addIsPR"><span class="id" title="instance">addIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.add"><span class="id" title="definition">Nat.add</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.pi2_5IsPR" class="idref" href="#Alt.pi2_5IsPR"><span class="id" title="instance">pi2_5IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 5 (<span class="id" title="keyword">fun</span> <a id="a:88" class="idref" href="#a:88"><span class="id" title="binder">a</span></a> <a id="b:89" class="idref" href="#b:89"><span class="id" title="binder">b</span></a> <a id="c:90" class="idref" href="#c:90"><span class="id" title="binder">c</span></a> <a id="d:91" class="idref" href="#d:91"><span class="id" title="binder">d</span></a> <a id="e:92" class="idref" href="#e:92"><span class="id" title="binder">e</span></a> ⇒ <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#b:89"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#composeFunc"><span class="id" title="constructor">composeFunc</span></a> 0 1.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Alt.compose_01" class="idref" href="#Alt.compose_01"><span class="id" title="lemma">compose_01</span></a>  (<a id="x:93" class="idref" href="#x:93"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 0) (<a id="t:94" class="idref" href="#t:94"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#PrimRec"><span class="id" title="inductive">PrimRec</span></a> 1) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.PRcomp"><span class="id" title="abbreviation">PRcomp</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#t:94"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:93"><span class="id" title="variable">x</span></a><a class="idref" href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e"><span class="id" title="notation">]</span></a>)%<span class="id" title="var">pr</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#t:94"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#PReval"><span class="id" title="abbreviation">PReval</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:93"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span>  <a id="Alt.const0_NIsPR" class="idref" href="#Alt.const0_NIsPR"><span class="id" title="instance">const0_NIsPR</span></a> <a id="n:95" class="idref" href="#n:95"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:95"><span class="id" title="variable">n</span></a>.  
+<br/>
+<span class="id" title="keyword">Search</span> (<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#isPRextEqual"><span class="id" title="instance">isPRextEqual</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#filter010IsPR"><span class="id" title="instance">filter010IsPR</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Alt.add'" class="idref" href="#Alt.add'"><span class="id" title="definition">add'</span></a> <a id="x:96" class="idref" href="#x:96"><span class="id" title="binder">x</span></a> <a id="y:97" class="idref" href="#y:97"><span class="id" title="binder">y</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <a id="n:100" class="idref" href="#n:100"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:97"><span class="id" title="variable">y</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="z:98" class="idref" href="#z:98"><span class="id" title="binder">z</span></a> <a id="t:99" class="idref" href="#t:99"><span class="id" title="binder">t</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#t:99"><span class="id" title="variable">t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:96"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Alt.add'_ok" class="idref" href="#Alt.add'_ok"><span class="id" title="lemma">add'_ok</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 2 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Alt.add'"><span class="id" title="definition">add'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.add"><span class="id" title="definition">Nat.add</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.addIsPR'" class="idref" href="#Alt.addIsPR'"><span class="id" title="instance">addIsPR'</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.add"><span class="id" title="definition">Nat.add</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Alt.xpred" class="idref" href="#Alt.xpred"><span class="id" title="definition">xpred</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#primRecFunc"><span class="id" title="constructor">primRecFunc</span></a> 0 <a class="idref" href="hydras.Ackermann.primRec.html#zeroFunc"><span class="id" title="constructor">zeroFunc</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#PRNotations.pi1_2"><span class="id" title="abbreviation">pi1_2</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="Alt.predIsPR" class="idref" href="#Alt.predIsPR"><span class="id" title="instance">predIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a>.  
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Alt"><span class="id" title="module">Alt</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="double" class="idref" href="#double"><span class="id" title="definition">double</span></a> (<a id="n:101" class="idref" href="#n:101"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:101"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="doubleIsPR" class="idref" href="#doubleIsPR"><span class="id" title="instance">doubleIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#double"><span class="id" title="definition">double</span></a>.  
+<br/>
+</div>
+
+<div class="doc">
+using cPair 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Exs" class="idref" href="#Exs"><span class="id" title="section">Exs</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a id="Exs.f" class="idref" href="#Exs.f"><span class="id" title="variable">f</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2 := <span class="id" title="keyword">fun</span> <a id="x:102" class="idref" href="#x:102"><span class="id" title="binder">x</span></a> <a id="y:103" class="idref" href="#y:103"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#x:102"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#y:103"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Exs.ffib" class="idref" href="#Exs.ffib"><span class="id" title="variable">ffib</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2 :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="c:104" class="idref" href="#c:104"><span class="id" title="binder">c</span></a> <a id="A:105" class="idref" href="#A:105"><span class="id" title="binder">A</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#c:104"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 | 1 ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> 0 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#A:105"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#A:105"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Exs.fdiv2" class="idref" href="#Exs.fdiv2"><span class="id" title="variable">fdiv2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2 :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="n:107" class="idref" href="#n:107"><span class="id" title="binder">n</span></a> <a id="acc:108" class="idref" href="#acc:108"><span class="id" title="binder">acc</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#n:107"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 | 1 ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeNth"><span class="id" title="definition">codeNth</span></a> 1 <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#acc:108"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.MoreAck.PrimRecExamples.html#Exs"><span class="id" title="section">Exs</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,89 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.MoreAck.expressibleExamples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.MoreAck.expressibleExamples</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.expressible.html#"><span class="id" title="library">expressible</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.NN.html#"><span class="id" title="library">NN</span></a> <a class="idref" href="hydras.Ackermann.NewNotations.html#"><span class="id" title="library">NewNotations</span></a> <a class="idref" href="hydras.Ackermann.NNtheory.html#"><span class="id" title="library">NNtheory</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.LNN.html#NNnotations"><span class="id" title="module">NNnotations</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">RepresentableHalf1</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">NN</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">v</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">#0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">natToTerm</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)%</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">fol</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a>:   <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:2"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a>:  <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf2"><span class="id" title="definition">RepresentableHalf2</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:3"><span class="id" title="variable">n</span></a>) <br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L3" class="idref" href="#L3"><span class="id" title="lemma">L3</span></a>:  <a class="idref" href="hydras.Ackermann.expressible.html#Representable"><span class="id" title="definition">Representable</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:4"><span class="id" title="variable">n</span></a>)   (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L4" class="idref" href="#L4"><span class="id" title="lemma">L4</span></a> : <a class="idref" href="hydras.Ackermann.expressible.html#Representable"><span class="id" title="definition">Representable</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 1 (<span class="id" title="keyword">fun</span> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 2)  (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1 <a class="idref" href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Mind the order of variables <span class="inlinecode"><span class="id" title="var">v</span>#1</span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" title="var">v</span>#<span class="id" title="var">n</span></span>  
+ <span class="inlinecode"><span class="id" title="var">f</span></span> <span class="inlinecode"><span class="id" title="var">v</span>#<span class="id" title="var">n</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" title="var">v</span>#1</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">v</span>#0</span>
+
+<div class="paragraph"> </div>
+
+ 
+<div class="paragraph"> </div>
+
+ <span class="inlinecode"><span class="id" title="var">v</span>#2</span> is bound to n 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L5" class="idref" href="#L5"><span class="id" title="lemma">L5</span></a>:   <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="n:8" class="idref" href="#n:8"><span class="id" title="binder">n</span></a> <a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#n:8"><span class="id" title="variable">n</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>2)%<span class="id" title="var">fol</span>.<br/>
+ <span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> . &nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Search</span> (<a class="idref" href="hydras.Ackermann.LNN.html#SysPrf"><span class="id" title="definition">SysPrf</span></a> <span class="id" title="var">_</span> (?<span class="id" title="var">A</span> <a class="idref" href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> ?<span class="id" title="var">A</span>)%<span class="id" title="var">fol</span>).<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProp.html#freeVarT"><span class="id" title="definition">freeVarT</span></a> (<a class="idref" href="hydras.Ackermann.LNN.html#NNnotations.S_"><span class="id" title="abbreviation">S_</span></a> <span class="id" title="var">_</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">v</span>#1</span> is bound to p 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="L6" class="idref" href="#L6"><span class="id" title="lemma">L6</span></a>:   <a class="idref" href="hydras.Ackermann.expressible.html#RepresentableHalf1"><span class="id" title="definition">RepresentableHalf1</span></a> <a class="idref" href="hydras.Ackermann.NN.html#NN"><span class="id" title="definition">NN</span></a> 2 (<span class="id" title="keyword">fun</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a> ⇒ <a class="idref" href="hydras.MoreAck.expressibleExamples.html#p:11"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>0 <a class="idref" href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">v</span></a><a class="idref" href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71"><span class="id" title="notation">#</span></a>1)%<span class="id" title="var">fol</span>.<br/>
+ <span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.folProp.html#substT"><span class="id" title="definition">substT</span></a> .<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html b/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html
new file mode 100644
index 00000000..693cd941
--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html
@@ -0,0 +1,70 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.Example_3PlusOmega</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.Example_3PlusOmega</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+A proof of isomorhism between ordinal notations for 3+omega and omega 
+<div class="paragraph"> </div>
+
+ Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#"><span class="id" title="library">ON_plus</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#"><span class="id" title="library">ON_Finite</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#"><span class="id" title="library">ON_Omega</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="module">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="O3O" class="idref" href="#O3O"><span class="id" title="definition">O3O</span></a>  :=  <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#ON_plus"><span class="id" title="instance">ON_plus</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#FinOrd"><span class="id" title="instance">FinOrd</span></a> 3) <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a>.<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">O3O</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">lt</span> <span class="id" title="var">cmp</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Program Definition</span> <a id="f" class="idref" href="#f"><span class="id" title="definition">f</span></a> (<a id="z:1" class="idref" href="#z:1"><span class="id" title="binder">z</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#O3O"><span class="id" title="definition">O3O</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#z:1"><span class="id" title="variable">z</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">i</span> ⇒ <span class="id" title="var">i</span> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">j</span> ⇒ 3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><span class="id" title="var">j</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Program Definition</span> <a id="g" class="idref" href="#g"><span class="id" title="definition">g</span></a> (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#O3O"><span class="id" title="definition">O3O</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_lt_dec"><span class="id" title="definition">le_lt_dec</span></a> 3 <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#a:3"><span class="id" title="variable">a</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> (<a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> 3)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#a:3"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="L_3_plus_omega" class="idref" href="#L_3_plus_omega"><span class="id" title="instance">L_3_plus_omega</span></a> :  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_Iso"><span class="id" title="class">ON_Iso</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#f"><span class="id" title="definition">f</span></a> <a class="idref" href="hydras.OrdinalNotations.Example_3PlusOmega.html#g"><span class="id" title="definition">g</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.ON_Finite.html b/theories/html/hydras.OrdinalNotations.ON_Finite.html
new file mode 100644
index 00000000..fb6ee64d
--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.ON_Finite.html
@@ -0,0 +1,185 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_Finite</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_Finite</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ A notation system for finite ordinals  Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">Sortclass</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.ltb"><span class="id" title="definition">Nat.ltb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#i:2"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:1"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> {<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:3"><span class="id" title="variable">n</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:4" class="idref" href="#alpha:4"><span class="id" title="binder">alpha</span></a> <a id="beta:5" class="idref" href="#beta:5"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.ltb"><span class="id" title="definition">Nat.ltb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:4"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:5"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="t0_empty" class="idref" href="#t0_empty"><span class="id" title="lemma">t0_empty</span></a> (<a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> 0): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bad" class="idref" href="#bad"><span class="id" title="definition">bad</span></a> :  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> 10. <span class="id" title="keyword">Abort</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="compare_fin" class="idref" href="#compare_fin"><span class="id" title="instance">compare_fin</span></a> {<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:7"><span class="id" title="variable">n</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> <a id="beta:9" class="idref" href="#beta:9"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <span class="id" title="var">n</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:8"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:9"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> {<a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a>} (<a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> <a id="beta:12" class="idref" href="#beta:12"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:10"><span class="id" title="variable">n</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:12"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:12"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> {<a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> <a id="beta:15" class="idref" href="#beta:15"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:13"><span class="id" title="variable">n</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:15"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:14"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:15"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:15"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:15"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:14"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_wf" class="idref" href="#lt_wf"><span class="id" title="lemma">lt_wf</span></a> (<a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:16"><span class="id" title="variable">n</span></a>). 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="sto" class="idref" href="#sto"><span class="id" title="instance">sto</span></a> <a id="n:21" class="idref" href="#n:21"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:21"><span class="id" title="variable">n</span></a>). 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="comp" class="idref" href="#comp"><span class="id" title="instance">comp</span></a> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:22"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>. 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="FinOrd" class="idref" href="#FinOrd"><span class="id" title="instance">FinOrd</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:23"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_limit_succ_dec" class="idref" href="#Zero_limit_succ_dec"><span class="id" title="definition">Zero_limit_succ_dec</span></a> (<a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec"><span class="id" title="definition">ZeroLimitSucc_dec</span></a> (<span class="id" title="var">on</span> := <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#FinOrd"><span class="id" title="instance">FinOrd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:24"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sig_eq_intro" class="idref" href="#sig_eq_intro"><span class="id" title="lemma">sig_eq_intro</span></a> {<a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> <a id="y:27" class="idref" href="#y:27"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#n:25"><span class="id" title="variable">n</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#y:27"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#y:27"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Inclusion_ij" class="idref" href="#Inclusion_ij"><span class="id" title="section">Inclusion_ij</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Inclusion_ij.i" class="idref" href="#Inclusion_ij.i"><span class="id" title="variable">i</span></a> <a id="Inclusion_ij.j" class="idref" href="#Inclusion_ij.j"><span class="id" title="variable">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Inclusion_ij.Hij" class="idref" href="#Inclusion_ij.Hij"><span class="id" title="variable">Hij</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="Ltb_ij" class="idref" href="#Ltb_ij"><span class="id" title="lemma">Ltb_ij</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.ltb"><span class="id" title="definition">Nat.ltb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">program</span>] <span class="id" title="keyword">Definition</span> <a id="iota_ij" class="idref" href="#iota_ij"><span class="id" title="definition">iota_ij</span></a>  (<a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i"><span class="id" title="variable">i</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j"><span class="id" title="variable">j</span></a> :=  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:31"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Inclusion_ij.b" class="idref" href="#Inclusion_ij.b"><span class="id" title="variable">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j"><span class="id" title="variable">j</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Ltb_ij"><span class="id" title="lemma">Ltb_ij</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>]<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="F_incl_ij" class="idref" href="#F_incl_ij"><span class="id" title="instance">F_incl_ij</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#FinOrd"><span class="id" title="instance">FinOrd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#FinOrd"><span class="id" title="instance">FinOrd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.b"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_compare_commute" class="idref" href="#iota_compare_commute"><span class="id" title="lemma">iota_compare_commute</span></a> <a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> <a id="beta:33" class="idref" href="#beta:33"><span class="id" title="binder">beta</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:33"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:32"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:33"><span class="id" title="variable">beta</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_mono" class="idref" href="#iota_mono"><span class="id" title="lemma">iota_mono</span></a>  : <span class="id" title="keyword">∀</span> <a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a> <a id="beta:35" class="idref" href="#beta:35"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:34"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:35"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha:34"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta:35"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij"><span class="id" title="section">Inclusion_ij</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a> {<span class="id" title="var">i</span> <span class="id" title="var">j</span>}.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Program Example</span> <a id="alpha1" class="idref" href="#alpha1"><span class="id" title="definition">alpha1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> 7 := 2.<br/>
+
+<br/>
+<span class="id" title="keyword">Program Example</span> <a id="beta1" class="idref" href="#beta1"><span class="id" title="definition">beta1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> 7 := 5.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="i1" class="idref" href="#i1"><span class="id" title="definition">i1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha1"><span class="id" title="definition">alpha1</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta1"><span class="id" title="definition">beta1</span></a>.   
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Program Example</span> <a id="gamma1" class="idref" href="#gamma1"><span class="id" title="definition">gamma1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#t"><span class="id" title="definition">t</span></a> 8 := 7.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="i2" class="idref" href="#i2"><span class="id" title="definition">i2</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#iota_ij"><span class="id" title="definition">iota_ij</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le_n"><span class="id" title="constructor">le_n</span></a> 8) <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha1"><span class="id" title="definition">alpha1</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#gamma1"><span class="id" title="definition">gamma1</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a>  (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#bigO"><span class="id" title="definition">bigO</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#beta1"><span class="id" title="definition">beta1</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#alpha1"><span class="id" title="definition">alpha1</span></a>.<br/>
+ </div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_Generic</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_Generic</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Castéran, Univ. Bordeaux and LaBRI 
+<div class="paragraph"> </div>
+
+ This module defines a type class of ordinal notations, <i>i.e._, data types
+    with a well-founded strict order  and a <span class="inlinecode"><span class="id" title="var">compare</span></span> function 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Prelude.OrdNotations.html#"><span class="id" title="library">OrdNotations</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#"><span class="id" title="module">Relation_Definitions</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreOrders.html#"><span class="id" title="library">MoreOrders</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.Comparable.html#"><span class="id" title="library">Comparable</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">on</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+  Ordinal notation system on type <span class="inlinecode"><span class="id" title="var">A</span></span> :
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="ON" class="idref" href="#ON"><span class="id" title="record">ON</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="lt:2" class="idref" href="#lt:2"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="cmp:3" class="idref" href="#cmp:3"><span class="id" title="binder">cmp</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;<a id="ON_comp" class="idref" href="#ON_comp"><span class="id" title="projection">ON_comp</span></a> :&gt; <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#lt:2"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#cmp:3"><span class="id" title="variable">cmp</span></a>;<br/>
+&nbsp;&nbsp;<a id="ON_wf" class="idref" href="#ON_wf"><span class="id" title="projection">ON_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#lt:2"><span class="id" title="variable">lt</span></a>;<br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="var">Existing</span> <span class="id" title="keyword">Instance</span> <span class="id" title="var">ON_comp</span>.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_wf"><span class="id" title="method">ON_wf</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_wf"><span class="id" title="method">:</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_wf"><span class="id" title="method">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_wf"><span class="id" title="method">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_wf"><span class="id" title="method">well_founded</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="rep" class="idref" href="#rep"><span class="id" title="definition">rep</span></a> {<a id="A:7" class="idref" href="#A:7"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="lt:8" class="idref" href="#lt:8"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:7"><span class="id" title="variable">A</span></a>} {<a id="cmp:9" class="idref" href="#cmp:9"><span class="id" title="binder">cmp</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:7"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="on:10" class="idref" href="#on:10"><span class="id" title="binder">on</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#lt:8"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#cmp:9"><span class="id" title="variable">cmp</span></a>) := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:7"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">Coercion</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">rep</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#rep"><span class="id" title="definition">Sortclass</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Definitions" class="idref" href="#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> {<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="lt:12" class="idref" href="#lt:12"><span class="id" title="binder">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:11"><span class="id" title="variable">A</span></a>} {<a id="cmp:13" class="idref" href="#cmp:13"><span class="id" title="binder">cmp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:11"><span class="id" title="variable">A</span></a>} {<a id="on:14" class="idref" href="#on:14"><span class="id" title="binder">on</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#lt:12"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#cmp:13"><span class="id" title="variable">cmp</span></a>}.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"] <span class="id" title="keyword">Definition</span> <a id="ON_t" class="idref" href="#ON_t"><span class="id" title="definition">ON_t</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"] <span class="id" title="keyword">Definition</span> <a id="ON_compare" class="idref" href="#ON_compare"><span class="id" title="definition">ON_compare</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a> := <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"] <span class="id" title="keyword">Definition</span> <a id="ON_lt" class="idref" href="#ON_lt"><span class="id" title="definition">ON_lt</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.lt"><span class="id" title="variable">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"] <span class="id" title="keyword">Definition</span> <a id="ON_le" class="idref" href="#ON_le"><span class="id" title="definition">ON_le</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> := <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.lt"><span class="id" title="variable">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="measure_lt" class="idref" href="#measure_lt"><span class="id" title="definition">measure_lt</span></a> {<a id="B:15" class="idref" href="#B:15"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="m:16" class="idref" href="#m:16"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:15"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:15"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="x:17" class="idref" href="#x:17"><span class="id" title="binder">x</span></a> <a id="y:18" class="idref" href="#y:18"><span class="id" title="binder">y</span></a> ⇒  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_lt"><span class="id" title="definition">ON_lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#m:16"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:17"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#m:16"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:18"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="wf_measure" class="idref" href="#wf_measure"><span class="id" title="lemma">wf_measure</span></a> {<a id="B:19" class="idref" href="#B:19"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="m:20" class="idref" href="#m:20"><span class="id" title="binder">m</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:19"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#measure_lt"><span class="id" title="definition">measure_lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#m:20"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ZeroLimitSucc_dec" class="idref" href="#ZeroLimitSucc_dec"><span class="id" title="definition">ZeroLimitSucc_dec</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:25" class="idref" href="#alpha:25"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:25"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:25"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:26" class="idref" href="#beta:26"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#beta:26"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The segment called <span class="inlinecode"><span class="id" title="var">O</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> in Schutte's book 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="bigO" class="idref" href="#bigO"><span class="id" title="definition">bigO</span></a> (<a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> := <span class="id" title="keyword">fun</span> <a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.A"><span class="id" title="variable">A</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:27"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="::ON_scope:x_'o&lt;'_x" class="idref" href="#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&quot;</span></a>o&lt;" := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_lt"><span class="id" title="definition">ON_lt</span></a> : <span class="id" title="var">ON_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="9e4e840fd05a61043174d8c81c890780" class="idref" href="#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">&quot;</span></a>o&lt;=" := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_le"><span class="id" title="definition">ON_le</span></a> : <span class="id" title="var">ON_scope</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="147b23beaf5fa5d05316eba5f8182d20" class="idref" href="#147b23beaf5fa5d05316eba5f8182d20"><span class="id" title="notation">&quot;</span></a>o?=" := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_compare"><span class="id" title="definition">ON_compare</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70) : <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">wf_measure</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The segment associated with nA is isomorphic to
+    the segment of ordinals strictly less than b 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span>  <a id="SubON" class="idref" href="#SubON"><span class="id" title="record">SubON</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="OA:32" class="idref" href="#OA:32"><span class="id" title="binder">OA</span></a>: @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:29" class="idref" href="#A:29"><span class="id" title="binder">A</span></a> <a id="ltA:30" class="idref" href="#ltA:30"><span class="id" title="binder">ltA</span></a> <a id="compareA:31" class="idref" href="#compareA:31"><span class="id" title="binder">compareA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="OB:36" class="idref" href="#OB:36"><span class="id" title="binder">OB</span></a>: @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:33" class="idref" href="#B:33"><span class="id" title="binder">B</span></a> <a id="ltB:34" class="idref" href="#ltB:34"><span class="id" title="binder">ltB</span></a> <a id="compareB:35" class="idref" href="#compareB:35"><span class="id" title="binder">compareB</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="alpha:37" class="idref" href="#alpha:37"><span class="id" title="binder">alpha</span></a>:  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:33"><span class="id" title="variable">B</span></a>) (<a id="iota:38" class="idref" href="#iota:38"><span class="id" title="binder">iota</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:29"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:33"><span class="id" title="variable">B</span></a>):=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="SubON_compare" class="idref" href="#SubON_compare"><span class="id" title="projection">SubON_compare</span></a>: <span class="id" title="keyword">∀</span> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a> <a id="y:41" class="idref" href="#y:41"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:29"><span class="id" title="variable">A</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#compareB:35"><span class="id" title="variable">compareB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:38"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:40"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:38"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:41"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#compareA:31"><span class="id" title="variable">compareA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:41"><span class="id" title="variable">y</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="SubON_incl" class="idref" href="#SubON_incl"><span class="id" title="projection">SubON_incl</span></a> : <span class="id" title="keyword">∀</span> <a id="x:43" class="idref" href="#x:43"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ltB:34"><span class="id" title="variable">ltB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:38"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:43"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:37"><span class="id" title="variable">alpha</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="SubON_onto" class="idref" href="#SubON_onto"><span class="id" title="projection">SubON_onto</span></a> : <span class="id" title="keyword">∀</span> <a id="y:45" class="idref" href="#y:45"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ltB:34"><span class="id" title="variable">ltB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:45"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:37"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:46" class="idref" href="#x:46"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:29"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:38"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:46"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:45"><span class="id" title="variable">y</span></a>}.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">OA</span></span> and <span class="inlinecode"><span class="id" title="var">OB</span></span> are order-isomporphic 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span>  <a id="ON_Iso" class="idref" href="#ON_Iso"><span class="id" title="record">ON_Iso</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="OA:51" class="idref" href="#OA:51"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:48" class="idref" href="#A:48"><span class="id" title="binder">A</span></a> <a id="ltA:49" class="idref" href="#ltA:49"><span class="id" title="binder">ltA</span></a> <a id="compareA:50" class="idref" href="#compareA:50"><span class="id" title="binder">compareA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="OB:55" class="idref" href="#OB:55"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:52" class="idref" href="#B:52"><span class="id" title="binder">B</span></a> <a id="ltB:53" class="idref" href="#ltB:53"><span class="id" title="binder">ltB</span></a> <a id="compareB:54" class="idref" href="#compareB:54"><span class="id" title="binder">compareB</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:56" class="idref" href="#f:56"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:48"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:52"><span class="id" title="variable">B</span></a>) (<a id="g:57" class="idref" href="#g:57"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:52"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:48"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="iso_compare" class="idref" href="#iso_compare"><span class="id" title="projection">iso_compare</span></a> :<span class="id" title="keyword">∀</span> <a id="x:59" class="idref" href="#x:59"><span class="id" title="binder">x</span></a> <a id="y:60" class="idref" href="#y:60"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:48"><span class="id" title="variable">A</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#compareB:54"><span class="id" title="variable">compareB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:56"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:59"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:56"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:60"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#compareA:50"><span class="id" title="variable">compareA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:59"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:60"><span class="id" title="variable">y</span></a>;<br/>
+&nbsp;&nbsp;<a id="iso_inv1" class="idref" href="#iso_inv1"><span class="id" title="projection">iso_inv1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:57"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:56"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:62"><span class="id" title="variable">a</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:62"><span class="id" title="variable">a</span></a>;<br/>
+&nbsp;&nbsp;<a id="iso_inv2" class="idref" href="#iso_inv2"><span class="id" title="projection">iso_inv2</span></a> : <span class="id" title="keyword">∀</span> <a id="b:64" class="idref" href="#b:64"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:56"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:57"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:64"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:64"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+OA is an ordinal notation for alpha (in Schutte's model) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="ON_correct" class="idref" href="#ON_correct"><span class="id" title="record">ON_correct</span></a> `(<a id="alpha:66" class="idref" href="#alpha:66"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="OA:70" class="idref" href="#OA:70"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:67" class="idref" href="#A:67"><span class="id" title="binder">A</span></a> <a id="ltA:68" class="idref" href="#ltA:68"><span class="id" title="binder">ltA</span></a> <a id="compareA:69" class="idref" href="#compareA:69"><span class="id" title="binder">compareA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="iota:71" class="idref" href="#iota:71"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;{ <a id="ON_correct_inj" class="idref" href="#ON_correct_inj"><span class="id" title="projection">ON_correct_inj</span></a> : <span class="id" title="keyword">∀</span> <a id="a:73" class="idref" href="#a:73"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:73"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:66"><span class="id" title="variable">alpha</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="ON_correct_onto" class="idref" href="#ON_correct_onto"><span class="id" title="projection">ON_correct_onto</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:75" class="idref" href="#beta:75"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#beta:75"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:66"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:76" class="idref" href="#b:76"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#beta:75"><span class="id" title="variable">beta</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="On_compare_spec" class="idref" href="#On_compare_spec"><span class="id" title="projection">On_compare_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="a:78" class="idref" href="#a:78"><span class="id" title="binder">a</span></a> <a id="b:79" class="idref" href="#b:79"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:67"><span class="id" title="variable">A</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#compareA:69"><span class="id" title="variable">compareA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:79"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:78"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:79"><span class="id" title="variable">b</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Datatypes.Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:79"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Datatypes.Gt</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:79"><span class="id" title="variable">b</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:71"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:78"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab1"></a><h2 class="section">Relative correctness of a constant or a function</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="SubON_same_cst" class="idref" href="#SubON_same_cst"><span class="id" title="definition">SubON_same_cst</span></a> `{<a id="OA:84" class="idref" href="#OA:84"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:81" class="idref" href="#A:81"><span class="id" title="binder">A</span></a> <a id="ltA:82" class="idref" href="#ltA:82"><span class="id" title="binder">ltA</span></a>  <a id="compareA:83" class="idref" href="#compareA:83"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:88" class="idref" href="#OB:88"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:85" class="idref" href="#B:85"><span class="id" title="binder">B</span></a> <a id="ltB:86" class="idref" href="#ltB:86"><span class="id" title="binder">ltB</span></a>  <a id="compareB:87" class="idref" href="#compareB:87"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:89" class="idref" href="#iota:89"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:81"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:85"><span class="id" title="variable">B</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="alpha:90" class="idref" href="#alpha:90"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:85"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:84"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:88"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:90"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:89"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:91" class="idref" href="#a:91"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:81"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="b:92" class="idref" href="#b:92"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:85"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:89"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:91"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:92"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="SubON_same_fun" class="idref" href="#SubON_same_fun"><span class="id" title="definition">SubON_same_fun</span></a> `{<a id="OA:96" class="idref" href="#OA:96"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:93" class="idref" href="#A:93"><span class="id" title="binder">A</span></a> <a id="ltA:94" class="idref" href="#ltA:94"><span class="id" title="binder">ltA</span></a>  <a id="compareA:95" class="idref" href="#compareA:95"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:100" class="idref" href="#OB:100"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:97" class="idref" href="#B:97"><span class="id" title="binder">B</span></a> <a id="ltB:98" class="idref" href="#ltB:98"><span class="id" title="binder">ltB</span></a>  <a id="compareB:99" class="idref" href="#compareB:99"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:101" class="idref" href="#iota:101"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:93"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:97"><span class="id" title="variable">B</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="alpha:102" class="idref" href="#alpha:102"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:97"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:96"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:100"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:102"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:101"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:103" class="idref" href="#f:103"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:93"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:93"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:104" class="idref" href="#g:104"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:97"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:97"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">∀</span> <a id="x:105" class="idref" href="#x:105"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:101"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:103"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:105"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:104"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:101"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:105"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="SubON_same_op" class="idref" href="#SubON_same_op"><span class="id" title="definition">SubON_same_op</span></a> `{<a id="OA:109" class="idref" href="#OA:109"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:106" class="idref" href="#A:106"><span class="id" title="binder">A</span></a> <a id="ltA:107" class="idref" href="#ltA:107"><span class="id" title="binder">ltA</span></a> <a id="compareA:108" class="idref" href="#compareA:108"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:113" class="idref" href="#OB:113"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:110" class="idref" href="#B:110"><span class="id" title="binder">B</span></a> <a id="ltB:111" class="idref" href="#ltB:111"><span class="id" title="binder">ltB</span></a>  <a id="compareB:112" class="idref" href="#compareB:112"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:114" class="idref" href="#iota:114"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:106"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:110"><span class="id" title="variable">B</span></a>} {<a id="alpha:115" class="idref" href="#alpha:115"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:110"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:109"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:113"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:115"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:114"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:116" class="idref" href="#f:116"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:106"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:106"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:106"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:117" class="idref" href="#g:117"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:110"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:110"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:110"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">∀</span> <a id="x:118" class="idref" href="#x:118"><span class="id" title="binder">x</span></a> <a id="y:119" class="idref" href="#y:119"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:114"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:116"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:118"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:119"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:117"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:114"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:118"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:114"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:119"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Correctness w.r.t. Schutte's model 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ON_cst_ok" class="idref" href="#ON_cst_ok"><span class="id" title="definition">ON_cst_ok</span></a>  {<a id="alpha:120" class="idref" href="#alpha:120"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} `{<a id="OA:124" class="idref" href="#OA:124"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:121" class="idref" href="#A:121"><span class="id" title="binder">A</span></a> <a id="ltA:122" class="idref" href="#ltA:122"><span class="id" title="binder">ltA</span></a> <a id="compareA:123" class="idref" href="#compareA:123"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:128" class="idref" href="#OB:128"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:125" class="idref" href="#B:125"><span class="id" title="binder">B</span></a> <a id="ltB:126" class="idref" href="#ltB:126"><span class="id" title="binder">ltB</span></a>  <a id="compareB:127" class="idref" href="#compareB:127"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:129" class="idref" href="#iota:129"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:121"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_correct"><span class="id" title="class">ON_correct</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:120"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:124"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:129"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:130" class="idref" href="#a:130"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:121"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="b:131" class="idref" href="#b:131"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:129"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:130"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:131"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ON_fun_ok" class="idref" href="#ON_fun_ok"><span class="id" title="definition">ON_fun_ok</span></a>  {<a id="alpha:132" class="idref" href="#alpha:132"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} `{<a id="OA:136" class="idref" href="#OA:136"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:133" class="idref" href="#A:133"><span class="id" title="binder">A</span></a> <a id="ltA:134" class="idref" href="#ltA:134"><span class="id" title="binder">ltA</span></a>   <a id="compareA:135" class="idref" href="#compareA:135"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:140" class="idref" href="#OB:140"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:137" class="idref" href="#B:137"><span class="id" title="binder">B</span></a> <a id="ltB:138" class="idref" href="#ltB:138"><span class="id" title="binder">ltB</span></a>   <a id="compareB:139" class="idref" href="#compareB:139"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:141" class="idref" href="#iota:141"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:133"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_correct"><span class="id" title="class">ON_correct</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:132"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:136"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:141"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:142" class="idref" href="#f:142"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:133"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:133"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:143" class="idref" href="#g:143"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:144" class="idref" href="#x:144"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:141"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:142"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:144"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:143"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:141"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:144"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ON_op_ok" class="idref" href="#ON_op_ok"><span class="id" title="definition">ON_op_ok</span></a>  {<a id="alpha:145" class="idref" href="#alpha:145"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} `{<a id="OA:149" class="idref" href="#OA:149"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:146" class="idref" href="#A:146"><span class="id" title="binder">A</span></a> <a id="ltA:147" class="idref" href="#ltA:147"><span class="id" title="binder">ltA</span></a>  <a id="compareA:148" class="idref" href="#compareA:148"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:153" class="idref" href="#OB:153"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:150" class="idref" href="#B:150"><span class="id" title="binder">B</span></a> <a id="ltB:151" class="idref" href="#ltB:151"><span class="id" title="binder">ltB</span></a> <a id="compareB:152" class="idref" href="#compareB:152"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="iota:154" class="idref" href="#iota:154"><span class="id" title="binder">iota</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:146"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_correct"><span class="id" title="class">ON_correct</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:145"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:149"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:154"><span class="id" title="variable">iota</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:155" class="idref" href="#f:155"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:146"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:146"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:146"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="g:156" class="idref" href="#g:156"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:157" class="idref" href="#x:157"><span class="id" title="binder">x</span></a> <a id="y:158" class="idref" href="#y:158"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:154"><span class="id" title="variable">iota</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:155"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:158"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:156"><span class="id" title="variable">g</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:154"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:157"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#iota:154"><span class="id" title="variable">iota</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:158"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Iso_same_cst" class="idref" href="#Iso_same_cst"><span class="id" title="definition">Iso_same_cst</span></a>  `{<a id="OA:162" class="idref" href="#OA:162"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:159" class="idref" href="#A:159"><span class="id" title="binder">A</span></a> <a id="ltA:160" class="idref" href="#ltA:160"><span class="id" title="binder">ltA</span></a>  <a id="compareA:161" class="idref" href="#compareA:161"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:166" class="idref" href="#OB:166"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:163" class="idref" href="#B:163"><span class="id" title="binder">B</span></a> <a id="ltB:164" class="idref" href="#ltB:164"><span class="id" title="binder">ltB</span></a>  <a id="compareB:165" class="idref" href="#compareB:165"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="f:167" class="idref" href="#f:167"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:159"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:163"><span class="id" title="variable">B</span></a>} {<a id="g:168" class="idref" href="#g:168"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:163"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:159"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_Iso"><span class="id" title="class">ON_Iso</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:162"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:166"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:167"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:168"><span class="id" title="variable">g</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:169" class="idref" href="#a:169"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:159"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="b:170" class="idref" href="#b:170"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:163"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:167"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:169"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:170"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Iso_same_fun" class="idref" href="#Iso_same_fun"><span class="id" title="definition">Iso_same_fun</span></a>  `{<a id="OA:174" class="idref" href="#OA:174"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:171" class="idref" href="#A:171"><span class="id" title="binder">A</span></a> <a id="ltA:172" class="idref" href="#ltA:172"><span class="id" title="binder">ltA</span></a>  <a id="compareA:173" class="idref" href="#compareA:173"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:178" class="idref" href="#OB:178"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:175" class="idref" href="#B:175"><span class="id" title="binder">B</span></a> <a id="ltB:176" class="idref" href="#ltB:176"><span class="id" title="binder">ltB</span></a>  <a id="compareB:177" class="idref" href="#compareB:177"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="f:179" class="idref" href="#f:179"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:171"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:175"><span class="id" title="variable">B</span></a>} {<a id="g:180" class="idref" href="#g:180"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:175"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:171"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_Iso"><span class="id" title="class">ON_Iso</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:174"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:178"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:179"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:180"><span class="id" title="variable">g</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fA:181" class="idref" href="#fA:181"><span class="id" title="binder">fA</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:171"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:171"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fB:182" class="idref" href="#fB:182"><span class="id" title="binder">fB</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:175"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:175"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:183" class="idref" href="#x:183"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:179"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#fA:181"><span class="id" title="variable">fA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:183"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#fB:182"><span class="id" title="variable">fB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:179"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:183"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Iso_same_op" class="idref" href="#Iso_same_op"><span class="id" title="definition">Iso_same_op</span></a>  `{<a id="OA:187" class="idref" href="#OA:187"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:184" class="idref" href="#A:184"><span class="id" title="binder">A</span></a> <a id="ltA:185" class="idref" href="#ltA:185"><span class="id" title="binder">ltA</span></a>  <a id="compareA:186" class="idref" href="#compareA:186"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:191" class="idref" href="#OB:191"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:188" class="idref" href="#B:188"><span class="id" title="binder">B</span></a> <a id="ltB:189" class="idref" href="#ltB:189"><span class="id" title="binder">ltB</span></a>  <a id="compareB:190" class="idref" href="#compareB:190"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="f:192" class="idref" href="#f:192"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:188"><span class="id" title="variable">B</span></a>} {<a id="g:193" class="idref" href="#g:193"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:188"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:184"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="var">_</span> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_Iso"><span class="id" title="class">ON_Iso</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:187"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:191"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#g:193"><span class="id" title="variable">g</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="opA:194" class="idref" href="#opA:194"><span class="id" title="binder">opA</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:184"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="opB:195" class="idref" href="#opB:195"><span class="id" title="binder">opB</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:188"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:188"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:188"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:196" class="idref" href="#x:196"><span class="id" title="binder">x</span></a> <a id="y:197" class="idref" href="#y:197"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:192"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#opA:194"><span class="id" title="variable">opA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:196"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:197"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#opB:195"><span class="id" title="variable">opB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#x:196"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#y:197"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="SubON_properties" class="idref" href="#SubON_properties"><span class="id" title="section">SubON_properties</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> `{<a id="OA:201" class="idref" href="#OA:201"><span class="id" title="binder">OA</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="A:198" class="idref" href="#A:198"><span class="id" title="binder">A</span></a> <a id="ltA:199" class="idref" href="#ltA:199"><span class="id" title="binder">ltA</span></a> <a id="compareA:200" class="idref" href="#compareA:200"><span class="id" title="binder">compareA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="OB:205" class="idref" href="#OB:205"><span class="id" title="binder">OB</span></a> : @<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a id="B:202" class="idref" href="#B:202"><span class="id" title="binder">B</span></a> <a id="ltB:203" class="idref" href="#ltB:203"><span class="id" title="binder">ltB</span></a> <a id="compareB:204" class="idref" href="#compareB:204"><span class="id" title="binder">compareB</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:206" class="idref" href="#f:206"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#A:198"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:202"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="alpha:207" class="idref" href="#alpha:207"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#B:202"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Su:208" class="idref" href="#Su:208"><span class="id" title="binder">Su</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OA:201"><span class="id" title="variable">OA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#OB:205"><span class="id" title="variable">OB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#alpha:207"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#f:206"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SubON_mono" class="idref" href="#SubON_mono"><span class="id" title="lemma">SubON_mono</span></a> <a id="a:209" class="idref" href="#a:209"><span class="id" title="binder">a</span></a> <a id="b:210" class="idref" href="#b:210"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.ltA"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:209"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:210"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.ltB"><span class="id" title="variable">ltB</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:209"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:210"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SubON_inj" class="idref" href="#SubON_inj"><span class="id" title="lemma">SubON_inj</span></a> : <span class="id" title="keyword">∀</span> <a id="a:211" class="idref" href="#a:211"><span class="id" title="binder">a</span></a> <a id="b:212" class="idref" href="#b:212"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:211"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:212"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:211"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:212"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SubON_successor" class="idref" href="#SubON_successor"><span class="id" title="lemma">SubON_successor</span></a> : <span class="id" title="keyword">∀</span> <a id="a:213" class="idref" href="#a:213"><span class="id" title="binder">a</span></a> <a id="b:214" class="idref" href="#b:214"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:213"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:214"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:213"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#b:214"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SubON_limit" class="idref" href="#SubON_limit"><span class="id" title="lemma">SubON_limit</span></a> : <span class="id" title="keyword">∀</span> <a id="a:215" class="idref" href="#a:215"><span class="id" title="binder">a</span></a> ,  <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:215"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:215"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="SubON_least" class="idref" href="#SubON_least"><span class="id" title="lemma">SubON_least</span></a> : <span class="id" title="keyword">∀</span> <a id="a:216" class="idref" href="#a:216"><span class="id" title="binder">a</span></a> ,  <a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:216"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#a:216"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties"><span class="id" title="section">SubON_properties</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_O</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_O</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Ordinal Notation for a segment <span class="inlinecode"><span class="id" title="var">O</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>  Pierre Castéran, Unviv. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">Sortclass</span></a>.<br/>
+<span class="id" title="keyword">Generalizable Variables</span> <span class="id" title="var">A</span> <span class="id" title="var">Lt</span> <span class="id" title="var">comp</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="OA_given" class="idref" href="#OA_given"><span class="id" title="section">OA_given</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="Lt:2" class="idref" href="#Lt:2"><span class="id" title="binder">Lt</span></a> <a id="Le:3" class="idref" href="#Le:3"><span class="id" title="binder">Le</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#A:1"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="cmpA:4" class="idref" href="#cmpA:4"><span class="id" title="binder">cmpA</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#A:1"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="OA:5" class="idref" href="#OA:5"><span class="id" title="binder">OA</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#Lt:2"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#cmpA:4"><span class="id" title="variable">cmpA</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The type of ordinals less than <span class="inlinecode"><span class="id" title="var">a</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> (<a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b:7" class="idref" href="#b:7"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#b:7"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> {<a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:8"><span class="id" title="variable">a</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_lt"><span class="id" title="definition">ON_lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:9"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:10"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="le" class="idref" href="#le"><span class="id" title="definition">le</span></a> {<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:11"><span class="id" title="variable">a</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:11"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>]<br/>
+<span class="id" title="keyword">Instance</span> <a id="compare_O" class="idref" href="#compare_O"><span class="id" title="instance">compare_O</span></a> {<a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>} : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:12"><span class="id" title="variable">a</span></a>) :=<br/>
+<span class="id" title="keyword">fun</span> (<a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <span class="id" title="var">a</span>) ⇒<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:13"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:14"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> {<a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a>} (<a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a> <a id="beta:17" class="idref" href="#beta:17"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:15"><span class="id" title="variable">a</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:16"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:17"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:16"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:17"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> {<a id="a:18" class="idref" href="#a:18"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>} (<a id="alpha:19" class="idref" href="#alpha:19"><span class="id" title="binder">alpha</span></a> <a id="beta:20" class="idref" href="#beta:20"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_O.html#t"><span class="id" title="definition">t</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:18"><span class="id" title="variable">a</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:20"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:19"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:20"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:20"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#beta:20"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_O.html#alpha:19"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_wf" class="idref" href="#lt_wf"><span class="id" title="lemma">lt_wf</span></a> (<a id="a:21" class="idref" href="#a:21"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:21"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="sto" class="idref" href="#sto"><span class="id" title="instance">sto</span></a> <a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:22"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="ON_O_comp" class="idref" href="#ON_O_comp"><span class="id" title="instance">ON_O_comp</span></a> (<a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:23"><span class="id" title="variable">a</span></a>)  <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> .<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+We have now an ordinal notation 
+</div>
+<div class="code">
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="ON_O" class="idref" href="#ON_O"><span class="id" title="instance">ON_O</span></a>  (<a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> (@<a class="idref" href="hydras.OrdinalNotations.ON_O.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#a:24"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_O.html#OA_given"><span class="id" title="section">OA_given</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega.html b/theories/html/hydras.OrdinalNotations.ON_Omega.html
new file mode 100644
index 00000000..508dc02d
--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.ON_Omega.html
@@ -0,0 +1,76 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_Omega</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_Omega</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ A notation system for the ordinal <span class="inlinecode"><span class="id" title="var">omega</span></span>   Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#"><span class="id" title="library">ON_Finite</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.Schutte.html#"><span class="id" title="library">Schutte</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="module">RelationClasses</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>]<br/>
+<span class="id" title="keyword">Instance</span> <a id="compare_nat" class="idref" href="#compare_nat"><span class="id" title="instance">compare_nat</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Omega_comp" class="idref" href="#Omega_comp"><span class="id" title="instance">Omega_comp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Omega" class="idref" href="#Omega"><span class="id" title="instance">Omega</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_limit_succ_dec" class="idref" href="#Zero_limit_succ_dec"><span class="id" title="definition">Zero_limit_succ_dec</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec"><span class="id" title="definition">ZeroLimitSucc_dec</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="FinOrd_Omega" class="idref" href="#FinOrd_Omega"><span class="id" title="instance">FinOrd_Omega</span></a> (<a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Finite.html#FinOrd"><span class="id" title="instance">FinOrd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#i:1"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#i:1"><span class="id" title="variable">i</span></a> (<span class="id" title="keyword">fun</span> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#alpha:2"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="omega_ok" class="idref" href="#omega_ok"><span class="id" title="instance">omega_ok</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_correct"><span class="id" title="class">ON_correct</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">Schutte_basics.omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">finite</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega2.html b/theories/html/hydras.OrdinalNotations.ON_Omega2.html
new file mode 100644
index 00000000..890557a8
--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.ON_Omega2.html
@@ -0,0 +1,383 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_Omega2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_Omega2</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ A notation system for the ordinal <span class="inlinecode"><span class="id" title="var">omega</span>^2</span>  
+<div class="paragraph"> </div>
+
+ Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_mult.html#"><span class="id" title="library">ON_mult</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#"><span class="id" title="library">ON_Omega</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="module">ON_Generic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="compare_omega2" class="idref" href="#compare_omega2"><span class="id" title="instance">compare_omega2</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#t"><span class="id" title="definition">ON_mult.t</span></a> :=<br/>
+&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_mult.html#compare_mult"><span class="id" title="instance">compare_mult</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Omega2" class="idref" href="#Omega2"><span class="id" title="instance">Omega2</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#compare_omega2"><span class="id" title="instance">compare_omega2</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#ON_mult"><span class="id" title="instance">ON_mult</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="omega" class="idref" href="#omega"><span class="id" title="abbreviation">omega</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="zero" class="idref" href="#zero"><span class="id" title="definition">zero</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fin" class="idref" href="#fin"><span class="id" title="definition">fin</span></a> (<a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:1"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">t</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex1" class="idref" href="#ex1"><span class="id" title="definition">ex1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>5<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>8<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>5<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>10<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_is_limit" class="idref" href="#omega_is_limit"><span class="id" title="lemma">omega_is_limit</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succ" class="idref" href="#succ"><span class="id" title="definition">succ</span></a> (<a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:2"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:2"><span class="id" title="variable">alpha</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eq_dec" class="idref" href="#eq_dec"><span class="id" title="lemma">eq_dec</span></a> (<a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> <a id="beta:4" class="idref" href="#beta:4"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:4"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:4"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_intror" class="idref" href="#le_intror"><span class="id" title="lemma">le_intror</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> <a id="j:6" class="idref" href="#j:6"><span class="id" title="binder">j</span></a> <a id="k:7" class="idref" href="#k:7"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:6"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#k:7"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:5"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:6"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:5"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#k:7"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_0" class="idref" href="#le_0"><span class="id" title="lemma">le_0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:8" class="idref" href="#p:8"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#p:8"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le" class="idref" href="#lt_succ_le"><span class="id" title="lemma">lt_succ_le</span></a> <a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:9"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:10"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:9"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:10"><span class="id" title="variable">beta</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ" class="idref" href="#lt_succ"><span class="id" title="lemma">lt_succ</span></a> <a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:11"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#lexico"><span class="id" title="inductive">lexico</span></a> : <span class="id" title="var">O2</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#le"><span class="id" title="definition">le</span></a> : <span class="id" title="var">O2</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> <a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a> <a id="beta:13" class="idref" href="#beta:13"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_compare"><span class="id" title="definition">ON_compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:13"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:13"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:12"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:13"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:13"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:12"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> <a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> <a id="beta:15" class="idref" href="#beta:15"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_lt"><span class="id" title="definition">ON_lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:15"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_compare"><span class="id" title="definition">ON_compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:15"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_le" class="idref" href="#zero_le"><span class="id" title="lemma">zero_le</span></a> <a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:16"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Least_zero" class="idref" href="#Least_zero"><span class="id" title="lemma">Least_zero</span></a> <a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_limit_succ_dec" class="idref" href="#Zero_limit_succ_dec"><span class="id" title="definition">Zero_limit_succ_dec</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec"><span class="id" title="definition">ZeroLimitSucc_dec</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="lemma">lt_eq_lt_dec</span></a> <a id="alpha:18" class="idref" href="#alpha:18"><span class="id" title="binder">alpha</span></a> <a id="beta:19" class="idref" href="#beta:19"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:19"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:19"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:19"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:18"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_inv" class="idref" href="#Successor_inv"><span class="id" title="lemma">Successor_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="i:20" class="idref" href="#i:20"><span class="id" title="binder">i</span></a> <a id="j:21" class="idref" href="#j:21"><span class="id" title="binder">j</span></a> <a id="k:22" class="idref" href="#k:22"><span class="id" title="binder">k</span></a> <a id="l:23" class="idref" href="#l:23"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#k:22"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#l:23"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:20"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:21"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:20"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#k:22"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#l:23"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:21"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Successor_not" class="idref" href="#Successor_not"><span class="id" title="lemma">Successor_not</span></a> (<a id="i:24" class="idref" href="#i:24"><span class="id" title="binder">i</span></a> <a id="j:25" class="idref" href="#j:25"><span class="id" title="binder">j</span></a> <a id="k:26" class="idref" href="#k:26"><span class="id" title="binder">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#k:26"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:24"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:25"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_succ" class="idref" href="#Successor_succ"><span class="id" title="lemma">Successor_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:27"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:27"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_ok" class="idref" href="#succ_ok"><span class="id" title="lemma">succ_ok</span></a> <a id="alpha:28" class="idref" href="#alpha:28"><span class="id" title="binder">alpha</span></a> <a id="beta:29" class="idref" href="#beta:29"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:29"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:28"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:29"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:28"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succb" class="idref" href="#succb"><span class="id" title="definition">succb</span></a> (<a id="alpha:30" class="idref" href="#alpha:30"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:30"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="limitb" class="idref" href="#limitb"><span class="id" title="definition">limitb</span></a> (<a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:32"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Omega_limit_limitb" class="idref" href="#Omega_limit_limitb"><span class="id" title="lemma">Omega_limit_limitb</span></a> <a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a> <a id="s:35" class="idref" href="#s:35"><span class="id" title="binder">s</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Omega_limit"><span class="id" title="definition">Omega_limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#s:35"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:34"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:34"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Canonical sequences 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="canon" class="idref" href="#canon"><span class="id" title="definition">canon</span></a>  <a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> <a id="i:37" class="idref" href="#i:37"><span class="id" title="binder">i</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:36"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:37"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limitb_limit" class="idref" href="#limitb_limit"><span class="id" title="lemma">limitb_limit</span></a> <a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Omega_limit"><span class="id" title="definition">Omega_limit</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:39"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:39"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_is_lub_0" class="idref" href="#limit_is_lub_0"><span class="id" title="lemma">limit_is_lub_0</span></a> : <span class="id" title="keyword">∀</span> <a id="i:40" class="idref" href="#i:40"><span class="id" title="binder">i</span></a> <a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="j:42" class="idref" href="#j:42"><span class="id" title="binder">j</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:40"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:42"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:41"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:40"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:41"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_is_lub" class="idref" href="#limit_is_lub"><span class="id" title="lemma">limit_is_lub</span></a> <a id="beta:43" class="idref" href="#beta:43"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:43"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#canon"><span class="id" title="definition">canon</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:43"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:45"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:44"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:43"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:44"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="zero_limit_succ_dec" class="idref" href="#zero_limit_succ_dec"><span class="id" title="definition">zero_limit_succ_dec</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:46" class="idref" href="#alpha:46"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:46"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:46"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:47" class="idref" href="#beta:47"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:46"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:47"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="plus" class="idref" href="#plus"><span class="id" title="definition">plus</span></a> (<a id="alpha:48" class="idref" href="#alpha:48"><span class="id" title="binder">alpha</span></a> <a id="beta:49" class="idref" href="#beta:49"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:48"><span class="id" title="variable">alpha</span></a>,<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:49"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <span class="id" title="var">y</span>  ⇒ <span class="id" title="var">y</span><br/>
+&nbsp;&nbsp;| <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>  ⇒ <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">b'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="a050364cdd242546ff34b16b41a206b9" class="idref" href="#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#plus"><span class="id" title="definition">plus</span></a> : <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_compat" class="idref" href="#plus_compat"><span class="id" title="lemma">plus_compat</span></a> (<a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> <a id="p:53" class="idref" href="#p:53"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">fin</span></a>  (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#p:53"><span class="id" title="variable">p</span></a> )%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#p:53"><span class="id" title="variable">p</span></a>.   
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="non_commutativity_of_plus" class="idref" href="#non_commutativity_of_plus"><span class="id" title="definition">non_commutativity_of_plus</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 3 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.   
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+multiplication of an ordinal by a natural number 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mult_fin_r" class="idref" href="#mult_fin_r"><span class="id" title="definition">mult_fin_r</span></a>  (<a id="alpha:54" class="idref" href="#alpha:54"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) (<a id="p:55" class="idref" href="#p:55"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:54"><span class="id" title="variable">alpha</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#p:55"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <span class="id" title="var">_</span>  ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">_</span>, 0 ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#p:55"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <span class="id" title="var">n'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a>  <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="4a4f2f510af1c6a5e896f63b5eead422" class="idref" href="#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#mult_fin_r"><span class="id" title="definition">mult_fin_r</span></a> : <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+multiplication of  a natural number by an ordinal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mult_fin_l" class="idref" href="#mult_fin_l"><span class="id" title="definition">mult_fin_l</span></a> (<a id="n:58" class="idref" href="#n:58"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="alpha:59" class="idref" href="#alpha:59"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:58"><span class="id" title="variable">n</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:59"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;|  0, <span class="id" title="var">_</span>  ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">_</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">n</span> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:58"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><span class="id" title="var">n'</span>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;|  <span class="id" title="var">n</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">p'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:58"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <span class="id" title="var">p'</span>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="e1" class="idref" href="#e1"><span class="id" title="definition">e1</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> 7 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 15<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> 21 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 15.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="e2" class="idref" href="#e2"><span class="id" title="definition">e2</span></a> :  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#mult_fin_l"><span class="id" title="definition">mult_fin_l</span></a> 3 (<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> 7 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 15) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> 7 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 45.<br/>
+ 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="plus_assoc" class="idref" href="#plus_assoc"><span class="id" title="instance">plus_assoc</span></a>: <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#plus"><span class="id" title="definition">plus</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_is_plus_1" class="idref" href="#succ_is_plus_1"><span class="id" title="lemma">succ_is_plus_1</span></a>  <a id="alpha:62" class="idref" href="#alpha:62"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:62"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:62"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega" class="idref" href="#lt_omega"><span class="id" title="lemma">lt_omega</span></a> <a id="alpha:63" class="idref" href="#alpha:63"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:64" class="idref" href="#n:64"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#n:64"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decompose" class="idref" href="#decompose"><span class="id" title="lemma">decompose</span></a> (<a id="i:65" class="idref" href="#i:65"><span class="id" title="binder">i</span></a> <a id="j:66" class="idref" href="#j:66"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:65"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:66"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:65"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:66"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unique_decomposition" class="idref" href="#unique_decomposition"><span class="id" title="lemma">unique_decomposition</span></a> (<a id="alpha:67" class="idref" href="#alpha:67"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="i:68" class="idref" href="#i:68"><span class="id" title="binder">i</span></a> <a id="j:69" class="idref" href="#j:69"><span class="id" title="binder">j</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:68"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:69"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab4"></a><h2 class="section">Additive principal ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ap" class="idref" href="#ap"><span class="id" title="definition">ap</span></a> (<a id="alpha:70" class="idref" href="#alpha:70"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:71" class="idref" href="#beta:71"><span class="id" title="binder">beta</span></a> <a id="gamma:72" class="idref" href="#gamma:72"><span class="id" title="binder">gamma</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:71"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#gamma:72"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#beta:71"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#gamma:72"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:70"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_ap" class="idref" href="#omega_ap"><span class="id" title="lemma">omega_ap</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#ap"><span class="id" title="definition">ap</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ap_cases" class="idref" href="#ap_cases"><span class="id" title="lemma">ap_cases</span></a> <a id="alpha:73" class="idref" href="#alpha:73"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#ap"><span class="id" title="definition">ap</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="L_3_plus_omega" class="idref" href="#L_3_plus_omega"><span class="id" title="definition">L_3_plus_omega</span></a> :  3 <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Equations</span>.<br/>
+<span class="id" title="keyword">Section</span> <a id="A_def" class="idref" href="#A_def"><span class="id" title="section">A_def</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="A_def.m" class="idref" href="#A_def.m"><span class="id" title="variable">m</span></a> (<a id="x:74" class="idref" href="#x:74"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#x:74"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#x:74"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">local</span> ] <span class="id" title="keyword">Instance</span> <a id="WF" class="idref" href="#WF"><span class="id" title="instance">WF</span></a> : <span class="id" title="class">WellFounded</span> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#measure_lt"><span class="id" title="definition">measure_lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A_def.m"><span class="id" title="variable">m</span></a>):=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#wf_measure"><span class="id" title="lemma">wf_measure</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A_def.m"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Equations</span> <a id="A" class="idref" href="#A"><span class="id" title="definition">A</span></a> (<a id="p:75" class="idref" href="#p:75"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span> <span class="id" title="var">p</span> (<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#measure_lt"><span class="id" title="definition">measure_lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A_def.m"><span class="id" title="variable">m</span></a>):=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:78"><span class="id" title="definition">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:77"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:80"><span class="id" title="definition">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:79"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:76"><span class="id" title="definition">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:83"><span class="id" title="definition">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#i:81"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#j:82"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:76"><span class="id" title="definition">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A:76"><span class="id" title="definition">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega2.html#A_def"><span class="id" title="section">A_def</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html b/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html
@@ -0,0 +1,221 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_Omega_plus_omega</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_Omega_plus_omega</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ A notation system for the ordinal <span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">omega</span></span>   Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Comparable.html#"><span class="id" title="library">Comparable</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_plus.html#"><span class="id" title="library">ON_plus</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#"><span class="id" title="library">ON_Omega</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">opo_scope</span>.<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">opo_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">opo</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ON_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">opo_scope</span>.<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="compare_nat_nat" class="idref" href="#compare_nat_nat"><span class="id" title="instance">compare_nat_nat</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_plus.html#compare_plus"><span class="id" title="instance">compare_plus</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#compare_nat"><span class="id" title="instance">compare_nat</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Omega_plus_Omega" class="idref" href="#Omega_plus_Omega"><span class="id" title="instance">Omega_plus_Omega</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#compare_nat_nat"><span class="id" title="instance">compare_nat_nat</span></a> :=<br/>
+&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_plus.html#ON_plus"><span class="id" title="instance">ON_plus</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex1" class="idref" href="#ex1"><span class="id" title="definition">ex1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> 7 <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> 8.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">opo_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fin" class="idref" href="#fin"><span class="id" title="definition">fin</span></a> (<a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:1"><span class="id" title="variable">i</span></a>.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="omega" class="idref" href="#omega"><span class="id" title="abbreviation">omega</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> 0:<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_t"><span class="id" title="definition">ON_t</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="ex2" class="idref" href="#ex2"><span class="id" title="definition">ex2</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> 7 <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_is_limit" class="idref" href="#omega_is_limit"><span class="id" title="lemma">omega_is_limit</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_is_omega" class="idref" href="#limit_is_omega"><span class="id" title="lemma">limit_is_omega</span></a> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="limit_iff" class="idref" href="#limit_iff"><span class="id" title="lemma">limit_iff</span></a> (<a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_inv1" class="idref" href="#Successor_inv1"><span class="id" title="lemma">Successor_inv1</span></a> : <span class="id" title="keyword">∀</span> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a> <a id="j:5" class="idref" href="#j:5"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:5"><span class="id" title="variable">j</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:4"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:5"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:4"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_inv2" class="idref" href="#Successor_inv2"><span class="id" title="lemma">Successor_inv2</span></a> : <span class="id" title="keyword">∀</span> <a id="i:6" class="idref" href="#i:6"><span class="id" title="binder">i</span></a> <a id="j:7" class="idref" href="#j:7"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:7"><span class="id" title="variable">j</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:6"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:7"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:6"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_inv3" class="idref" href="#Successor_inv3"><span class="id" title="lemma">Successor_inv3</span></a> : <span class="id" title="keyword">∀</span> <a id="i:8" class="idref" href="#i:8"><span class="id" title="binder">i</span></a> <a id="j:9" class="idref" href="#j:9"><span class="id" title="binder">j</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:9"><span class="id" title="variable">j</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:8"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_inv4" class="idref" href="#Successor_inv4"><span class="id" title="lemma">Successor_inv4</span></a> : <span class="id" title="keyword">∀</span> <a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a> <a id="j:11" class="idref" href="#j:11"><span class="id" title="binder">j</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:11"><span class="id" title="variable">j</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:10"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succ" class="idref" href="#succ"><span class="id" title="definition">succ</span></a> (<a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:12"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_succ" class="idref" href="#Successor_succ"><span class="id" title="lemma">Successor_succ</span></a> <a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> (<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:14"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:14"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_correct" class="idref" href="#succ_correct"><span class="id" title="lemma">succ_correct</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="beta:16" class="idref" href="#beta:16"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:16"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:16"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:15"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succb" class="idref" href="#succb"><span class="id" title="definition">succb</span></a> (<a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><br/>
+&nbsp;&nbsp;:= <span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:17"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>  <span class="id" title="var">_</span>) | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succb_correct" class="idref" href="#succb_correct"><span class="id" title="lemma">succb_correct</span></a> (<a id="alpha:19" class="idref" href="#alpha:19"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succb"><span class="id" title="definition">succb</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:20" class="idref" href="#beta:20"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:20"><span class="id" title="variable">beta</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_not_succ" class="idref" href="#omega_not_succ"><span class="id" title="lemma">omega_not_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:21" class="idref" href="#alpha:21"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:21"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Least_is_0" class="idref" href="#Least_is_0"><span class="id" title="lemma">Least_is_0</span></a> (<a id="alpha:22" class="idref" href="#alpha:22"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:22"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:22"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ZLS_dec" class="idref" href="#ZLS_dec"><span class="id" title="lemma">ZLS_dec</span></a> (<a id="alpha:23" class="idref" href="#alpha:23"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:23"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:23"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:23"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:24"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Zero_limit_succ_dec" class="idref" href="#Zero_limit_succ_dec"><span class="id" title="definition">Zero_limit_succ_dec</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec"><span class="id" title="definition">ON_Generic.ZeroLimitSucc_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="limitb" class="idref" href="#limitb"><span class="id" title="definition">limitb</span></a> (<a id="alpha:25" class="idref" href="#alpha:25"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_compare"><span class="id" title="definition">ON_compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eq_dec" class="idref" href="#eq_dec"><span class="id" title="lemma">eq_dec</span></a> (<a id="alpha:26" class="idref" href="#alpha:26"><span class="id" title="binder">alpha</span></a> <a id="beta:27" class="idref" href="#beta:27"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:27"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:27"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_introl" class="idref" href="#le_introl"><span class="id" title="lemma">le_introl</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:28" class="idref" href="#i:28"><span class="id" title="binder">i</span></a> <a id="j:29" class="idref" href="#j:29"><span class="id" title="binder">j</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:28"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:29"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:28"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:29"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_intror" class="idref" href="#le_intror"><span class="id" title="lemma">le_intror</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:30" class="idref" href="#i:30"><span class="id" title="binder">i</span></a> <a id="j:31" class="idref" href="#j:31"><span class="id" title="binder">j</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, (<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:30"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:31"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:30"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:31"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_0" class="idref" href="#le_0"><span class="id" title="lemma">le_0</span></a> : <span class="id" title="keyword">∀</span> <a id="p:32" class="idref" href="#p:32"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a>,   <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">fin</span></a> 0 <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:32"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_trans" class="idref" href="#le_lt_trans"><span class="id" title="lemma">le_lt_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="p:33" class="idref" href="#p:33"><span class="id" title="binder">p</span></a> <a id="q:34" class="idref" href="#q:34"><span class="id" title="binder">q</span></a> <a id="r:35" class="idref" href="#r:35"><span class="id" title="binder">r</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:33"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#q:34"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#q:34"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#r:35"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:33"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#r:35"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_le_trans" class="idref" href="#lt_le_trans"><span class="id" title="lemma">lt_le_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a> <a id="q:37" class="idref" href="#q:37"><span class="id" title="binder">q</span></a> <a id="r:38" class="idref" href="#r:38"><span class="id" title="binder">r</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:36"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#q:37"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#q:37"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#r:38"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:36"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#r:38"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le" class="idref" href="#lt_succ_le"><span class="id" title="lemma">lt_succ_le</span></a> <a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a> <a id="beta:40" class="idref" href="#beta:40"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:40"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#beta:40"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ" class="idref" href="#lt_succ"><span class="id" title="lemma">lt_succ</span></a> <a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:41"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:41"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega" class="idref" href="#lt_omega"><span class="id" title="lemma">lt_omega</span></a> <a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:43" class="idref" href="#n:43"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#n:43"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Omega_as_lub" class="idref" href="#Omega_as_lub"><span class="id" title="lemma">Omega_as_lub</span></a>  :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:44" class="idref" href="#alpha:44"><span class="id" title="binder">alpha</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:45"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:44"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#alpha:44"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Incl" class="idref" href="#Incl"><span class="id" title="instance">Incl</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Omega_plus_Omega"><span class="id" title="instance">Omega_plus_Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin"><span class="id" title="definition">fin</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="NotIncl" class="idref" href="#NotIncl"><span class="id" title="section">NotIncl</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="i:46" class="idref" href="#i:46"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:47" class="idref" href="#f:47"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hyp:48" class="idref" href="#Hyp:48"><span class="id" title="binder">Hyp</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#SubON"><span class="id" title="class">SubON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Omega_plus_Omega"><span class="id" title="instance">Omega_plus_Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega.html#Omega"><span class="id" title="instance">Omega</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#i:46"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#f:47"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a>: <span class="id" title="keyword">∀</span> <a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#n:49"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ExNotIncl" class="idref" href="#ExNotIncl"><span class="id" title="lemma">ExNotIncl</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#NotIncl"><span class="id" title="section">NotIncl</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.ON_mult.html b/theories/html/hydras.OrdinalNotations.ON_mult.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.ON_mult.html
@@ -0,0 +1,151 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_mult</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_mult</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Product of ordinal  notations 
+<div class="paragraph"> </div>
+
+ Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Comparable.html#"><span class="id" title="library">Comparable</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#"><span class="id" title="library">Simple_LexProd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">Sortclass</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab2"></a><h1 class="section">Definitions</h1>
+
+<div class="paragraph"> </div>
+
+  The product of two notation systems NA and NB is defined as the 
+    lexicographic product of the order on NB by the order on NA 
+    (in this order ! ) 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Defs" class="idref" href="#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> `(<a id="ltA:2" class="idref" href="#ltA:2"><span class="id" title="binder">ltA</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmpA:3" class="idref" href="#cmpA:3"><span class="id" title="binder">cmpA</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="NA:4" class="idref" href="#NA:4"><span class="id" title="binder">NA</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#ltA:2"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#cmpA:3"><span class="id" title="variable">cmpA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`(<a id="ltB:6" class="idref" href="#ltB:6"><span class="id" title="binder">ltB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a id="B:5" class="idref" href="#B:5"><span class="id" title="binder">B</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmpB:7" class="idref" href="#cmpB:7"><span class="id" title="binder">cmpB</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#B:5"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="NB:8" class="idref" href="#NB:8"><span class="id" title="binder">NB</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#ltB:6"><span class="id" title="variable">ltB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#cmpB:7"><span class="id" title="variable">cmpB</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> := (<a class="idref" href="hydras.OrdinalNotations.ON_mult.html#Defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#Defs.A"><span class="id" title="variable">A</span></a>)%<span class="id" title="keyword">type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#Defs.ltB"><span class="id" title="variable">ltB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#Defs.ltA"><span class="id" title="variable">ltA</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="le" class="idref" href="#le"><span class="id" title="definition">le</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="compare_mult" class="idref" href="#compare_mult"><span class="id" title="instance">compare_mult</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#t"><span class="id" title="definition">t</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:9"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:10"><span class="id" title="variable">beta</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:9"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:10"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">c</span> ⇒ <span class="id" title="var">c</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lexico"><span class="id" title="inductive">lexico</span></a>: <span class="id" title="var">core</span>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span>  <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#le"><span class="id" title="definition">le</span></a> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab3"></a><h1 class="section">Properties</h1>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="lt_strorder" class="idref" href="#lt_strorder"><span class="id" title="instance">lt_strorder</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_wf" class="idref" href="#lt_wf"><span class="id" title="lemma">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> <a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> <a id="beta:12" class="idref" href="#beta:12"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:12"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:11"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:12"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:12"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:12"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:11"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:14"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:14"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="mult_comp" class="idref" href="#mult_comp"><span class="id" title="instance">mult_comp</span></a>:  <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#compare_mult"><span class="id" title="instance">compare_mult</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="ON_mult" class="idref" href="#ON_mult"><span class="id" title="instance">ON_mult</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#compare_mult"><span class="id" title="instance">compare_mult</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="lemma">lt_eq_lt_dec</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="beta:16" class="idref" href="#beta:16"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:15"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:16"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:16"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#beta:16"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#alpha:15"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt_eq_lt_dec"><span class="id" title="lemma">lt_eq_lt_dec</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">ltA</span> <span class="id" title="var">cmpA</span>} <span class="id" title="var">_</span> {<span class="id" title="var">B</span> <span class="id" title="var">ltB</span> <span class="id" title="var">cmpB</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#ON_mult"><span class="id" title="instance">ON_mult</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">ltA</span> <span class="id" title="var">cmpA</span>} <span class="id" title="var">_</span> {<span class="id" title="var">B</span> <span class="id" title="var">ltB</span> <span class="id" title="var">cmpB</span>}.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_mult.html#lt_strorder"><span class="id" title="instance">lt_strorder</span></a> {<span class="id" title="var">A</span>} {<span class="id" title="var">ltA</span>} {<span class="id" title="var">cmpA</span>} <span class="id" title="var">_</span>  {<span class="id" title="var">B</span>} {<span class="id" title="var">ltB</span>} {<span class="id" title="var">cmpB</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+++ b/theories/html/hydras.OrdinalNotations.ON_plus.html
@@ -0,0 +1,148 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.ON_plus</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.ON_plus</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+The sum of two ordinal  notations 
+<div class="paragraph"> </div>
+
+ Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Disjoint_Union.html#"><span class="id" title="library">Disjoint_Union</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#"><span class="id" title="library">MoreOrders</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a>.<br/>
+<span class="id" title="keyword">Generalizable All Variables</span>.<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">Sortclass</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Defs" class="idref" href="#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> `(<a id="ltA:2" class="idref" href="#ltA:2"><span class="id" title="binder">ltA</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmpA:3" class="idref" href="#cmpA:3"><span class="id" title="binder">cmpA</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="NA:4" class="idref" href="#NA:4"><span class="id" title="binder">NA</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#ltA:2"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#cmpA:3"><span class="id" title="variable">cmpA</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> `(<a id="ltB:6" class="idref" href="#ltB:6"><span class="id" title="binder">ltB</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a id="B:5" class="idref" href="#B:5"><span class="id" title="binder">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmpB:7" class="idref" href="#cmpB:7"><span class="id" title="binder">cmpB</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#B:5"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="NB:8" class="idref" href="#NB:8"><span class="id" title="binder">NB</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#ltB:6"><span class="id" title="variable">ltB</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#cmpB:7"><span class="id" title="variable">cmpB</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> := (<a class="idref" href="hydras.OrdinalNotations.ON_plus.html#Defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e03f39daf98516fa530d3f6f5a1b4d92"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#Defs.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="keyword">type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} <span class="id" title="var">_</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#le_AsB"><span class="id" title="inductive">le_AsB</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#Defs.ltA"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#Defs.ltB"><span class="id" title="variable">ltB</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="compare_plus" class="idref" href="#compare_plus"><span class="id" title="instance">compare_plus</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#t"><span class="id" title="definition">t</span></a> :=<br/>
+<span class="id" title="keyword">fun</span> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#t"><span class="id" title="definition">t</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:9"><span class="id" title="variable">alpha</span></a>, <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:10"><span class="id" title="variable">beta</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">a</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">a'</span> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">a</span> <span class="id" title="var">a'</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">b</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">b'</span> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">b</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">_</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="le" class="idref" href="#le"><span class="id" title="definition">le</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#le"><span class="id" title="definition">le</span></a> : <span class="id" title="var">core</span>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#le_AsB"><span class="id" title="inductive">le_AsB</span></a>: <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Instance</span> <a id="lt_strorder" class="idref" href="#lt_strorder"><span class="id" title="instance">lt_strorder</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:14"><span class="id" title="variable">beta</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:14"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:14"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:14"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:13"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="compare_correct" class="idref" href="#compare_correct"><span class="id" title="lemma">compare_correct</span></a> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="beta:16" class="idref" href="#beta:16"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:16"><span class="id" title="variable">beta</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:16"><span class="id" title="variable">beta</span></a>). 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="plus_comp" class="idref" href="#plus_comp"><span class="id" title="instance">plus_comp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#compare_plus"><span class="id" title="instance">compare_plus</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_wf" class="idref" href="#lt_wf"><span class="id" title="lemma">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="ON_plus" class="idref" href="#ON_plus"><span class="id" title="instance">ON_plus</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#compare_plus"><span class="id" title="instance">compare_plus</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="lemma">lt_eq_lt_dec</span></a> <a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a> <a id="beta:18" class="idref" href="#beta:18"><span class="id" title="binder">beta</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:17"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:18"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:18"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#beta:18"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#alpha:17"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#lt_eq_lt_dec"><span class="id" title="lemma">lt_eq_lt_dec</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">ltA</span> <span class="id" title="var">cmpA</span>} <span class="id" title="var">_</span> {<span class="id" title="var">B</span> <span class="id" title="var">ltB</span>  <span class="id" title="var">cmpB</span>} <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.ON_plus.html#ON_plus"><span class="id" title="instance">ON_plus</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">ltA</span> <span class="id" title="var">cmpA</span>} <span class="id" title="var">_</span> {<span class="id" title="var">B</span> <span class="id" title="var">ltB</span> <span class="id" title="var">cmpB</span>}.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.OrdinalNotations.OmegaOmega.html b/theories/html/hydras.OrdinalNotations.OmegaOmega.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.OrdinalNotations.OmegaOmega.html
@@ -0,0 +1,481 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.OrdinalNotations.OmegaOmega</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.OrdinalNotations.OmegaOmega</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab5"></a><h1 class="section">An implementation of  &omega;<sup>&omega;</sup></h1>
+
+<div class="paragraph"> </div>
+
+ New implementation as a refinement of epsilon0 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#"><span class="id" title="library">Comparable</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Eqdep_dec.html#"><span class="id" title="library">Logic.Eqdep_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> .<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab6"></a><h1 class="section">Representation by lists of pairs of integers</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="LO" class="idref" href="#LO"><span class="id" title="module">LO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="LO.t" class="idref" href="#LO.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="LO.zero" class="idref" href="#LO.zero"><span class="id" title="definition">zero</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+omega^ i * S n + alpha 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="LO.cons" class="idref" href="#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span> <span class="id" title="var">alpha</span> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">alpha</span>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Finite ordinals 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span>  <a id="LO.FS" class="idref" href="#LO.FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">n</span> := (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">n</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="LO.fin" class="idref" href="#LO.fin"><span class="id" title="definition">fin</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.FS"><span class="id" title="abbreviation">FS</span></a> <span class="id" title="var">p</span> <span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">fin</span></a>  <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">t</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">^<span class="id" title="var">i</span></span> <span class="inlinecode"></span> 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="LO.phi0" class="idref" href="#LO.phi0"><span class="id" title="abbreviation">phi0</span></a> <span class="id" title="var">i</span> := (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">i</span> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="LO.omega" class="idref" href="#LO.omega"><span class="id" title="abbreviation">omega</span></a> := (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0"><span class="id" title="abbreviation">phi0</span></a> 1).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+data refinement 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.refine" class="idref" href="#LO.refine"><span class="id" title="definition">refine</span></a> (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1.T1</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:3"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <span class="id" title="var">i</span>)%<span class="id" title="var">t1</span> <span class="id" title="var">n</span> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#refine:4"><span class="id" title="definition">refine</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="LO.ap" class="idref" href="#LO.ap"><span class="id" title="inductive">ap</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="LO.ap_intro" class="idref" href="#LO.ap_intro"><span class="id" title="constructor">ap_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#ap:6"><span class="id" title="inductive">ap</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:8"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Successor and limits (syntactic definitions) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.succb" class="idref" href="#LO.succb"><span class="id" title="definition">succb</span></a> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:9"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#succb:10"><span class="id" title="definition">succb</span></a> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.limitb" class="idref" href="#LO.limitb"><span class="id" title="definition">limitb</span></a> (<a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:12"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#limitb:13"><span class="id" title="definition">limitb</span></a> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.succb_ref" class="idref" href="#LO.succb_ref"><span class="id" title="lemma">succb_ref</span></a> (<a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.succb"><span class="id" title="definition">succb</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1is_succ"><span class="id" title="definition">T1is_succ</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:15"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.limitb_ref" class="idref" href="#LO.limitb_ref"><span class="id" title="lemma">limitb_ref</span></a> (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.limitb"><span class="id" title="definition">limitb</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:16"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="LO.compare_oo" class="idref" href="#LO.compare_oo"><span class="id" title="instance">compare_oo</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">cmp</span> (<a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a> <a id="b:18" class="idref" href="#b:18"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:17"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:18"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>   , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>),(<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:17"><span class="id" title="variable">a</span></a> <span class="id" title="var">a'</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ (<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a> <span class="id" title="var">n</span> <span class="id" title="var">n'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#cmp:19"><span class="id" title="variable">cmp</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:18"><span class="id" title="variable">b</span></a> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">comp</span> ⇒ <span class="id" title="var">comp</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.compare_ref" class="idref" href="#LO.compare_ref"><span class="id" title="lemma">compare_ref</span></a> (<a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a> <a id="b:23" class="idref" href="#b:23"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:22"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:22"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:23"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="LO.lt" class="idref" href="#LO.lt"><span class="id" title="definition">lt</span></a> (<a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:25"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.compare_rev" class="idref" href="#LO.compare_rev"><span class="id" title="lemma">compare_rev</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompOpp"><span class="id" title="definition">CompOpp</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:27"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.compare_reflect" class="idref" href="#LO.compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:29"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:29"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:29"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:28"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.compare_correct" class="idref" href="#LO.compare_correct"><span class="id" title="lemma">compare_correct</span></a> (<a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a> <a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:31"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:31"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.lt_ref" class="idref" href="#LO.lt_ref"><span class="id" title="lemma">lt_ref</span></a> (<a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a> <a id="b:33" class="idref" href="#b:33"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:33"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:32"><span class="id" title="variable">a</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:33"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.eq_ref" class="idref" href="#LO.eq_ref"><span class="id" title="lemma">eq_ref</span></a>  (<a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a> <a id="b:35" class="idref" href="#b:35"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:35"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.lt_irrefl" class="idref" href="#LO.lt_irrefl"><span class="id" title="lemma">lt_irrefl</span></a> (<a id="a:36" class="idref" href="#a:36"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:36"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.lt_trans" class="idref" href="#LO.lt_trans"><span class="id" title="lemma">lt_trans</span></a> (<a id="a:37" class="idref" href="#a:37"><span class="id" title="binder">a</span></a> <a id="b:38" class="idref" href="#b:38"><span class="id" title="binder">b</span></a> <a id="c:39" class="idref" href="#c:39"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:37"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:38"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:38"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#c:39"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:37"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#c:39"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="LO.lo_strorder" class="idref" href="#LO.lo_strorder"><span class="id" title="instance">lo_strorder</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="LO.lo_comparable" class="idref" href="#LO.lo_comparable"><span class="id" title="instance">lo_comparable</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.nf_b" class="idref" href="#LO.nf_b"><span class="id" title="definition">nf_b</span></a> (<a id="alpha:40" class="idref" href="#alpha:40"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:40"><span class="id" title="variable">alpha</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> ((<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">b</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#nf_b:41"><span class="id" title="definition">nf_b</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.ltb"><span class="id" title="definition">Nat.ltb</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">a</span>)%<span class="id" title="var">bool</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="LO.nf" class="idref" href="#LO.nf"><span class="id" title="definition">nf</span></a> <a id="alpha:43" class="idref" href="#alpha:43"><span class="id" title="binder">alpha</span></a> :<span class="id" title="keyword">Prop</span> := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf_b"><span class="id" title="definition">nf_b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:43"><span class="id" title="variable">alpha</span></a>.<br/>
+</div>
+
+<div class="doc">
+refinements of T1's lemmas 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.zero_nf" class="idref" href="#LO.zero_nf"><span class="id" title="lemma">zero_nf</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.fin_nf" class="idref" href="#LO.fin_nf"><span class="id" title="lemma">fin_nf</span></a> (<a id="i:44" class="idref" href="#i:44"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">fin</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:44"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.single_nf" class="idref" href="#LO.single_nf"><span class="id" title="lemma">single_nf</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a> <a id="n:46" class="idref" href="#n:46"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:45"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n:46"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.cons_nf" class="idref" href="#LO.cons_nf"><span class="id" title="lemma">cons_nf</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:47" class="idref" href="#i:47"><span class="id" title="binder">i</span></a> <a id="n:48" class="idref" href="#n:48"><span class="id" title="binder">n</span></a> <a id="j:49" class="idref" href="#j:49"><span class="id" title="binder">j</span></a> <a id="n':50" class="idref" href="#n':50"><span class="id" title="binder">n'</span></a> <a id="b:51" class="idref" href="#b:51"><span class="id" title="binder">b</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt"><span class="id" title="definition">Nat.lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#j:49"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:47"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a>(<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#j:49"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n':50"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:51"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a>(<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:47"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n:48"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#j:49"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n':50"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:51"><span class="id" title="variable">b</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_nf</span>  <span class="id" title="var">single_nf</span> <span class="id" title="var">cons_nf</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.nf_inv2" class="idref" href="#LO.nf_inv2"><span class="id" title="lemma">nf_inv2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:52" class="idref" href="#i:52"><span class="id" title="binder">i</span></a> <a id="n:53" class="idref" href="#n:53"><span class="id" title="binder">n</span></a> <a id="b:54" class="idref" href="#b:54"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:52"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:54"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:54"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.nf_inv3" class="idref" href="#LO.nf_inv3"><span class="id" title="lemma">nf_inv3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a> <a id="n:56" class="idref" href="#n:56"><span class="id" title="binder">n</span></a>  <a id="j:57" class="idref" href="#j:57"><span class="id" title="binder">j</span></a> <a id="n':58" class="idref" href="#n':58"><span class="id" title="binder">n'</span></a> <a id="b':59" class="idref" href="#b':59"><span class="id" title="binder">b'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:55"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n:56"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#j:57"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#n':58"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b':59"><span class="id" title="variable">b'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.lt"><span class="id" title="definition">Nat.lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#j:57"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:55"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.nf_ref" class="idref" href="#LO.nf_ref"><span class="id" title="lemma">nf_ref</span></a>: <span class="id" title="keyword">∀</span> <a id="a:60" class="idref" href="#a:60"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:60"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:60"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">lo_scope</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">lo_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">lo</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">lo_scope</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.succ" class="idref" href="#LO.succ"><span class="id" title="definition">succ</span></a> (<a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:61"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin"><span class="id" title="definition">fin</span></a> 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">n</span> <span class="id" title="var">_</span>  ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:61"><span class="id" title="variable">a</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#succ:62"><span class="id" title="definition">succ</span></a> <span class="id" title="var">b</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.plus" class="idref" href="#LO.plus"><span class="id" title="definition">plus</span></a> (<a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a> <a id="b:65" class="idref" href="#b:65"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:64"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:65"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <span class="id" title="var">y</span>  ⇒ <span class="id" title="var">y</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>  ⇒ <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.compare"><span class="id" title="definition">Nat.compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:64"><span class="id" title="variable">a</span></a> <span class="id" title="var">a'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Datatypes.Lt</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:64"><span class="id" title="variable">a</span></a> <span class="id" title="var">n</span> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#plus:66"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:65"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span>)))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>  ⇒ (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:64"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><span class="id" title="var">n'</span>)) <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">where</span> <a id="055aadbfc80fa55b1fc005bff6efe02e" class="idref" href="#055aadbfc80fa55b1fc005bff6efe02e"><span class="id" title="notation">&quot;</span></a>a + b" := (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#plus:66"><span class="id" title="definition">plus</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">lo_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="LO.mult" class="idref" href="#LO.mult"><span class="id" title="definition">mult</span></a> (<a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a> <a id="b:70" class="idref" href="#b:70"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:69"><span class="id" title="variable">a</span></a>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:70"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <span class="id" title="var">_</span>  ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <span class="id" title="var">_</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">n</span> <span class="id" title="var">_</span>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)) <span class="id" title="var">b'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> 0 <span class="id" title="var">n'</span> <span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:69"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">Peano.pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>)) <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:70"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;|  <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span>, <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <span class="id" title="var">a'</span> <span class="id" title="var">n'</span> <span class="id" title="var">b'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">a'</span>)%<span class="id" title="var">nat</span> <span class="id" title="var">n'</span> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons"><span class="id" title="abbreviation">cons</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:69"><span class="id" title="variable">a</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:70"><span class="id" title="variable">b</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <span class="id" title="var">b'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">where</span> <a id="e6b32a3f2141890c87e9ae54bcac37eb" class="idref" href="#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">&quot;</span></a>a * b" := (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#mult:71"><span class="id" title="definition">mult</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) : <span class="id" title="var">lo_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.phi0_ref" class="idref" href="#LO.phi0_ref"><span class="id" title="lemma">phi0_ref</span></a> (<a id="i:74" class="idref" href="#i:74"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:74"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:74"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.phi0_nf" class="idref" href="#LO.phi0_nf"><span class="id" title="lemma">phi0_nf</span></a>  (<a id="i:75" class="idref" href="#i:75"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#i:75"><span class="id" title="variable">i</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.succ_ref" class="idref" href="#LO.succ_ref"><span class="id" title="lemma">succ_ref</span></a> (<a id="alpha:76" class="idref" href="#alpha:76"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:76"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#succ"><span class="id" title="definition">T1.succ</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:76"><span class="id" title="variable">alpha</span></a>).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.succ_nf" class="idref" href="#LO.succ_nf"><span class="id" title="lemma">succ_nf</span></a> <a id="alpha:77" class="idref" href="#alpha:77"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:77"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:77"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.plus_ref" class="idref" href="#LO.plus_ref"><span class="id" title="lemma">plus_ref</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:78" class="idref" href="#alpha:78"><span class="id" title="binder">alpha</span></a> <a id="beta:79" class="idref" href="#beta:79"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:78"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#055aadbfc80fa55b1fc005bff6efe02e"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:79"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1add"><span class="id" title="definition">T1.T1add</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:78"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:79"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.plus_nf" class="idref" href="#LO.plus_nf"><span class="id" title="lemma">plus_nf</span></a> <a id="alpha:80" class="idref" href="#alpha:80"><span class="id" title="binder">alpha</span></a> <a id="beta:81" class="idref" href="#beta:81"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:81"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:80"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#055aadbfc80fa55b1fc005bff6efe02e"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:81"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.mult_ref" class="idref" href="#LO.mult_ref"><span class="id" title="lemma">mult_ref</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:82" class="idref" href="#alpha:82"><span class="id" title="binder">alpha</span></a> <a id="beta:83" class="idref" href="#beta:83"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:82"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:83"><span class="id" title="variable">beta</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1mul"><span class="id" title="definition">T1.T1mul</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:82"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine"><span class="id" title="definition">refine</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:83"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.mult_nf" class="idref" href="#LO.mult_nf"><span class="id" title="lemma">mult_nf</span></a> <a id="alpha:84" class="idref" href="#alpha:84"><span class="id" title="binder">alpha</span></a> <a id="beta:85" class="idref" href="#beta:85"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:84"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:85"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:84"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:85"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="LO.plus_assoc" class="idref" href="#LO.plus_assoc"><span class="id" title="instance">plus_assoc</span></a>: <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus"><span class="id" title="definition">plus</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LO.mult_plus_distr_l" class="idref" href="#LO.mult_plus_distr_l"><span class="id" title="lemma">mult_plus_distr_l</span></a> (<a id="a:86" class="idref" href="#a:86"><span class="id" title="binder">a</span></a> <a id="b:87" class="idref" href="#b:87"><span class="id" title="binder">b</span></a> <a id="c:88" class="idref" href="#c:88"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#c:88"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#055aadbfc80fa55b1fc005bff6efe02e"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#c:88"><span class="id" title="variable">c</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#055aadbfc80fa55b1fc005bff6efe02e"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#e6b32a3f2141890c87e9ae54bcac37eb"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#c:88"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO"><span class="id" title="module">LO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">OO_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">oo</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">OO_scope</span>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO"><span class="id" title="module">LO</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab7"></a><h1 class="section">well formed ordinal representation</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="OO" class="idref" href="#OO"><span class="id" title="module">OO</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a id="OO.OO" class="idref" href="#OO.OO"><span class="id" title="record">OO</span></a> : <span class="id" title="keyword">Type</span> := <a id="OO.mkord" class="idref" href="#OO.mkord"><span class="id" title="constructor">mkord</span></a> {<a id="OO.data" class="idref" href="#OO.data"><span class="id" title="projection">data</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">LO.t</span></a> ; <a id="OO.data_ok" class="idref" href="#OO.data_ok"><span class="id" title="projection">data_ok</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">LO.nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#data:90"><span class="id" title="method">data</span></a>}.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">data_ok</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="OO.lt" class="idref" href="#OO.lt"><span class="id" title="definition">lt</span></a> (<a id="alpha:92" class="idref" href="#alpha:92"><span class="id" title="binder">alpha</span></a> <a id="beta:93" class="idref" href="#beta:93"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt"><span class="id" title="definition">LO.lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:92"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:93"><span class="id" title="variable">beta</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="OO.le" class="idref" href="#OO.le"><span class="id" title="definition">le</span></a> := <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.compare_OO" class="idref" href="#OO.compare_OO"><span class="id" title="instance">compare_OO</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:94" class="idref" href="#alpha:94"><span class="id" title="binder">alpha</span></a> <a id="beta:95" class="idref" href="#beta:95"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) ⇒ <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_oo"><span class="id" title="instance">LO.compare_oo</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:94"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:95"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.Zero" class="idref" href="#OO.Zero"><span class="id" title="instance">Zero</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a> := @<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.mkord"><span class="id" title="constructor">mkord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO._Omega" class="idref" href="#OO._Omega"><span class="id" title="instance">_Omega</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.Fin" class="idref" href="#OO.Fin"><span class="id" title="instance">Fin</span></a> (<a id="i:96" class="idref" href="#i:96"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="OO.omega" class="idref" href="#OO.omega"><span class="id" title="abbreviation">omega</span></a>  := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO._Omega"><span class="id" title="instance">_Omega</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.Succ" class="idref" href="#OO.Succ"><span class="id" title="instance">Succ</span></a> (<a id="alpha:97" class="idref" href="#alpha:97"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.plus" class="idref" href="#OO.plus"><span class="id" title="instance">plus</span></a> (<a id="alpha:98" class="idref" href="#alpha:98"><span class="id" title="binder">alpha</span></a> <a id="beta:99" class="idref" href="#beta:99"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="f0892e4f827b9668045c34fe437da503" class="idref" href="#f0892e4f827b9668045c34fe437da503"><span class="id" title="notation">&quot;</span></a>+" := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.plus"><span class="id" title="instance">plus</span></a> : <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.mult" class="idref" href="#OO.mult"><span class="id" title="instance">mult</span></a> (<a id="alpha:100" class="idref" href="#alpha:100"><span class="id" title="binder">alpha</span></a> <a id="beta:101" class="idref" href="#beta:101"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="3e75d3b5e7edc50985c52d117590a4bb" class="idref" href="#3e75d3b5e7edc50985c52d117590a4bb"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.mult"><span class="id" title="instance">mult</span></a> : <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.phi0" class="idref" href="#OO.phi0"><span class="id" title="instance">phi0</span></a> (<a id="i:102" class="idref" href="#i:102"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="1a72aa5d15ab5318d1af7e45f6175964" class="idref" href="#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">&quot;</span></a>'omega^'" := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.phi0"><span class="id" title="instance">phi0</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="3e75d3b5e7edc50985c52d117590a4bb" class="idref" href="#3e75d3b5e7edc50985c52d117590a4bb"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.mult"><span class="id" title="instance">mult</span></a> : <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.embed" class="idref" href="#OO.embed"><span class="id" title="instance">embed</span></a> (<a id="alpha:103" class="idref" href="#alpha:103"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>) : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0.E0</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="OO.lt_embed" class="idref" href="#OO.lt_embed"><span class="id" title="lemma">lt_embed</span></a> (<a id="alpha:104" class="idref" href="#alpha:104"><span class="id" title="binder">alpha</span></a> <a id="beta:105" class="idref" href="#beta:105"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>): <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:105"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0lt"><span class="id" title="definition">E0lt</span></a> (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.embed"><span class="id" title="instance">embed</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:104"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.embed"><span class="id" title="instance">embed</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:105"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.oo_str" class="idref" href="#OO.oo_str"><span class="id" title="instance">oo_str</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="OO.nf_proof_unicity" class="idref" href="#OO.nf_proof_unicity"><span class="id" title="lemma">nf_proof_unicity</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="alpha:106" class="idref" href="#alpha:106"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.t"><span class="id" title="definition">t</span></a>) (<a id="H:107" class="idref" href="#H:107"><span class="id" title="binder">H</span></a> <a id="H':108" class="idref" href="#H':108"><span class="id" title="binder">H'</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:106"><span class="id" title="variable">alpha</span></a>), <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#H:107"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#H':108"><span class="id" title="variable">H'</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="OO.OO_eq_intro" class="idref" href="#OO.OO_eq_intro"><span class="id" title="lemma">OO_eq_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:109" class="idref" href="#alpha:109"><span class="id" title="binder">alpha</span></a> <a id="beta:110" class="idref" href="#beta:110"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO"><span class="id" title="class">OO</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.data"><span class="id" title="method">data</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:110"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#beta:110"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.OO_comp" class="idref" href="#OO.OO_comp"><span class="id" title="instance">OO_comp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="OO.lt_wf" class="idref" href="#OO.lt_wf"><span class="id" title="lemma">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="module">ON_Generic</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="OO.ON_OO" class="idref" href="#OO.ON_OO"><span class="id" title="instance">ON_OO</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO"><span class="id" title="module">OO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#OO"><span class="id" title="module">OO</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">OO_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#phi0"><span class="id" title="instance">phi0</span></a>  7.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">Coercion</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">Fin</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">:</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">nat</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">&gt;-&gt;</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#Fin"><span class="id" title="instance">OO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex42" class="idref" href="#Ex42"><span class="id" title="definition">Ex42</span></a>: <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#f0892e4f827b9668045c34fe437da503"><span class="id" title="notation">+</span></a> 42 <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#f0892e4f827b9668045c34fe437da503"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">^</span></a> <a class="idref" href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964"><span class="id" title="notation">2</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Comparable</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Comparable</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreOrders.html#"><span class="id" title="library">MoreOrders</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#"><span class="id" title="library">STDPP_compat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Compare" class="idref" href="#Compare"><span class="id" title="record">Compare</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) := <a id="compare" class="idref" href="#compare"><span class="id" title="projection">compare</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Comparable" class="idref" href="#Comparable"><span class="id" title="record">Comparable</span></a> {<a id="A:4" class="idref" href="#A:4"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="lt:5" class="idref" href="#lt:5"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#A:4"><span class="id" title="variable">A</span></a>) (<a id="cmp:6" class="idref" href="#cmp:6"><span class="id" title="binder">cmp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#A:4"><span class="id" title="variable">A</span></a>) :=<br/>
+{<br/>
+&nbsp;&nbsp;<a id="comparable_sto" class="idref" href="#comparable_sto"><span class="id" title="projection">comparable_sto</span></a> :&gt; <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#lt:5"><span class="id" title="variable">lt</span></a>;<br/>
+&nbsp;&nbsp;<a id="comparable_comp_spec" class="idref" href="#comparable_comp_spec"><span class="id" title="projection">comparable_comp_spec</span></a> : <span class="id" title="keyword">∀</span> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#A:4"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#lt:5"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:10"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:10"><span class="id" title="variable">b</span></a>)<br/>
+}.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="var">Mode</span> <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> ! : <span class="id" title="var">typeclass_instances</span>.<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="var">Mode</span> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="class">Comparable</span></a> ! - - : <span class="id" title="var">typeclass_instances</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Comparable" class="idref" href="#Comparable"><span class="id" title="section">Comparable</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> {<a id="A:12" class="idref" href="#A:12"><span class="id" title="binder">A</span></a>: <span class="id" title="keyword">Type</span>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="lt:13" class="idref" href="#lt:13"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#A:12"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="cmp:14" class="idref" href="#cmp:14"><span class="id" title="binder">cmp</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#A:12"><span class="id" title="variable">A</span></a>} `{<a id="Comparable0:15" class="idref" href="#Comparable0:15"><span class="id" title="binder">!</span></a><a id="Comparable0:15" class="idref" href="#Comparable0:15"><span class="id" title="binder">Comparable</span></a> <a id="Comparable0:15" class="idref" href="#Comparable0:15"><span class="id" title="binder">lt</span></a> <a id="Comparable0:15" class="idref" href="#Comparable0:15"><span class="id" title="binder">cmp</span></a>}.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="le" class="idref" href="#le"><span class="id" title="abbreviation">le</span></a> := (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use StrictOrder_Transitive")]<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="lt_trans" class="idref" href="#lt_trans"><span class="id" title="abbreviation">lt_trans</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder_Transitive"><span class="id" title="method">StrictOrder_Transitive</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">deprecated</span>(<span class="id" title="var">note</span>="use StrictOrder_Irreflexive")] <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="lt_irrefl" class="idref" href="#lt_irrefl"><span class="id" title="abbreviation">lt_irrefl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder_Irreflexive"><span class="id" title="method">StrictOrder_Irreflexive</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_not_gt" class="idref" href="#lt_not_gt"><span class="id" title="lemma">lt_not_gt</span></a> (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a> <a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:17"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:17"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:16"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_not_ge" class="idref" href="#lt_not_ge"><span class="id" title="lemma">lt_not_ge</span></a> (<a id="a:18" class="idref" href="#a:18"><span class="id" title="binder">a</span></a> <a id="b:19" class="idref" href="#b:19"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:18"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:19"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:19"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:18"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_lt_iff" class="idref" href="#compare_lt_iff"><span class="id" title="lemma">compare_lt_iff</span></a> (<a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> <a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:21"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_lt_trans" class="idref" href="#compare_lt_trans"><span class="id" title="lemma">compare_lt_trans</span></a> (<a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a> <a id="b:23" class="idref" href="#b:23"><span class="id" title="binder">b</span></a> <a id="c:24" class="idref" href="#c:24"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:22"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:24"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:22"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:24"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_lt_irrefl" class="idref" href="#compare_lt_irrefl"><span class="id" title="lemma">compare_lt_irrefl</span></a> (<a id="a:25" class="idref" href="#a:25"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_eq_iff" class="idref" href="#compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a> (<a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:27"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_refl" class="idref" href="#compare_refl"><span class="id" title="lemma">compare_refl</span></a> (<a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_eq_trans" class="idref" href="#compare_eq_trans"><span class="id" title="lemma">compare_eq_trans</span></a> (<a id="a:29" class="idref" href="#a:29"><span class="id" title="binder">a</span></a> <a id="b:30" class="idref" href="#b:30"><span class="id" title="binder">b</span></a> <a id="c:31" class="idref" href="#c:31"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:30"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:30"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:31"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:29"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:31"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_gt_iff" class="idref" href="#compare_gt_iff"><span class="id" title="lemma">compare_gt_iff</span></a> (<a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a> <a id="b:33" class="idref" href="#b:33"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:33"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:33"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:32"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_gt_irrefl" class="idref" href="#compare_gt_irrefl"><span class="id" title="lemma">compare_gt_irrefl</span></a> (<a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_gt_trans" class="idref" href="#compare_gt_trans"><span class="id" title="lemma">compare_gt_trans</span></a> (<a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a> <a id="b:36" class="idref" href="#b:36"><span class="id" title="binder">b</span></a> <a id="c:37" class="idref" href="#c:37"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:36"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:36"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:37"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:35"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:37"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_lt_not_gt" class="idref" href="#compare_lt_not_gt"><span class="id" title="lemma">compare_lt_not_gt</span></a> (<a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a> <a id="b:39" class="idref" href="#b:39"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:39"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:39"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_gt_not_lt" class="idref" href="#compare_gt_not_lt"><span class="id" title="lemma">compare_gt_not_lt</span></a> (<a id="a:40" class="idref" href="#a:40"><span class="id" title="binder">a</span></a> <a id="b:41" class="idref" href="#b:41"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:40"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:41"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:40"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:41"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_refl" class="idref" href="#le_refl"><span class="id" title="lemma">le_refl</span></a> (<a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:42"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_le_iff_refl" class="idref" href="#compare_le_iff_refl"><span class="id" title="lemma">compare_le_iff_refl</span></a> (<a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_le_iff" class="idref" href="#compare_le_iff"><span class="id" title="lemma">compare_le_iff</span></a> (<a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> <a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_ge_iff" class="idref" href="#compare_ge_iff"><span class="id" title="lemma">compare_ge_iff</span></a> (<a id="a:46" class="idref" href="#a:46"><span class="id" title="binder">a</span></a> <a id="b:47" class="idref" href="#b:47"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:47"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_trans" class="idref" href="#le_trans"><span class="id" title="lemma">le_trans</span></a> (<a id="a:48" class="idref" href="#a:48"><span class="id" title="binder">a</span></a> <a id="b:49" class="idref" href="#b:49"><span class="id" title="binder">b</span></a> <a id="c:50" class="idref" href="#c:50"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:49"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:49"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:50"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:48"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:50"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_lt_trans" class="idref" href="#le_lt_trans"><span class="id" title="lemma">le_lt_trans</span></a> (<a id="a:51" class="idref" href="#a:51"><span class="id" title="binder">a</span></a> <a id="b:52" class="idref" href="#b:52"><span class="id" title="binder">b</span></a> <a id="c:53" class="idref" href="#c:53"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:51"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:52"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:52"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:53"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:51"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:53"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_le_trans" class="idref" href="#lt_le_trans"><span class="id" title="lemma">lt_le_trans</span></a> (<a id="a:54" class="idref" href="#a:54"><span class="id" title="binder">a</span></a> <a id="b:55" class="idref" href="#b:55"><span class="id" title="binder">b</span></a> <a id="c:56" class="idref" href="#c:56"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:54"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:55"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:55"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:56"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:54"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:56"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_incl_le" class="idref" href="#lt_incl_le"><span class="id" title="lemma">lt_incl_le</span></a> (<a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a> <a id="b:58" class="idref" href="#b:58"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:58"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:58"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_not_gt" class="idref" href="#le_not_gt"><span class="id" title="lemma">le_not_gt</span></a> (<a id="a:59" class="idref" href="#a:59"><span class="id" title="binder">a</span></a> <a id="b:60" class="idref" href="#b:60"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:59"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:60"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:60"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:59"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="max" class="idref" href="#max"><span class="id" title="definition">max</span></a> (<a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> <a id="b:62" class="idref" href="#b:62"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:62"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#a:61"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#b:62"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="max_dec" class="idref" href="#max_dec"><span class="id" title="lemma">max_dec</span></a> (<a id="a:63" class="idref" href="#a:63"><span class="id" title="binder">a</span></a> <a id="b:64" class="idref" href="#b:64"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:63"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:64"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:63"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:63"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:64"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:64"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="max_comm" class="idref" href="#max_comm"><span class="id" title="lemma">max_comm</span></a> (<a id="a:65" class="idref" href="#a:65"><span class="id" title="binder">a</span></a> <a id="b:66" class="idref" href="#b:66"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:65"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:66"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:66"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:65"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="max_ge_a" class="idref" href="#max_ge_a"><span class="id" title="lemma">max_ge_a</span></a> (<a id="a:67" class="idref" href="#a:67"><span class="id" title="binder">a</span></a> <a id="b:68" class="idref" href="#b:68"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:68"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:68"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:67"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="max_ge_b" class="idref" href="#max_ge_b"><span class="id" title="lemma">max_ge_b</span></a> (<a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a> <a id="b:70" class="idref" href="#b:70"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:70"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="max_refl" class="idref" href="#max_refl"><span class="id" title="lemma">max_refl</span></a> (<a id="a:71" class="idref" href="#a:71"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):  <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:71"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:71"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_max_a" class="idref" href="#le_max_a"><span class="id" title="lemma">le_max_a</span></a> (<a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> <a id="b:73" class="idref" href="#b:73"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:72"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:73"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_max_b" class="idref" href="#le_max_b"><span class="id" title="lemma">le_max_b</span></a> (<a id="a:74" class="idref" href="#a:74"><span class="id" title="binder">a</span></a> <a id="b:75" class="idref" href="#b:75"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:75"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:74"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:75"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="max_assoc" class="idref" href="#max_assoc"><span class="id" title="instance">max_assoc</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="keyword">using</span>="All"]<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="min" class="idref" href="#min"><span class="id" title="definition">min</span></a> (<a id="a:76" class="idref" href="#a:76"><span class="id" title="binder">a</span></a> <a id="b:77" class="idref" href="#b:77"><span class="id" title="binder">b</span></a> :<a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>): <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:76"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:77"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#a:76"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#b:77"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_max_iff" class="idref" href="#min_max_iff"><span class="id" title="lemma">min_max_iff</span></a> (<a id="a:78" class="idref" href="#a:78"><span class="id" title="binder">a</span></a> <a id="b:79" class="idref" href="#b:79"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:79"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:79"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:79"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_comm" class="idref" href="#min_comm"><span class="id" title="lemma">min_comm</span></a> (<a id="a:80" class="idref" href="#a:80"><span class="id" title="binder">a</span></a> <a id="b:81" class="idref" href="#b:81"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:80"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:81"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:81"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:80"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_dec" class="idref" href="#min_dec"><span class="id" title="lemma">min_dec</span></a> (<a id="a:82" class="idref" href="#a:82"><span class="id" title="binder">a</span></a> <a id="b:83" class="idref" href="#b:83"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:82"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:83"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:82"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:82"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:83"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:83"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_le_ad" class="idref" href="#min_le_ad"><span class="id" title="lemma">min_le_ad</span></a> (<a id="a:84" class="idref" href="#a:84"><span class="id" title="binder">a</span></a> <a id="b:85" class="idref" href="#b:85"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:84"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:85"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:84"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:85"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:84"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_le_b" class="idref" href="#min_le_b"><span class="id" title="lemma">min_le_b</span></a> (<a id="a:86" class="idref" href="#a:86"><span class="id" title="binder">a</span></a> <a id="b:87" class="idref" href="#b:87"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:86"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:87"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:87"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_refl" class="idref" href="#min_refl"><span class="id" title="lemma">min_refl</span></a> (<a id="a:88" class="idref" href="#a:88"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:88"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_min_a" class="idref" href="#le_min_a"><span class="id" title="lemma">le_min_a</span></a> (<a id="a:89" class="idref" href="#a:89"><span class="id" title="binder">a</span></a> <a id="b:90" class="idref" href="#b:90"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:89"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:90"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Prelude.Comparable.html#a:89"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="le_min_bd" class="idref" href="#le_min_bd"><span class="id" title="lemma">le_min_bd</span></a> (<a id="a:91" class="idref" href="#a:91"><span class="id" title="binder">a</span></a> <a id="b:92" class="idref" href="#b:92"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#le"><span class="id" title="abbreviation">le</span></a> (<a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:91"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:92"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Prelude.Comparable.html#b:92"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span>  <a id="min_assoc" class="idref" href="#min_assoc"><span class="id" title="instance">min_assoc</span></a>: <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#min"><span class="id" title="definition">min</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_trans" class="idref" href="#compare_trans"><span class="id" title="lemma">compare_trans</span></a> (<a id="a:93" class="idref" href="#a:93"><span class="id" title="binder">a</span></a> <a id="b:94" class="idref" href="#b:94"><span class="id" title="binder">b</span></a> <a id="c:95" class="idref" href="#c:95"><span class="id" title="binder">c</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>) (<a id="comp_res:96" class="idref" href="#comp_res:96"><span class="id" title="binder">comp_res</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:93"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:94"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#comp_res:96"><span class="id" title="variable">comp_res</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:94"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:95"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#comp_res:96"><span class="id" title="variable">comp_res</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:93"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#c:95"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#comp_res:96"><span class="id" title="variable">comp_res</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="compare_reflect" class="idref" href="#compare_reflect"><span class="id" title="lemma">compare_reflect</span></a> (<a id="a:97" class="idref" href="#a:97"><span class="id" title="binder">a</span></a> <a id="b:98" class="idref" href="#b:98"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:98"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:98"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:98"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#b:98"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#a:97"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="lt_eq_lt" class="idref" href="#lt_eq_lt"><span class="id" title="lemma">lt_eq_lt</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:99" class="idref" href="#alpha:99"><span class="id" title="binder">alpha</span></a> <a id="beta:100" class="idref" href="#beta:100"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:99"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Prelude.Comparable.html#beta:100"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:99"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#beta:100"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#beta:100"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:99"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="lt_eq_lt_dec" class="idref" href="#lt_eq_lt_dec"><span class="id" title="definition">lt_eq_lt_dec</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="alpha:101" class="idref" href="#alpha:101"><span class="id" title="binder">alpha</span></a> <a id="beta:102" class="idref" href="#beta:102"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:101"><span class="id" title="variable">alpha</span></a>  <a class="idref" href="hydras.Prelude.Comparable.html#beta:102"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.Comparable.html#alpha:101"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#beta:102"><span class="id" title="variable">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.Comparable.html#Comparable.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#beta:102"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Prelude.Comparable.html#alpha:101"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#53e8e3b7ca0342c8b99a777b3c5faf3a"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LimitNotSucc" class="idref" href="#LimitNotSucc"><span class="id" title="lemma">LimitNotSucc</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="alpha:103" class="idref" href="#alpha:103"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Prelude.Comparable.html#Comparable.A"><span class="id" title="variable">A</span></a>)  :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:103"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="beta:104" class="idref" href="#beta:104"><span class="id" title="binder">beta</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#alpha:103"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Prelude.Comparable.html#beta:104"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Comparable.html#Comparable"><span class="id" title="section">Comparable</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Ltac</span> <span class="id" title="var">compare_trans</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> <span class="id" title="var">intropattern</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">lazymatch</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">H1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">H2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">a</span> ?<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">comp_res</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">b</span> ?<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">comp_res</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">compare</span> <span class="id" title="var">a</span> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">comp_res</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">intropattern</span> <span class="id" title="tactic">by</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_trans"><span class="id" title="lemma">compare_trans</span></a> <span class="id" title="keyword">with</span> <span class="id" title="var">b</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[ <span class="id" title="tactic">exact</span> <span class="id" title="var">H1</span> | <span class="id" title="tactic">exact</span> <span class="id" title="var">H2</span> ])<br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">a</span> ?<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">comp_res</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">b</span> ?<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">compare</span> <span class="id" title="var">a</span> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">comp_res</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">intropattern</span> <span class="id" title="tactic">by</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">assert</span> (<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">c</span>) <span class="id" title="keyword">as</span> → <span class="id" title="tactic">by</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a>; <span class="id" title="tactic">exact</span> <span class="id" title="var">H2</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> <span class="id" title="var">H1</span>)<br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">a</span> ?<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> ?<span class="id" title="var">b</span> ?<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">comp_res</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assert</span> (<span class="id" title="var">compare</span> <span class="id" title="var">a</span> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">comp_res</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">intropattern</span> <span class="id" title="tactic">by</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">assert</span> (<span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">b</span>) <span class="id" title="keyword">as</span> → <span class="id" title="tactic">by</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a>; <span class="id" title="tactic">exact</span> <span class="id" title="var">H1</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> <span class="id" title="var">H2</span>)<br/>
+&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a>?<span class="id" title="var">compare</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span> ⇒ <span class="id" title="tactic">fail</span> "Not a supported case."<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <span class="id" title="tactic">fail</span> "Did not find hypotheses talking about compare: did you declare an instance of Comparable?"<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> "compare" "trans" <span class="id" title="keyword">constr</span>(<span class="id" title="var">H1</span>) <span class="id" title="keyword">constr</span>(<span class="id" title="var">H2</span>) "as" <span class="id" title="var">simple_intropattern</span>(<span class="id" title="var">intropattern</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="var">compare_trans</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> <span class="id" title="var">intropattern</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">compare_destruct_eqn</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">H</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.Prelude.Comparable.html#compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>) <span class="id" title="var">eqn</span>: <span class="id" title="var">H</span>;<br/>
+&nbsp;&nbsp;[ <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H</span> <span class="id" title="keyword">as</span> &lt;-<br/>
+&nbsp;&nbsp;| <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_lt_iff"><span class="id" title="lemma">compare_lt_iff</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H</span><br/>
+&nbsp;&nbsp;| <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Comparable.html#compare_gt_iff"><span class="id" title="lemma">compare_gt_iff</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H</span><br/>
+&nbsp;&nbsp;].<br/>
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> "compare" "destruct" <span class="id" title="keyword">constr</span>(<span class="id" title="var">a</span>) <span class="id" title="keyword">constr</span>(<span class="id" title="var">b</span>) "as" <span class="id" title="var">ident</span>(<span class="id" title="var">H</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="var">compare_destruct_eqn</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">H</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Compat815.html b/theories/html/hydras.Prelude.Compat815.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Prelude.Compat815.html
@@ -0,0 +1,109 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Compat815</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Compat815</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+provisionally fixes some compatibilty issues 8.15 -&gt; 8.16 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Compat815" class="idref" href="#Compat815"><span class="id" title="module">Compat815</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.le_n_0_eq" class="idref" href="#Compat815.le_n_0_eq"><span class="id" title="lemma">le_n_0_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:1"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.le_lt_or_eq" class="idref" href="#Compat815.le_lt_or_eq"><span class="id" title="lemma">le_lt_or_eq</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> <a id="m:3" class="idref" href="#m:3"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:3"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:3"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:3"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.lt_n_Sm_le" class="idref" href="#Compat815.lt_n_Sm_le"><span class="id" title="lemma">lt_n_Sm_le</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> <a id="m:5" class="idref" href="#m:5"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:5"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:5"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Compat815.ind_0_1_SS" class="idref" href="#Compat815.ind_0_1_SS"><span class="id" title="definition">ind_0_1_SS</span></a> (<a id="P:6" class="idref" href="#P:6"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="H0:7" class="idref" href="#H0:7"><span class="id" title="binder">H0</span></a> : <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> 0) (<a id="H1:8" class="idref" href="#H1:8"><span class="id" title="binder">H1</span></a> : <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> 1)<br/>
+&nbsp;&nbsp;(<a id="H2:10" class="idref" href="#H2:10"><span class="id" title="binder">H2</span></a> : <span class="id" title="keyword">∀</span> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:9"><span class="id" title="variable">n</span></a>))) : <span class="id" title="keyword">∀</span> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:11"><span class="id" title="variable">n</span></a> :=<br/>
+<br/>
+<span class="id" title="keyword">fix</span> <span class="id" title="var">ind_0_1_SS</span> (<a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:12"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Compat815.html#n:12"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">n0</span> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n0:14"><span class="id" title="variable">n0</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="hydras.Prelude.Compat815.html#H0:7"><span class="id" title="variable">H0</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n0</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="n1:15" class="idref" href="#n1:15"><span class="id" title="binder">n1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Compat815.html#n1:15"><span class="id" title="variable">n1</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">n2</span> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Prelude.Compat815.html#P:6"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n2:16"><span class="id" title="variable">n2</span></a>)) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="hydras.Prelude.Compat815.html#H1:8"><span class="id" title="variable">H1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n2</span> ⇒ (<span class="id" title="keyword">fun</span> <a id="n3:17" class="idref" href="#n3:17"><span class="id" title="binder">n3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Prelude.Compat815.html#H2:10"><span class="id" title="variable">H2</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n3:17"><span class="id" title="variable">n3</span></a> (<a class="idref" href="hydras.Prelude.Compat815.html#ind_0_1_SS:13"><span class="id" title="variable">ind_0_1_SS</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n3:17"><span class="id" title="variable">n3</span></a>)) <span class="id" title="var">n2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="var">n0</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.lt_S_n" class="idref" href="#Compat815.lt_S_n"><span class="id" title="lemma">lt_S_n</span></a> (<a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a> <a id="m:19" class="idref" href="#m:19"><span class="id" title="binder">m</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:19"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:19"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.lt_n_S" class="idref" href="#Compat815.lt_n_S"><span class="id" title="lemma">lt_n_S</span></a> : <span class="id" title="keyword">∀</span> <a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> <a id="m:21" class="idref" href="#m:21"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:21"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:21"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span>  <a id="Compat815.le_lt_n_Sm" class="idref" href="#Compat815.le_lt_n_Sm"><span class="id" title="lemma">le_lt_n_Sm</span></a> : <span class="id" title="keyword">∀</span> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a> <a id="m:23" class="idref" href="#m:23"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:23"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:23"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.lt_not_le" class="idref" href="#Compat815.lt_not_le"><span class="id" title="lemma">lt_not_le</span></a> : <span class="id" title="keyword">∀</span> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> <a id="m:25" class="idref" href="#m:25"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:25"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:25"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:24"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Compat815.le_plus_r" class="idref" href="#Compat815.le_plus_r"><span class="id" title="lemma">le_plus_r</span></a> : <span class="id" title="keyword">∀</span> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> <a id="m:27" class="idref" href="#m:27"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#m:27"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:27"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Compat815.mult_O_le" class="idref" href="#Compat815.mult_O_le"><span class="id" title="lemma">mult_O_le</span></a> : <span class="id" title="keyword">∀</span> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> <a id="m:29" class="idref" href="#m:29"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#m:29"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:29"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:28"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Compat815.plus_Snm_nSm" class="idref" href="#Compat815.plus_Snm_nSm"><span class="id" title="lemma">plus_Snm_nSm</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;: <span class="id" title="keyword">∀</span> <a id="n:30" class="idref" href="#n:30"><span class="id" title="binder">n</span></a> <a id="m:31" class="idref" href="#m:31"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:31"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:31"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Compat815.n_SSSn" class="idref" href="#Compat815.n_SSSn"><span class="id" title="lemma">n_SSSn</span></a> : <span class="id" title="keyword">∀</span> <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:32"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:32"><span class="id" title="variable">n</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Compat815.n_SSn" class="idref" href="#Compat815.n_SSn"><span class="id" title="lemma">n_SSn</span></a> : <span class="id" title="keyword">∀</span> <a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:33"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:33"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Compat815.le_not_lt" class="idref" href="#Compat815.le_not_lt"><span class="id" title="lemma">le_not_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="n:34" class="idref" href="#n:34"><span class="id" title="binder">n</span></a> <a id="m:35" class="idref" href="#m:35"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Compat815.html#n:34"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:35"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#m:35"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Compat815.html#n:34"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Compat815.html#Compat815"><span class="id" title="module">Compat815</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.DecPreOrder.html b/theories/html/hydras.Prelude.DecPreOrder.html
new file mode 100644
index 00000000..0af73f62
--- /dev/null
+++ b/theories/html/hydras.Prelude.DecPreOrder.html
@@ -0,0 +1,180 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.DecPreOrder</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.DecPreOrder</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ Decidable, Total Pre-orders 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Total" class="idref" href="#Total"><span class="id" title="record">Total</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="R:2" class="idref" href="#R:2"><span class="id" title="binder">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a id="totalness" class="idref" href="#totalness"><span class="id" title="projection">totalness</span></a> : <span class="id" title="keyword">∀</span> <a id="a:4" class="idref" href="#a:4"><span class="id" title="binder">a</span></a> <a id="b:5" class="idref" href="#b:5"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder.html#A:1"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:4"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:5"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:5"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:4"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#"><span class="id" title="library">STDPP_compat</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="comparison_eq_dec" class="idref" href="#comparison_eq_dec"><span class="id" title="instance">comparison_eq_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#EqDecision"><span class="id" title="abbreviation">EqDecision</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a>.<br/>
+ 
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Total_Reflexive" class="idref" href="#Total_Reflexive"><span class="id" title="instance">Total_Reflexive</span></a>{<a id="A:7" class="idref" href="#A:7"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="R:8" class="idref" href="#R:8"><span class="id" title="binder">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:7"><span class="id" title="variable">A</span></a>}(<a id="rt:9" class="idref" href="#rt:9"><span class="id" title="binder">rt</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#Total"><span class="id" title="class">Total</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:8"><span class="id" title="variable">R</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:8"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Semibundled" class="idref" href="#Semibundled"><span class="id" title="module">Semibundled</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Semibundled.TotalDec" class="idref" href="#Semibundled.TotalDec"><span class="id" title="record">TotalDec</span></a> {<a id="A:10" class="idref" href="#A:10"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="R:11" class="idref" href="#R:11"><span class="id" title="binder">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:10"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;{ <a id="Semibundled.total_dec" class="idref" href="#Semibundled.total_dec"><span class="id" title="projection">total_dec</span></a> :&gt; <a class="idref" href="hydras.Prelude.DecPreOrder.html#Total"><span class="id" title="class">Total</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:11"><span class="id" title="variable">R</span></a> ;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Semibundled.dec_dec" class="idref" href="#Semibundled.dec_dec"><span class="id" title="projection">dec_dec</span></a> :&gt; <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:11"><span class="id" title="variable">R</span></a>}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A class of decidable total pre-orders 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Semibundled.TotalDecPreOrder" class="idref" href="#Semibundled.TotalDecPreOrder"><span class="id" title="record">TotalDecPreOrder</span></a> {<a id="A:15" class="idref" href="#A:15"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="le:16" class="idref" href="#le:16"><span class="id" title="binder">le</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:15"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Semibundled.total_dec_pre_order" class="idref" href="#Semibundled.total_dec_pre_order"><span class="id" title="projection">total_dec_pre_order</span></a> :&gt; <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:16"><span class="id" title="variable">le</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Semibundled.total_dec_total" class="idref" href="#Semibundled.total_dec_total"><span class="id" title="projection">total_dec_total</span></a>  :&gt; <a class="idref" href="hydras.Prelude.DecPreOrder.html#Semibundled.TotalDec"><span class="id" title="class">TotalDec</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:16"><span class="id" title="variable">le</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#Semibundled"><span class="id" title="module">Semibundled</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+when applied to a pre-order relation le, returns the equivalence and
+    the strict order associated to le 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="preorder_equiv" class="idref" href="#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> {<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:21" class="idref" href="#le:21"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:20"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="P:22" class="idref" href="#P:22"><span class="id" title="binder">P</span></a>:@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:21"><span class="id" title="variable">le</span></a> }(<a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:20"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#le:21"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:21"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:23"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lt" class="idref" href="#lt"><span class="id" title="definition">lt</span></a> {<a id="A:25" class="idref" href="#A:25"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:26" class="idref" href="#le:26"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:25"><span class="id" title="variable">A</span></a>} `{<a id="P:27" class="idref" href="#P:27"><span class="id" title="binder">P</span></a>:@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:25"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:26"><span class="id" title="variable">le</span></a> }(<a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:25"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#le:26"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:26"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:28"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A class of total pre-orders 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="TotalPreOrder" class="idref" href="#TotalPreOrder"><span class="id" title="record">TotalPreOrder</span></a> {<a id="A:30" class="idref" href="#A:30"><span class="id" title="binder">A</span></a>} (<a id="R:31" class="idref" href="#R:31"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:30"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := {<br/>
+&nbsp;&nbsp;<a id="total_pre_order_pre" class="idref" href="#total_pre_order_pre"><span class="id" title="projection">total_pre_order_pre</span></a> :&gt; <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:31"><span class="id" title="variable">R</span></a>;<br/>
+&nbsp;&nbsp;<a id="total_pre_order_total" class="idref" href="#total_pre_order_total"><span class="id" title="projection">total_pre_order_total</span></a> :&gt; <a class="idref" href="hydras.Prelude.DecPreOrder.html#Total"><span class="id" title="class">Total</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#R:31"><span class="id" title="variable">R</span></a><br/>
+}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties of decidable total pre-orders 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="lt_irreflexive" class="idref" href="#lt_irreflexive"><span class="id" title="lemma">lt_irreflexive</span></a>  {<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:36" class="idref" href="#le:36"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:35"><span class="id" title="variable">A</span></a>}{<a id="P:37" class="idref" href="#P:37"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:36"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:38"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:38"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_not_ge" class="idref" href="#lt_not_ge"><span class="id" title="lemma">lt_not_ge</span></a> {<a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:40" class="idref" href="#le:40"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:39"><span class="id" title="variable">A</span></a>}{<a id="P:41" class="idref" href="#P:41"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:40"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a> <a id="b:43" class="idref" href="#b:43"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:43"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:40"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:43"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:42"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_le_ge" class="idref" href="#not_le_ge"><span class="id" title="lemma">not_le_ge</span></a> {<a id="A:44" class="idref" href="#A:44"><span class="id" title="binder">A</span></a>} {<a id="le:45" class="idref" href="#le:45"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:44"><span class="id" title="variable">A</span></a>} {<a id="P0:46" class="idref" href="#P0:46"><span class="id" title="binder">P0</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:45"><span class="id" title="variable">le</span></a>} :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:47" class="idref" href="#a:47"><span class="id" title="binder">a</span></a> <a id="b:48" class="idref" href="#b:48"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:45"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:47"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:48"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:45"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:48"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:47"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_not_gt" class="idref" href="#le_not_gt"><span class="id" title="lemma">le_not_gt</span></a> {<a id="A:49" class="idref" href="#A:49"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:50" class="idref" href="#le:50"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:49"><span class="id" title="variable">A</span></a>}{<a id="P:51" class="idref" href="#P:51"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:50"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a> <a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:50"><span class="id" title="variable">le</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:52"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_not_equiv" class="idref" href="#lt_not_equiv"><span class="id" title="lemma">lt_not_equiv</span></a>  {<a id="A:54" class="idref" href="#A:54"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:55" class="idref" href="#le:55"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:54"><span class="id" title="variable">A</span></a>}{<a id="P:56" class="idref" href="#P:56"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:55"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a> <a id="b:58" class="idref" href="#b:58"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:58"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:58"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="equiv_not_lt" class="idref" href="#equiv_not_lt"><span class="id" title="lemma">equiv_not_lt</span></a>  {<a id="A:59" class="idref" href="#A:59"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:60" class="idref" href="#le:60"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:59"><span class="id" title="variable">A</span></a>}{<a id="P:61" class="idref" href="#P:61"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:60"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a> <a id="b:63" class="idref" href="#b:63"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:62"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:63"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:62"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:63"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_le_trans" class="idref" href="#lt_le_trans"><span class="id" title="lemma">lt_le_trans</span></a> {<a id="A:64" class="idref" href="#A:64"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:65" class="idref" href="#le:65"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:64"><span class="id" title="variable">A</span></a>}{<a id="P:66" class="idref" href="#P:66"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:65"><span class="id" title="variable">le</span></a>}: <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:67" class="idref" href="#a:67"><span class="id" title="binder">a</span></a> <a id="b:68" class="idref" href="#b:68"><span class="id" title="binder">b</span></a> <a id="c:69" class="idref" href="#c:69"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:68"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:65"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:68"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#c:69"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#c:69"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_trans" class="idref" href="#le_lt_trans"><span class="id" title="lemma">le_lt_trans</span></a> {<a id="A:70" class="idref" href="#A:70"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:71" class="idref" href="#le:71"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:70"><span class="id" title="variable">A</span></a>}{<a id="P:72" class="idref" href="#P:72"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:71"><span class="id" title="variable">le</span></a>}: <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:73" class="idref" href="#a:73"><span class="id" title="binder">a</span></a> <a id="b:74" class="idref" href="#b:74"><span class="id" title="binder">b</span></a> <a id="c:75" class="idref" href="#c:75"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:71"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:73"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:74"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:74"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#c:75"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:73"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#c:75"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_weak" class="idref" href="#le_lt_weak"><span class="id" title="lemma">le_lt_weak</span></a> {<a id="A:76" class="idref" href="#A:76"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:77" class="idref" href="#le:77"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:76"><span class="id" title="variable">A</span></a>}{<a id="P:78" class="idref" href="#P:78"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:77"><span class="id" title="variable">le</span></a>}: <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:79" class="idref" href="#a:79"><span class="id" title="binder">a</span></a> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:80"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:77"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:80"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="lt_transitive" class="idref" href="#lt_transitive"><span class="id" title="instance">lt_transitive</span></a>  {<a id="A:81" class="idref" href="#A:81"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:82" class="idref" href="#le:82"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:81"><span class="id" title="variable">A</span></a>}{<a id="P:83" class="idref" href="#P:83"><span class="id" title="binder">P</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:82"><span class="id" title="variable">le</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="equiv_equiv" class="idref" href="#equiv_equiv"><span class="id" title="instance">equiv_equiv</span></a> {<a id="A:84" class="idref" href="#A:84"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="le:85" class="idref" href="#le:85"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:84"><span class="id" title="variable">A</span></a>} `{<a id="P:86" class="idref" href="#P:86"><span class="id" title="binder">P</span></a>:@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:84"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:85"><span class="id" title="variable">le</span></a> }: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_equiv_dec" class="idref" href="#le_lt_equiv_dec"><span class="id" title="lemma">le_lt_equiv_dec</span></a>  {<a id="A:87" class="idref" href="#A:87"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="le:88" class="idref" href="#le:88"><span class="id" title="binder">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:87"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;{<a id="P0:89" class="idref" href="#P0:89"><span class="id" title="binder">P0</span></a>: <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:88"><span class="id" title="variable">le</span></a>} {<a id="dec:90" class="idref" href="#dec:90"><span class="id" title="binder">dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:88"><span class="id" title="variable">le</span></a>} (<a id="a:91" class="idref" href="#a:91"><span class="id" title="binder">a</span></a> <a id="b:92" class="idref" href="#b:92"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder.html#A:87"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#le:88"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:91"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:92"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:91"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:92"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:91"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:92"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_le_gt" class="idref" href="#not_le_gt"><span class="id" title="lemma">not_le_gt</span></a> {<a id="A:93" class="idref" href="#A:93"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="le:94" class="idref" href="#le:94"><span class="id" title="binder">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#A:93"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P0:95" class="idref" href="#P0:95"><span class="id" title="binder">P0</span></a>: <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:94"><span class="id" title="variable">le</span></a>) (<a id="a:96" class="idref" href="#a:96"><span class="id" title="binder">a</span></a> <a id="b:97" class="idref" href="#b:97"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder.html#A:93"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#le:94"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:96"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:97"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#b:97"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#a:96"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
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+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.DecPreOrder_Instances</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.DecPreOrder_Instances</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Finite_sets.html#"><span class="id" title="library">Sets.Finite_sets</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Sets.Ensembles</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Nat_le_dec" class="idref" href="#Nat_le_dec"><span class="id" title="instance">Nat_le_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#le_dec"><span class="id" title="lemma">le_dec</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Nat_le_TO" class="idref" href="#Nat_le_TO"><span class="id" title="instance">Nat_le_TO</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Z_le_dec" class="idref" href="#Z_le_dec"><span class="id" title="instance">Z_le_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.le"><span class="id" title="definition">Z.le</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.ZArith_dec.html#Z_le_dec"><span class="id" title="definition">Z_le_dec</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Z_le_TO" class="idref" href="#Z_le_TO"><span class="id" title="instance">Z_le_TO</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.ZArith.BinInt.html#Z.le"><span class="id" title="definition">Z.le</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="module">DecPreOrder</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Pre-order associated with an inverse function 
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Inverse_fun" class="idref" href="#Inverse_fun"><span class="id" title="instance">Inverse_fun</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> <a id="B:2" class="idref" href="#B:2"><span class="id" title="binder">B</span></a>:<span class="id" title="keyword">Type</span>}{<a id="f:3" class="idref" href="#f:3"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:2"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="leB:4" class="idref" href="#leB:4"><span class="id" title="binder">leB</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:2"><span class="id" title="variable">B</span></a>}{<a id="PB:5" class="idref" href="#PB:5"><span class="id" title="binder">PB</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:4"><span class="id" title="variable">leB</span></a>}:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> (<span class="id" title="keyword">fun</span> <a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a> <a id="y:7" class="idref" href="#y:7"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:4"><span class="id" title="variable">leB</span></a> (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#x:6"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#y:7"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Total_Inverse_fun" class="idref" href="#Total_Inverse_fun"><span class="id" title="instance">Total_Inverse_fun</span></a> {<a id="A:8" class="idref" href="#A:8"><span class="id" title="binder">A</span></a> <a id="B:9" class="idref" href="#B:9"><span class="id" title="binder">B</span></a>:<span class="id" title="keyword">Type</span>}{<a id="f:10" class="idref" href="#f:10"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:8"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:9"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="leB:11" class="idref" href="#leB:11"><span class="id" title="binder">leB</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:9"><span class="id" title="variable">B</span></a>}{<a id="TB:12" class="idref" href="#TB:12"><span class="id" title="binder">TB</span></a>: <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:11"><span class="id" title="variable">leB</span></a>} :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> (<span class="id" title="keyword">fun</span> <a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a> <a id="y:14" class="idref" href="#y:14"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:11"><span class="id" title="variable">leB</span></a> (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#x:13"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:10"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#y:14"><span class="id" title="variable">y</span></a>)).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="RelDecision_Inverse_fun" class="idref" href="#RelDecision_Inverse_fun"><span class="id" title="instance">RelDecision_Inverse_fun</span></a> {<a id="A:15" class="idref" href="#A:15"><span class="id" title="binder">A</span></a> <a id="B:16" class="idref" href="#B:16"><span class="id" title="binder">B</span></a>:<span class="id" title="keyword">Type</span>}{<a id="f:17" class="idref" href="#f:17"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:15"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:16"><span class="id" title="variable">B</span></a>}<br/>
+&nbsp;{<a id="leB:18" class="idref" href="#leB:18"><span class="id" title="binder">leB</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:16"><span class="id" title="variable">B</span></a>} {<a id="RDB:19" class="idref" href="#RDB:19"><span class="id" title="binder">RDB</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:18"><span class="id" title="variable">leB</span></a>} :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> (<span class="id" title="keyword">fun</span> <a id="x:20" class="idref" href="#x:20"><span class="id" title="binder">x</span></a> <a id="y:21" class="idref" href="#y:21"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#leB:18"><span class="id" title="variable">leB</span></a> (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#x:20"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#y:21"><span class="id" title="variable">y</span></a>)) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> <a id="y:23" class="idref" href="#y:23"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide_rel"><span class="id" title="definition">decide_rel</span></a> <span class="id" title="var">leB</span> (<span class="id" title="var">f</span> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#x:22"><span class="id" title="variable">x</span></a>) (<span class="id" title="var">f</span> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#y:23"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Pre-order associated with inclusion 
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="PrO_Included" class="idref" href="#PrO_Included"><span class="id" title="instance">PrO_Included</span></a> {<a id="U:24" class="idref" href="#U:24"><span class="id" title="binder">U</span></a>:<span class="id" title="keyword">Type</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#U:24"><span class="id" title="variable">U</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> {<span class="id" title="var">U</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Included_s" class="idref" href="#Included_s"><span class="id" title="definition">Included_s</span></a> {<a id="U:25" class="idref" href="#U:25"><span class="id" title="binder">U</span></a>}  (<a id="A:26" class="idref" href="#A:26"><span class="id" title="binder">A</span></a> <a id="B:27" class="idref" href="#B:27"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#U:25"><span class="id" title="variable">U</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:26"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:27"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:27"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:26"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'&lt;:'_x" class="idref" href="#:::x_'&lt;:'_x"><span class="id" title="notation">&quot;</span></a>&lt;:" := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'''&lt;s'''_x" class="idref" href="#:::x_'''&lt;s'''_x"><span class="id" title="notation">&quot;</span></a>'&lt;s'" := <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#Included_s"><span class="id" title="definition">Included_s</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 30).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Example 
+  Lists (pre-ordered by their length) 
+</div>
+<div class="code">
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span>  <a id="List_length" class="idref" href="#List_length"><span class="id" title="instance">List_length</span></a> {<a id="A:28" class="idref" href="#A:28"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}: <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> (<span class="id" title="keyword">fun</span> <a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a> <a id="l':30" class="idref" href="#l':30"><span class="id" title="binder">l'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:28"><span class="id" title="variable">A</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#l:29"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#l':30"><span class="id" title="variable">l'</span></a>)%<span class="id" title="var">nat</span>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Non dependent lexicographic product 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="lexprod" class="idref" href="#lexprod"><span class="id" title="section">lexprod</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="lexprod.A" class="idref" href="#lexprod.A"><span class="id" title="variable">A</span></a> <a id="lexprod.B" class="idref" href="#lexprod.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lexprod.leA" class="idref" href="#lexprod.leA"><span class="id" title="variable">leA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#A:32"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lexprod.leB" class="idref" href="#lexprod.leB"><span class="id" title="variable">leB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#B:31"><span class="id" title="variable">B</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Context</span> (<a id="TA:35" class="idref" href="#TA:35"><span class="id" title="binder">TA</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="TB:36" class="idref" href="#TB:36"><span class="id" title="binder">TB</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="DA:37" class="idref" href="#DA:37"><span class="id" title="binder">DA</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="DB:38" class="idref" href="#DB:38"><span class="id" title="binder">DB</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="PA" class="idref" href="#PA"><span class="id" title="definition">PA</span></a> := @<a class="idref" href="hydras.Prelude.DecPreOrder.html#total_pre_order_pre"><span class="id" title="method">total_pre_order_pre</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.TA"><span class="id" title="variable">TA</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="PB" class="idref" href="#PB"><span class="id" title="definition">PB</span></a> := @<a class="idref" href="hydras.Prelude.DecPreOrder.html#total_pre_order_pre"><span class="id" title="method">total_pre_order_pre</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.TB"><span class="id" title="variable">TB</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="776f9baa25bd513014f05975f276fe02" class="idref" href="#776f9baa25bd513014f05975f276fe02"><span class="id" title="notation">&quot;</span></a>x '&lt;=A' y" := (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="lexprod.:::x_'&lt;A'_x" class="idref" href="#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">&quot;</span></a>x '&lt;A' y" := (@<a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#PA"><span class="id" title="definition">PA</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="431a2798f744cb20088a983430e54725" class="idref" href="#431a2798f744cb20088a983430e54725"><span class="id" title="notation">&quot;</span></a>x '&lt;=B' y" := (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="lexprod.:::x_'&lt;B'_x" class="idref" href="#lexprod.:::x_'&lt;B'_x"><span class="id" title="notation">&quot;</span></a>x '&lt;B' y" := (@<a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#PB"><span class="id" title="definition">PB</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> ) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="6623eebf3a76064b3501b2e089b67493" class="idref" href="#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">&quot;</span></a>x =A= y" := (@<a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#PA"><span class="id" title="definition">PA</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="29d4d7c85e1747f3f20f3fb958853966" class="idref" href="#29d4d7c85e1747f3f20f3fb958853966"><span class="id" title="notation">&quot;</span></a>x =B= y" := (@<a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB"><span class="id" title="variable">leB</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#PB"><span class="id" title="definition">PB</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="lex_prod" class="idref" href="#lex_prod"><span class="id" title="inductive">lex_prod</span></a> (<a id="p:39" class="idref" href="#p:39"><span class="id" title="binder">p</span></a> <a id="p':40" class="idref" href="#p':40"><span class="id" title="binder">p'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>): <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="lex1" class="idref" href="#lex1"><span class="id" title="constructor">lex1</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p:39"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p':40"><span class="id" title="variable">p'</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod:41"><span class="id" title="inductive">lex_prod</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p:39"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p':40"><span class="id" title="variable">p'</span></a><br/>
+| <a id="lex2" class="idref" href="#lex2"><span class="id" title="constructor">lex2</span></a> :  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p:39"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">A</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p':40"><span class="id" title="variable">p'</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p:39"><span class="id" title="variable">p</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p':40"><span class="id" title="variable">p'</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod:41"><span class="id" title="inductive">lex_prod</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p:39"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#p':40"><span class="id" title="variable">p'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="e523b9bcd6ab1830bbf762a9717f65e6" class="idref" href="#e523b9bcd6ab1830bbf762a9717f65e6"><span class="id" title="notation">&quot;</span></a>x '&lt;=lex' y" := (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod"><span class="id" title="inductive">lex_prod</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Notation</span> <a id="lexprod.:::x_'&lt;lex'_x" class="idref" href="#lexprod.:::x_'&lt;lex'_x"><span class="id" title="notation">&quot;</span></a>x '&lt;lex' y" := (@<a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>)  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod"><span class="id" title="inductive">lex_prod</span></a> <span class="id" title="var">_</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="PO_lex_prod" class="idref" href="#PO_lex_prod"><span class="id" title="instance">PO_lex_prod</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod"><span class="id" title="inductive">lex_prod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_of_equiv" class="idref" href="#lex_of_equiv"><span class="id" title="lemma">lex_of_equiv</span></a> (<a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="a':44" class="idref" href="#a':44"><span class="id" title="binder">a'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a>)(<a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a> <a id="b':46" class="idref" href="#b':46"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">A</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':44"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#29d4d7c85e1747f3f20f3fb958853966"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#29d4d7c85e1747f3f20f3fb958853966"><span class="id" title="notation">B</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#29d4d7c85e1747f3f20f3fb958853966"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':46"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#preorder_equiv"><span class="id" title="definition">preorder_equiv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:43"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:45"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':44"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':46"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_inv_left" class="idref" href="#lex_inv_left"><span class="id" title="lemma">lex_inv_left</span></a> (<a id="a:47" class="idref" href="#a:47"><span class="id" title="binder">a</span></a> <a id="a':48" class="idref" href="#a':48"><span class="id" title="binder">a'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a>)(<a id="b:49" class="idref" href="#b:49"><span class="id" title="binder">b</span></a> <a id="b':50" class="idref" href="#b':50"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:47"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:49"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#e523b9bcd6ab1830bbf762a9717f65e6"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#e523b9bcd6ab1830bbf762a9717f65e6"><span class="id" title="notation">lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':48"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':50"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:47"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#776f9baa25bd513014f05975f276fe02"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#776f9baa25bd513014f05975f276fe02"><span class="id" title="notation">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':48"><span class="id" title="variable">a'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_inv_right" class="idref" href="#lex_inv_right"><span class="id" title="lemma">lex_inv_right</span></a> (<a id="a:51" class="idref" href="#a:51"><span class="id" title="binder">a</span></a> <a id="a':52" class="idref" href="#a':52"><span class="id" title="binder">a'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a>)(<a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a> <a id="b':54" class="idref" href="#b':54"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:51"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:53"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#e523b9bcd6ab1830bbf762a9717f65e6"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#e523b9bcd6ab1830bbf762a9717f65e6"><span class="id" title="notation">lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':52"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':54"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:51"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">A</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':52"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725"><span class="id" title="notation">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':54"><span class="id" title="variable">b'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_strict_intro_right" class="idref" href="#lex_strict_intro_right"><span class="id" title="lemma">lex_strict_intro_right</span></a>  (<a id="a:55" class="idref" href="#a:55"><span class="id" title="binder">a</span></a> <a id="a':56" class="idref" href="#a':56"><span class="id" title="binder">a'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a>)(<a id="b:57" class="idref" href="#b:57"><span class="id" title="binder">b</span></a> <a id="b':58" class="idref" href="#b':58"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:55"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">A</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':56"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;B'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;B'_x"><span class="id" title="notation">B</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':58"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:55"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:57"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;lex'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;lex'_x"><span class="id" title="notation">lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':56"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':58"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_strict_intro_left" class="idref" href="#lex_strict_intro_left"><span class="id" title="lemma">lex_strict_intro_left</span></a>  (<a id="a:59" class="idref" href="#a:59"><span class="id" title="binder">a</span></a> <a id="a':60" class="idref" href="#a':60"><span class="id" title="binder">a'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A"><span class="id" title="variable">A</span></a>)(<a id="b:61" class="idref" href="#b:61"><span class="id" title="binder">b</span></a> <a id="b':62" class="idref" href="#b':62"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:59"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x"><span class="id" title="notation">A</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':60"><span class="id" title="variable">a'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a:59"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b:61"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;lex'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;lex'_x"><span class="id" title="notation">lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#a':60"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#b':62"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="To_lex_prod" class="idref" href="#To_lex_prod"><span class="id" title="instance">To_lex_prod</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod"><span class="id" title="inductive">lex_prod</span></a>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="lex_prod_dec" class="idref" href="#lex_prod_dec"><span class="id" title="instance">lex_prod_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod"><span class="id" title="inductive">lex_prod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#lexprod"><span class="id" title="section">lexprod</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Exp2.html b/theories/html/hydras.Prelude.Exp2.html
new file mode 100644
index 00000000..d6ba75c6
--- /dev/null
+++ b/theories/html/hydras.Prelude.Exp2.html
@@ -0,0 +1,56 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Exp2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Exp2</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="exp2" class="idref" href="#exp2"><span class="id" title="definition">exp2</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Exp2.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> ⇒ 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2:2"><span class="id" title="definition">exp2</span></a> <span class="id" title="var">i</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_positive" class="idref" href="#exp2_positive"><span class="id" title="lemma">exp2_positive</span></a> : <span class="id" title="keyword">∀</span> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#i:4"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_not_zero" class="idref" href="#exp2_not_zero"><span class="id" title="lemma">exp2_not_zero</span></a> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#i:5"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_gt_id" class="idref" href="#exp2_gt_id"><span class="id" title="lemma">exp2_gt_id</span></a> : <span class="id" title="keyword">∀</span> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Exp2.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#n:6"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2S" class="idref" href="#exp2S"><span class="id" title="lemma">exp2S</span></a> : <span class="id" title="keyword">∀</span> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#n:7"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#n:7"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.First_toggle.html b/theories/html/hydras.Prelude.First_toggle.html
new file mode 100644
index 00000000..393b6265
--- /dev/null
+++ b/theories/html/hydras.Prelude.First_toggle.html
@@ -0,0 +1,75 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.First_toggle</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.First_toggle</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ First_toggle 
+
+<div class="paragraph"> </div>
+
+Computes the first  <span class="inlinecode"><span class="id" title="var">l</span></span>  between <span class="inlinecode"><span class="id" title="var">n</span></span> and <span class="inlinecode"><span class="id" title="var">p</span></span> (excluded) such that 
+       <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">true</span></span> and  <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">l</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">false</span></span>.
+ 
+<div class="paragraph"> </div>
+
+ Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Inclusion</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Inverse_Image</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Hypos" class="idref" href="#Hypos"><span class="id" title="section">Hypos</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="P:1" class="idref" href="#P:1"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) `{<a id="Pdec:3" class="idref" href="#Pdec:3"><span class="id" title="binder">Pdec</span></a>: <span class="id" title="keyword">∀</span> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.STDPP_compat.html#Decision"><span class="id" title="class">Decision</span></a> (<a class="idref" href="hydras.Prelude.First_toggle.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#n:2"><span class="id" title="variable">n</span></a>)} (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypotheses</span> (<a id="Hypos.Hnp" class="idref" href="#Hypos.Hnp"><span class="id" title="variable">Hnp</span></a> : <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.p"><span class="id" title="variable">p</span></a>) (<a id="Hypos.Hn" class="idref" href="#Hypos.Hn"><span class="id" title="variable">Hn</span></a> : <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.n"><span class="id" title="variable">n</span></a>) (<a id="Hypos.Hp" class="idref" href="#Hypos.Hp"><span class="id" title="variable">Hp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Hypos.spec" class="idref" href="#Hypos.spec"><span class="id" title="variable">spec</span></a>  := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="l:9" class="idref" href="#l:9"><span class="id" title="binder">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#l:9"><span class="id" title="variable">l</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#l:9"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:10" class="idref" href="#i:10"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#i:10"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#i:10"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#l:9"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#i:10"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.First_toggle.html#l:9"><span class="id" title="variable">l</span></a> )<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="first_toggle" class="idref" href="#first_toggle"><span class="id" title="definition">first_toggle</span></a> : <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos.spec"><span class="id" title="variable">spec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.First_toggle.html#Hypos"><span class="id" title="section">Hypos</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.First_toggle.html#first_toggle"><span class="id" title="definition">first_toggle</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Fuel.html b/theories/html/hydras.Prelude.Fuel.html
new file mode 100644
index 00000000..175d320c
--- /dev/null
+++ b/theories/html/hydras.Prelude.Fuel.html
@@ -0,0 +1,117 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Fuel</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Fuel</h1>
+
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.FunInd.html#"><span class="id" title="library">FunInd</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="var">zero</span> (<a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  {<span class="id" title="keyword">wf</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <span class="id" title="var">n</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Fuel.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span> <span class="id" title="keyword">as</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#zero:2"><span class="id" title="definition">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#div2"><span class="id" title="definition">Nat.div2</span></a> <span class="id" title="var">p</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#lt_wf"><span class="id" title="lemma">lt_wf</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let's see whqat happens if lt_wf was opaque 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="OpaqueWf" class="idref" href="#OpaqueWf"><span class="id" title="module">OpaqueWf</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="OpaqueWf.lt_wf" class="idref" href="#OpaqueWf.lt_wf"><span class="id" title="lemma">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Function</span> <span class="id" title="var">zero</span> (<a id="n:29" class="idref" href="#n:29"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  {<span class="id" title="keyword">wf</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <span class="id" title="var">n</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Fuel.html#n:20"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#zero:21"><span class="id" title="definition">zero</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#div2"><span class="id" title="definition">Nat.div2</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#n:20"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+From Init.Wf 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc_intro_generator"><span class="id" title="definition">Acc_intro_generator</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Function</span> <span class="id" title="var">zero'</span> (<a id="n:49" class="idref" href="#n:49"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)  {<span class="id" title="keyword">wf</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <span class="id" title="var">n</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Fuel.html#n:40"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 0<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#zero':41"><span class="id" title="definition">zero'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#div2"><span class="id" title="definition">Nat.div2</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#n:40"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Fuel.html#OpaqueWf"><span class="id" title="module">OpaqueWf</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="fuel" class="idref" href="#fuel"><span class="id" title="inductive">fuel</span></a> :=<br/>
+&nbsp;| <a id="FO" class="idref" href="#FO"><span class="id" title="constructor">FO</span></a> : <a class="idref" href="hydras.Prelude.Fuel.html#fuel:60"><span class="id" title="inductive">fuel</span></a><br/>
+&nbsp;| <a id="FS" class="idref" href="#FS"><span class="id" title="constructor">FS</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#fuel:60"><span class="id" title="inductive">fuel</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#fuel:60"><span class="id" title="inductive">fuel</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="foo" class="idref" href="#foo"><span class="id" title="definition">foo</span></a> (<a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="x:63" class="idref" href="#x:63"><span class="id" title="binder">x</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Fuel.html#n:62"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#FS"><span class="id" title="constructor">FS</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#foo:64"><span class="id" title="definition">foo</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#n:62"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.Fuel.html#foo:64"><span class="id" title="definition">foo</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Fuel.html#x:63"><span class="id" title="variable">x</span></a>))<br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Prelude.Fuel.html#x:63"><span class="id" title="variable">x</span></a><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Iterates.html b/theories/html/hydras.Prelude.Iterates.html
new file mode 100644
index 00000000..b1c8f5df
--- /dev/null
+++ b/theories/html/hydras.Prelude.Iterates.html
@@ -0,0 +1,390 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Iterates</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Iterates</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Iteration of a function (similar to <span class="inlinecode"><span class="id" title="var">Nat.iter</span></span>) 
+     Abstract Properties
+
+<div class="paragraph"> </div>
+
+Experimental use of LibHyps 
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Max.html#"><span class="id" title="library">Max</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="library">Exp2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a>.<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">rename_hyp</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span> ::= <span class="id" title="var">rename_short</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="iterate" class="idref" href="#iterate"><span class="id" title="definition">iterate</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="f:2" class="idref" href="#f:2"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="x:4" class="idref" href="#x:4"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Iterates.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Iterates.html#n:3"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0 ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#x:4"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#f:2"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate:5"><span class="id" title="definition">iterate</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#f:2"><span class="id" title="variable">f</span></a> <span class="id" title="var">p</span> <a class="idref" href="hydras.Prelude.Iterates.html#x:4"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_comm" class="idref" href="#iterate_comm"><span class="id" title="lemma">iterate_comm</span></a> {<a id="A:7" class="idref" href="#A:7"><span class="id" title="binder">A</span></a>: <span class="id" title="keyword">Type</span>} <a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> (<a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Iterates.html#A:7"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:9"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:10"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:8"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:10"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Compatibility with Ackermann Library's definition 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_compat" class="idref" href="#iterate_compat"><span class="id" title="lemma">iterate_compat</span></a> {<a id="f:11" class="idref" href="#f:11"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}(<a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:11"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#x:16"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="rec:14" class="idref" href="#rec:14"><span class="id" title="binder">rec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#f:11"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#rec:14"><span class="id" title="variable">rec</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:15"><span class="id" title="variable">x</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:13"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_compat2" class="idref" href="#iterate_compat2"><span class="id" title="lemma">iterate_compat2</span></a> {<a id="A:17" class="idref" href="#A:17"><span class="id" title="binder">A</span></a>} (<a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:17"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:17"><span class="id" title="variable">A</span></a>) <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:20" class="idref" href="#x:20"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:20"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.iter"><span class="id" title="definition">Nat.iter</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:20"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+TODO : move to more generic libraries 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_compat3" class="idref" href="#iterate_compat3"><span class="id" title="lemma">iterate_compat3</span></a> <a id="f:21" class="idref" href="#f:21"><span class="id" title="binder">f</span></a> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:23"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#x:22"><span class="id" title="variable">x</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="y:24" class="idref" href="#y:24"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#f:21"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:24"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#n:23"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab9"></a><h2 class="section">Abstract properties of arithmetic functions</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strict_mono" class="idref" href="#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a id="f:25" class="idref" href="#f:25"><span class="id" title="binder">f</span></a> := <span class="id" title="keyword">∀</span> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> <a id="p:27" class="idref" href="#p:27"><span class="id" title="binder">p</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:27"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:26"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:25"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:27"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates_from" class="idref" href="#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a> <a id="g:29" class="idref" href="#g:29"><span class="id" title="binder">g</span></a> <a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> := <span class="id" title="keyword">∀</span> <a id="p:31" class="idref" href="#p:31"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:31"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:31"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:29"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:31"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fun_le" class="idref" href="#fun_le"><span class="id" title="definition">fun_le</span></a> <a id="f:32" class="idref" href="#f:32"><span class="id" title="binder">f</span></a> <a id="g:33" class="idref" href="#g:33"><span class="id" title="binder">g</span></a> := <span class="id" title="keyword">∀</span> <a id="n:34" class="idref" href="#n:34"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#f:32"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:34"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#g:33"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:34"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="25e51f6c9a7647f5a562b237127007c4" class="idref" href="#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&quot;</span></a>&lt;&lt;=" := <a class="idref" href="hydras.Prelude.Iterates.html#fun_le"><span class="id" title="definition">fun_le</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates" class="idref" href="#dominates"><span class="id" title="definition">dominates</span></a> <a id="g:35" class="idref" href="#g:35"><span class="id" title="binder">g</span></a> <a id="f:36" class="idref" href="#f:36"><span class="id" title="binder">f</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n:37" class="idref" href="#n:37"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:37"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:35"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:36"><span class="id" title="variable">f</span></a> .<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'&gt;&gt;'_x" class="idref" href="#:::x_'&gt;&gt;'_x"><span class="id" title="notation">&quot;</span></a>&gt;&gt;" := <a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="dominates_strong" class="idref" href="#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> <a id="g:38" class="idref" href="#g:38"><span class="id" title="binder">g</span></a> <a id="f:39" class="idref" href="#f:39"><span class="id" title="binder">f</span></a>  := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="n:40" class="idref" href="#n:40"><span class="id" title="binder">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:40"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:38"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:39"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id=":::x_'&gt;&gt;s'_x" class="idref" href="#:::x_'&gt;&gt;s'_x"><span class="id" title="notation">&quot;</span></a>&gt;&gt;s" := <a class="idref" href="hydras.Prelude.Iterates.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="S_pred_rw" class="idref" href="#S_pred_rw"><span class="id" title="lemma">S_pred_rw</span></a> (<a id="f:41" class="idref" href="#f:41"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:42" class="idref" href="#x:42"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:42"><span class="id" title="variable">x</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:42"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fun_le_trans" class="idref" href="#fun_le_trans"><span class="id" title="lemma">fun_le_trans</span></a> <a id="f:43" class="idref" href="#f:43"><span class="id" title="binder">f</span></a> <a id="g:44" class="idref" href="#g:44"><span class="id" title="binder">g</span></a> <a id="h:45" class="idref" href="#h:45"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#f:43"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:44"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#g:44"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:45"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:43"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:45"><span class="id" title="variable">h</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mono_le" class="idref" href="#mono_le"><span class="id" title="lemma">mono_le</span></a> <a id="f:46" class="idref" href="#f:46"><span class="id" title="binder">f</span></a> (<a id="Hf:47" class="idref" href="#Hf:47"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:46"><span class="id" title="variable">f</span></a>) :  <span class="id" title="keyword">∀</span> <a id="n:48" class="idref" href="#n:48"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#n:48"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:46"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:48"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mono_injective" class="idref" href="#mono_injective"><span class="id" title="lemma">mono_injective</span></a> <a id="f:49" class="idref" href="#f:49"><span class="id" title="binder">f</span></a> (<a id="Hf:50" class="idref" href="#Hf:50"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:49"><span class="id" title="variable">f</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:51" class="idref" href="#n:51"><span class="id" title="binder">n</span></a> <a id="p:52" class="idref" href="#p:52"><span class="id" title="binder">p</span></a> , <a class="idref" href="hydras.Prelude.Iterates.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:51"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:49"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:52"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:51"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:52"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mono_weak" class="idref" href="#mono_weak"><span class="id" title="lemma">mono_weak</span></a> <a id="f:53" class="idref" href="#f:53"><span class="id" title="binder">f</span></a> (<a id="H:54" class="idref" href="#H:54"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:53"><span class="id" title="variable">f</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:55" class="idref" href="#n:55"><span class="id" title="binder">n</span></a> <a id="p:56" class="idref" href="#p:56"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#n:55"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:56"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:53"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:55"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:53"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:56"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dominates_from_trans" class="idref" href="#dominates_from_trans"><span class="id" title="lemma">dominates_from_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:57" class="idref" href="#f:57"><span class="id" title="binder">f</span></a> <a id="g:58" class="idref" href="#g:58"><span class="id" title="binder">g</span></a> <a id="h:59" class="idref" href="#h:59"><span class="id" title="binder">h</span></a> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a> <a id="j:61" class="idref" href="#j:61"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:58"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:57"><span class="id" title="variable">f</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:61"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:59"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:58"><span class="id" title="variable">g</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:61"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#h:59"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:57"><span class="id" title="variable">f</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dominates_trans" class="idref" href="#dominates_trans"><span class="id" title="lemma">dominates_trans</span></a> <a id="f:62" class="idref" href="#f:62"><span class="id" title="binder">f</span></a> <a id="g:63" class="idref" href="#g:63"><span class="id" title="binder">g</span></a> <a id="h:64" class="idref" href="#h:64"><span class="id" title="binder">h</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:63"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:62"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:64"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:63"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates"><span class="id" title="definition">dominates</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:64"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:62"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dominates_trans_strong" class="idref" href="#dominates_trans_strong"><span class="id" title="lemma">dominates_trans_strong</span></a> : <span class="id" title="keyword">∀</span> <a id="f:65" class="idref" href="#f:65"><span class="id" title="binder">f</span></a> <a id="g:66" class="idref" href="#g:66"><span class="id" title="binder">g</span></a> <a id="h:67" class="idref" href="#h:67"><span class="id" title="binder">h</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:66"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:65"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:67"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:66"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_strong"><span class="id" title="definition">dominates_strong</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:67"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:65"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab10"></a><h2 class="section">Abstract properties of iterate</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_S_eqn" class="idref" href="#iterate_S_eqn"><span class="id" title="lemma">iterate_S_eqn</span></a> {<a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="f:69" class="idref" href="#f:69"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:68"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:68"><span class="id" title="variable">A</span></a>) (<a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="x:71" class="idref" href="#x:71"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Iterates.html#A:68"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:69"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:70"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#x:71"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:69"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:69"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:71"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_S_eqn2" class="idref" href="#iterate_S_eqn2"><span class="id" title="lemma">iterate_S_eqn2</span></a> {<a id="A:72" class="idref" href="#A:72"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="f:73" class="idref" href="#f:73"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:72"><span class="id" title="variable">A</span></a>) (<a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Iterates.html#A:72"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:73"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:74"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:74"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:73"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:75"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_rw" class="idref" href="#iterate_rw"><span class="id" title="lemma">iterate_rw</span></a> {<a id="A:76" class="idref" href="#A:76"><span class="id" title="binder">A</span></a>} {<a id="f:77" class="idref" href="#f:77"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:76"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:76"><span class="id" title="variable">A</span></a>} <a id="n:78" class="idref" href="#n:78"><span class="id" title="binder">n</span></a>  :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:79" class="idref" href="#x:79"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:77"><span class="id" title="variable">f</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:78"><span class="id" title="variable">n</span></a>)  <a class="idref" href="hydras.Prelude.Iterates.html#x:79"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:78"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:77"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:79"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ext" class="idref" href="#iterate_ext"><span class="id" title="lemma">iterate_ext</span></a> {<a id="A:80" class="idref" href="#A:80"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="f:81" class="idref" href="#f:81"><span class="id" title="binder">f</span></a> <a id="g:82" class="idref" href="#g:82"><span class="id" title="binder">g</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#A:80"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:80"><span class="id" title="variable">A</span></a>) (<a id="H:84" class="idref" href="#H:84"><span class="id" title="binder">H</span></a>: <span class="id" title="keyword">∀</span> <a id="x:83" class="idref" href="#x:83"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#f:81"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:83"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:82"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:83"><span class="id" title="variable">x</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:85" class="idref" href="#n:85"><span class="id" title="binder">n</span></a> <a id="x:86" class="idref" href="#x:86"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:81"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:85"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:86"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:82"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:85"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:86"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_le" class="idref" href="#iterate_le"><span class="id" title="lemma">iterate_le</span></a> <a id="f:87" class="idref" href="#f:87"><span class="id" title="binder">f</span></a> (<a id="Hf:88" class="idref" href="#Hf:88"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:87"><span class="id" title="variable">f</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:89" class="idref" href="#i:89"><span class="id" title="binder">i</span></a> <a id="j:90" class="idref" href="#j:90"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#i:89"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:90"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="z:91" class="idref" href="#z:91"><span class="id" title="binder">z</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:87"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:89"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:91"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:87"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:90"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:91"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_lt" class="idref" href="#iterate_lt"><span class="id" title="lemma">iterate_lt</span></a>  <a id="f:92" class="idref" href="#f:92"><span class="id" title="binder">f</span></a> (<a id="Hf:93" class="idref" href="#Hf:93"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:92"><span class="id" title="variable">f</span></a>)(<a id="Hf':94" class="idref" href="#Hf':94"><span class="id" title="binder">Hf'</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#fun_le"><span class="id" title="definition">fun_le</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:92"><span class="id" title="variable">f</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:95" class="idref" href="#i:95"><span class="id" title="binder">i</span></a> <a id="j:96" class="idref" href="#j:96"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#i:95"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:96"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="z:97" class="idref" href="#z:97"><span class="id" title="binder">z</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:92"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:95"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:97"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:92"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:96"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:97"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_lt_from" class="idref" href="#iterate_lt_from"><span class="id" title="lemma">iterate_lt_from</span></a> <a id="f:98" class="idref" href="#f:98"><span class="id" title="binder">f</span></a> <a id="k:99" class="idref" href="#k:99"><span class="id" title="binder">k</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:98"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a> <span class="id" title="keyword">∀</span> <a id="n:100" class="idref" href="#n:100"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#k:99"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:98"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:100"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:101" class="idref" href="#i:101"><span class="id" title="binder">i</span></a> <a id="j:102" class="idref" href="#j:102"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#i:101"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:102"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="z:103" class="idref" href="#z:103"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#k:99"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:103"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:98"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:101"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:103"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:98"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:102"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:103"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_le_n_Sn" class="idref" href="#iterate_le_n_Sn"><span class="id" title="lemma">iterate_le_n_Sn</span></a> (<a id="f:104" class="idref" href="#f:104"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:105" class="idref" href="#x:105"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#x:105"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:105"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:106" class="idref" href="#n:106"><span class="id" title="binder">n</span></a> <a id="x:107" class="idref" href="#x:107"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:106"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:104"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:106"><span class="id" title="variable">n</span></a>)  <a class="idref" href="hydras.Prelude.Iterates.html#x:107"><span class="id" title="variable">x</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_le_np_le" class="idref" href="#iterate_le_np_le"><span class="id" title="lemma">iterate_le_np_le</span></a> (<a id="f:108" class="idref" href="#f:108"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:109" class="idref" href="#x:109"><span class="id" title="binder">x</span></a>,  (<a class="idref" href="hydras.Prelude.Iterates.html#x:109"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:108"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:109"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">nat</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:110" class="idref" href="#n:110"><span class="id" title="binder">n</span></a> <a id="p:111" class="idref" href="#p:111"><span class="id" title="binder">p</span></a> <a id="x:112" class="idref" href="#x:112"><span class="id" title="binder">x</span></a>,  (<a class="idref" href="hydras.Prelude.Iterates.html#n:110"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:111"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:108"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:110"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:112"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:108"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:111"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:112"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_mono2" class="idref" href="#iterate_mono2"><span class="id" title="lemma">iterate_mono2</span></a> (<a id="f:113" class="idref" href="#f:113"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="x:114" class="idref" href="#x:114"><span class="id" title="binder">x</span></a> <a id="y:115" class="idref" href="#y:115"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#x:114"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:115"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:113"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:114"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:113"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:115"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:116" class="idref" href="#n:116"><span class="id" title="binder">n</span></a> <a id="x:117" class="idref" href="#x:117"><span class="id" title="binder">x</span></a> <a id="y:118" class="idref" href="#y:118"><span class="id" title="binder">y</span></a>,  (<a class="idref" href="hydras.Prelude.Iterates.html#x:117"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:118"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:113"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:117"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:113"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:116"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:118"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_mono" class="idref" href="#iterate_mono"><span class="id" title="lemma">iterate_mono</span></a> <a id="f:119" class="idref" href="#f:119"><span class="id" title="binder">f</span></a> (<a id="Hf:120" class="idref" href="#Hf:120"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:119"><span class="id" title="variable">f</span></a>) (<a id="Hf':121" class="idref" href="#Hf':121"><span class="id" title="binder">Hf'</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:119"><span class="id" title="variable">f</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:122" class="idref" href="#n:122"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:119"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:122"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ge" class="idref" href="#iterate_ge"><span class="id" title="lemma">iterate_ge</span></a> : <span class="id" title="keyword">∀</span> <a id="f:123" class="idref" href="#f:123"><span class="id" title="binder">f</span></a> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:123"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="j:124" class="idref" href="#j:124"><span class="id" title="binder">j</span></a> <a id="n:125" class="idref" href="#n:125"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#j:124"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:123"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:125"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:124"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_Sge" class="idref" href="#iterate_Sge"><span class="id" title="lemma">iterate_Sge</span></a> <a id="f:126" class="idref" href="#f:126"><span class="id" title="binder">f</span></a> <a id="j:127" class="idref" href="#j:127"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:126"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:126"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:127"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ge'" class="idref" href="#iterate_ge'"><span class="id" title="lemma">iterate_ge'</span></a> : <span class="id" title="keyword">∀</span> <a id="f:128" class="idref" href="#f:128"><span class="id" title="binder">f</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:128"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:129" class="idref" href="#n:129"><span class="id" title="binder">n</span></a> <a id="j:130" class="idref" href="#j:130"><span class="id" title="binder">j</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:129"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:130"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:128"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:129"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:130"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ge''" class="idref" href="#iterate_ge''"><span class="id" title="lemma">iterate_ge''</span></a> <a id="f:131" class="idref" href="#f:131"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:131"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:131"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="i:132" class="idref" href="#i:132"><span class="id" title="binder">i</span></a> <a id="k:133" class="idref" href="#k:133"><span class="id" title="binder">k</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#k:133"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<span class="id" title="keyword">fun</span> <a id="z:134" class="idref" href="#z:134"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:131"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:134"><span class="id" title="variable">z</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:132"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#k:133"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="strict_mono_iterate_S" class="idref" href="#strict_mono_iterate_S"><span class="id" title="lemma">strict_mono_iterate_S</span></a> <a id="f:141" class="idref" href="#f:141"><span class="id" title="binder">f</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:141"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:141"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:142" class="idref" href="#i:142"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="k:143" class="idref" href="#k:143"><span class="id" title="binder">k</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<span class="id" title="keyword">fun</span> <a id="z:144" class="idref" href="#z:144"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:141"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#z:144"><span class="id" title="variable">z</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:142"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.Iterates.html#k:143"><span class="id" title="variable">k</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_mono_1" class="idref" href="#iterate_mono_1"><span class="id" title="lemma">iterate_mono_1</span></a> (<a id="f:151" class="idref" href="#f:151"><span class="id" title="binder">f</span></a> <a id="g:152" class="idref" href="#g:152"><span class="id" title="binder">g</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="k:153" class="idref" href="#k:153"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="Hf:154" class="idref" href="#Hf:154"><span class="id" title="binder">Hf</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:151"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hf':155" class="idref" href="#Hf':155"><span class="id" title="binder">Hf'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:151"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="H:157" class="idref" href="#H:157"><span class="id" title="binder">H</span></a> : <span class="id" title="keyword">∀</span> <a id="n:156" class="idref" href="#n:156"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#k:153"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:156"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:151"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:156"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:152"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:156"><span class="id" title="variable">n</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:158" class="idref" href="#i:158"><span class="id" title="binder">i</span></a> <a id="n:159" class="idref" href="#n:159"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#k:153"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:159"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:151"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:158"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:159"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:152"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:158"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:159"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_dom_prop" class="idref" href="#iterate_dom_prop"><span class="id" title="lemma">iterate_dom_prop</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:160" class="idref" href="#f:160"><span class="id" title="binder">f</span></a> <a id="g:161" class="idref" href="#g:161"><span class="id" title="binder">g</span></a> <a id="i:162" class="idref" href="#i:162"><span class="id" title="binder">i</span></a> (<a id="Hgt:163" class="idref" href="#Hgt:163"><span class="id" title="binder">Hgt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:160"><span class="id" title="variable">f</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hm:164" class="idref" href="#Hm:164"><span class="id" title="binder">Hm</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:160"><span class="id" title="variable">f</span></a>) (<a id="Hm':165" class="idref" href="#Hm':165"><span class="id" title="binder">Hm'</span></a>: <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:161"><span class="id" title="variable">g</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:162"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:161"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:166" class="idref" href="#k:166"><span class="id" title="binder">k</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:166"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:162"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:161"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:166"><span class="id" title="variable">k</span></a>) (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:166"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dominates_from_le" class="idref" href="#dominates_from_le"><span class="id" title="lemma">dominates_from_le</span></a>  <a id="i:167" class="idref" href="#i:167"><span class="id" title="binder">i</span></a> <a id="j:168" class="idref" href="#j:168"><span class="id" title="binder">j</span></a> <a id="g:169" class="idref" href="#g:169"><span class="id" title="binder">g</span></a> <a id="f:170" class="idref" href="#f:170"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#i:167"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:168"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:167"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:169"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:170"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:168"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:169"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:170"><span class="id" title="variable">f</span></a> .<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="smono_Sle" class="idref" href="#smono_Sle"><span class="id" title="lemma">smono_Sle</span></a> <a id="f:171" class="idref" href="#f:171"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#f:171"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:171"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:171"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab11"></a><h2 class="section">Second-order iterate</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ext2" class="idref" href="#iterate_ext2"><span class="id" title="lemma">iterate_ext2</span></a> {<a id="A:172" class="idref" href="#A:172"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="f:173" class="idref" href="#f:173"><span class="id" title="binder">f</span></a> <a id="g:174" class="idref" href="#g:174"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="h:175" class="idref" href="#h:175"><span class="id" title="binder">h</span></a> <a id="i:176" class="idref" href="#i:176"><span class="id" title="binder">i</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Prelude.Iterates.html#A:172"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:177" class="idref" href="#x:177"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#h:175"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:177"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:176"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:177"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="h':178" class="idref" href="#h':178"><span class="id" title="binder">h'</span></a> <a id="i':179" class="idref" href="#i':179"><span class="id" title="binder">i'</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:180" class="idref" href="#x:180"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#h':178"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:180"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i':179"><span class="id" title="variable">i'</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:180"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:181" class="idref" href="#x:181"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#f:173"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h':178"><span class="id" title="variable">h'</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:181"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:174"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i':179"><span class="id" title="variable">i'</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:181"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:182" class="idref" href="#n:182"><span class="id" title="binder">n</span></a> <a id="x:183" class="idref" href="#x:183"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:173"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:182"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:175"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:183"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:174"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:182"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:176"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:183"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate2_mono" class="idref" href="#iterate2_mono"><span class="id" title="lemma">iterate2_mono</span></a> (<a id="f:184" class="idref" href="#f:184"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:185" class="idref" href="#g:185"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:185"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:185"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:184"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:185"><span class="id" title="variable">g</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:186" class="idref" href="#g:186"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:186"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:186"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#f:184"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:186"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:187" class="idref" href="#k:187"><span class="id" title="binder">k</span></a> <a id="g:188" class="idref" href="#g:188"><span class="id" title="binder">g</span></a> <a id="x:189" class="idref" href="#x:189"><span class="id" title="binder">x</span></a>  <a id="y:190" class="idref" href="#y:190"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:188"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:188"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#x:189"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:190"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:184"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:187"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:188"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:189"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:184"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:187"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:188"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:190"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate2_mono_weak" class="idref" href="#iterate2_mono_weak"><span class="id" title="lemma">iterate2_mono_weak</span></a> (<a id="f:191" class="idref" href="#f:191"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:192" class="idref" href="#g:192"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:192"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:192"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#f:191"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:192"><span class="id" title="variable">g</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:193" class="idref" href="#g:193"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:193"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:193"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#f:191"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:193"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:194" class="idref" href="#k:194"><span class="id" title="binder">k</span></a> <a id="g:195" class="idref" href="#g:195"><span class="id" title="binder">g</span></a> <a id="x:196" class="idref" href="#x:196"><span class="id" title="binder">x</span></a>  <a id="y:197" class="idref" href="#y:197"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:195"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:195"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#x:196"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:197"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:191"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:194"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:195"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:196"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:191"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:194"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:195"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:197"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate2_mono3" class="idref" href="#iterate2_mono3"><span class="id" title="lemma">iterate2_mono3</span></a> (<a id="phi:198" class="idref" href="#phi:198"><span class="id" title="binder">phi</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:199" class="idref" href="#g:199"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:199"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:199"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:199"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:199"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="f:200" class="idref" href="#f:200"><span class="id" title="binder">f</span></a> <a id="g:201" class="idref" href="#g:201"><span class="id" title="binder">g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:200"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:200"><span class="id" title="variable">f</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:201"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:201"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">((</span></a><span class="id" title="keyword">∀</span> <a id="x:202" class="idref" href="#x:202"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#f:200"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:202"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:201"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:202"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:203" class="idref" href="#x:203"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:200"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:203"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:201"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:203"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="g:204" class="idref" href="#g:204"><span class="id" title="binder">g</span></a> <a id="h:205" class="idref" href="#h:205"><span class="id" title="binder">h</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:204"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:204"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:205"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:205"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:206" class="idref" href="#x:206"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#g:204"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:206"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:205"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:206"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:207" class="idref" href="#k:207"><span class="id" title="binder">k</span></a> <a id="x:208" class="idref" href="#x:208"><span class="id" title="binder">x</span></a> <a id="y:209" class="idref" href="#y:209"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#x:208"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:209"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:207"><span class="id" title="variable">k</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#g:204"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:208"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:198"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:207"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#h:205"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:209"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate2_mono2" class="idref" href="#iterate2_mono2"><span class="id" title="lemma">iterate2_mono2</span></a> (<a id="phi:214" class="idref" href="#phi:214"><span class="id" title="binder">phi</span></a> <a id="psi:215" class="idref" href="#psi:215"><span class="id" title="binder">psi</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:216" class="idref" href="#g:216"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:216"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:216"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#phi:214"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:216"><span class="id" title="variable">g</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:217" class="idref" href="#g:217"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:217"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:217"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#phi:214"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:217"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:218" class="idref" href="#g:218"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:218"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:218"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:218"><span class="id" title="variable">g</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:219" class="idref" href="#g:219"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:219"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:219"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:219"><span class="id" title="variable">g</span></a><a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="g:220" class="idref" href="#g:220"><span class="id" title="binder">g</span></a> <a id="x:221" class="idref" href="#x:221"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:220"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#fun_le"><span class="id" title="definition">fun_le</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:220"><span class="id" title="variable">g</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:214"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:220"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:221"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:220"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:221"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:222" class="idref" href="#f:222"><span class="id" title="binder">f</span></a> <a id="g:223" class="idref" href="#g:223"><span class="id" title="binder">g</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:222"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:223"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:222"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:223"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:224" class="idref" href="#x:224"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#f:222"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:224"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:223"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:224"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:225" class="idref" href="#x:225"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:222"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:225"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:223"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:225"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="k:226" class="idref" href="#k:226"><span class="id" title="binder">k</span></a> <a id="g:227" class="idref" href="#g:227"><span class="id" title="binder">g</span></a> <a id="x:228" class="idref" href="#x:228"><span class="id" title="binder">x</span></a>  <a id="y:229" class="idref" href="#y:229"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:227"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:227"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#x:228"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:229"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#phi:214"><span class="id" title="variable">phi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:226"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:227"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#x:228"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#psi:215"><span class="id" title="variable">psi</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:226"><span class="id" title="variable">k</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#g:227"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#y:229"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab12"></a><h2 class="section">Exponential and hyper exponential of base 2</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_ge_S" class="idref" href="#exp2_ge_S"><span class="id" title="lemma">exp2_ge_S</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a>  <a class="idref" href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4"><span class="id" title="notation">&lt;&lt;=</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_mono" class="idref" href="#exp2_mono"><span class="id" title="lemma">exp2_mono</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_mono_weak" class="idref" href="#exp2_mono_weak"><span class="id" title="lemma">exp2_mono_weak</span></a> : <span class="id" title="keyword">∀</span> <a id="n:230" class="idref" href="#n:230"><span class="id" title="binder">n</span></a> <a id="p:231" class="idref" href="#p:231"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#n:230"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:231"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:230"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#p:231"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exp2_as_iterate" class="idref" href="#exp2_as_iterate"><span class="id" title="lemma">exp2_as_iterate</span></a> <a id="n:232" class="idref" href="#n:232"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:232"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<span class="id" title="keyword">fun</span> <a id="i:233" class="idref" href="#i:233"><span class="id" title="binder">i</span></a> ⇒ 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:233"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="hydras.Prelude.Iterates.html#n:232"><span class="id" title="variable">n</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="hyper_exp2" class="idref" href="#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> <a id="k:234" class="idref" href="#k:234"><span class="id" title="binder">k</span></a> := <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:234"><span class="id" title="variable">k</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="hyper_exp2_S" class="idref" href="#hyper_exp2_S"><span class="id" title="lemma">hyper_exp2_S</span></a> : <span class="id" title="keyword">∀</span> <a id="n:235" class="idref" href="#n:235"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:235"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#hyper_exp2"><span class="id" title="definition">hyper_exp2</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:235"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iterate_ge_from" class="idref" href="#iterate_ge_from"><span class="id" title="lemma">iterate_ge_from</span></a> : <span class="id" title="keyword">∀</span> <a id="f:236" class="idref" href="#f:236"><span class="id" title="binder">f</span></a> <a id="i:237" class="idref" href="#i:237"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:237"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:236"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="j:238" class="idref" href="#j:238"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#i:237"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:238"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:239" class="idref" href="#n:239"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#j:238"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:236"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:239"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:238"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="dominates_iterate" class="idref" href="#dominates_iterate"><span class="id" title="lemma">dominates_iterate</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="i:240" class="idref" href="#i:240"><span class="id" title="binder">i</span></a> <a id="f:241" class="idref" href="#f:241"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:240"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:241"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:241"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:242" class="idref" href="#n:242"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:243" class="idref" href="#j:243"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:240"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:243"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:243"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:241"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:242"><span class="id" title="variable">n</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="iterate_gt_diag'" class="idref" href="#iterate_gt_diag'"><span class="id" title="lemma">iterate_gt_diag'</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="i:244" class="idref" href="#i:244"><span class="id" title="binder">i</span></a> <a id="f:245" class="idref" href="#f:245"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:244"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:245"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:245"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:246" class="idref" href="#n:246"><span class="id" title="binder">n</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:246"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:247" class="idref" href="#j:247"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:244"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:247"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:247"><span class="id" title="variable">j</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:245"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:246"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="iterate_ge_diag'" class="idref" href="#iterate_ge_diag'"><span class="id" title="lemma">iterate_ge_diag'</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="i:248" class="idref" href="#i:248"><span class="id" title="binder">i</span></a> <a id="f:249" class="idref" href="#f:249"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:248"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:249"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Iterates.html#strict_mono"><span class="id" title="definition">strict_mono</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:249"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:250" class="idref" href="#n:250"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="j:251" class="idref" href="#j:251"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#i:248"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#j:251"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="k:252" class="idref" href="#k:252"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Prelude.Iterates.html#j:251"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:252"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:252"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#f:249"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#n:250"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#k:252"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.LibHyps_Experiments.html b/theories/html/hydras.Prelude.LibHyps_Experiments.html
new file mode 100644
index 00000000..c7192956
--- /dev/null
+++ b/theories/html/hydras.Prelude.LibHyps_Experiments.html
@@ -0,0 +1,57 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.LibHyps_Experiments</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.LibHyps_Experiments</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="library">LibHyps</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.MoreLibHyps.html#"><span class="id" title="library">MoreLibHyps</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">autonaming_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">rename_hyp</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span> ::= <span class="id" title="var">rename_short</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> <a id="p:2" class="idref" href="#p:2"><span class="id" title="binder">p</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="q:3" class="idref" href="#q:3"><span class="id" title="binder">q</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">p</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">q</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">∀</span></a> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> <a id="p:5" class="idref" href="#p:5"><span class="id" title="binder">p</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">p</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">∀</span></a> <a id="q:6" class="idref" href="#q:6"><span class="id" title="binder">q</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">p</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">q</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">n</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">≤</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">Parameters</span> <a id="f" class="idref" href="#f"><span class="id" title="axiom">f</span></a> <a id="g" class="idref" href="#g"><span class="id" title="axiom">g</span></a> <a id="h" class="idref" href="#h"><span class="id" title="axiom">h</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Parameter</span> <a id="P" class="idref" href="#P"><span class="id" title="axiom">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">∀</span></a> <a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">,</span></a>  <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">h</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">))</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">h</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">g</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">(</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">y</span></a><a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">))</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">y</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">&lt;</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">y</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">P</span></a>  <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">y</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a>  <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">→</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">y</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">≠</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">f</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">x</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Merge_Sort</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Merge_Sort</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+P. Casteran, S. Salvati 
+<div class="paragraph"> </div>
+
+ Maybe already done in StdLib ???? 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.funind.Recdef.html#"><span class="id" title="library">Recdef</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#"><span class="id" title="library">Peano_dec</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorting.Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#"><span class="id" title="library">Sorting.Permutation</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#"><span class="id" title="library">DecPreOrder_Instances</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#"><span class="id" title="library">Sort_spec</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="module">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#"><span class="id" title="module">Morphisms</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Generic" class="idref" href="#Generic"><span class="id" title="section">Generic</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Generic.A" class="idref" href="#Generic.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Generic.Splitting" class="idref" href="#Generic.Splitting"><span class="id" title="section">Splitting</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A function that splits any list in two lists of (almost) the same length
+     
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Function</span> <span class="id" title="tactic">split</span> (<a id="l:14" class="idref" href="#l:14"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:2"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l:2"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="l1:17" class="idref" href="#l1:17"><span class="id" title="binder">l1</span></a>,<a id="l2:18" class="idref" href="#l2:18"><span class="id" title="binder">l2</span></a>):= <a class="idref" href="hydras.Prelude.Merge_Sort.html#split:3"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:2"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:5"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:6"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Function</span> <span class="id" title="var">split'_aux</span> (<a id="l:51" class="idref" href="#l:51"><span class="id" title="binder">l</span></a> <a id="l':52" class="idref" href="#l':52"><span class="id" title="binder">l'</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:33"><span class="id" title="variable">l</span></a>,<a class="idref" href="hydras.Prelude.Merge_Sort.html#l':34"><span class="id" title="variable">l'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span>,<span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="l1:56" class="idref" href="#l1:56"><span class="id" title="binder">l1</span></a>,<a id="l2:57" class="idref" href="#l2:57"><span class="id" title="binder">l2</span></a>) := <a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux:35"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:33"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':34"><span class="id" title="variable">l'</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:38"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:39"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>,<span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l:33"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Function</span> <span class="id" title="var">split'</span> (<a id="l:86" class="idref" href="#l:86"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:80"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:80"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Applying split  to a list l returns a pair of two strictly shorter lists 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split_decr" class="idref" href="#split_decr"><span class="id" title="lemma">split_decr</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:91" class="idref" href="#l1:91"><span class="id" title="binder">l1</span></a> <a id="l2:92" class="idref" href="#l2:92"><span class="id" title="binder">l2</span></a> <a id="a:93" class="idref" href="#a:93"><span class="id" title="binder">a</span></a> <a id="b:94" class="idref" href="#b:94"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:91"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a:93"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#b:94"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:92"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:91"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:91"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:91"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:91"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split_permutation" class="idref" href="#split_permutation"><span class="id" title="lemma">split_permutation</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:95" class="idref" href="#l:95"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:95"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)++</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:95"><span class="id" title="variable">l</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:95"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split'_aux_length_preserve" class="idref" href="#split'_aux_length_preserve"><span class="id" title="lemma">split'_aux_length_preserve</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:96" class="idref" href="#l:96"><span class="id" title="binder">l</span></a> <a id="l':97" class="idref" href="#l':97"><span class="id" title="binder">l'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:96"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':97"><span class="id" title="variable">l'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:96"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':97"><span class="id" title="variable">l'</span></a>))  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:96"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split'_aux_length_fst" class="idref" href="#split'_aux_length_fst"><span class="id" title="lemma">split'_aux_length_fst</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:98" class="idref" href="#l:98"><span class="id" title="binder">l</span></a> <a id="l':99" class="idref" href="#l':99"><span class="id" title="binder">l'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:98"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':99"><span class="id" title="variable">l'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">Nat.div2</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':99"><span class="id" title="variable">l'</span></a>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:98"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split'_decr" class="idref" href="#split'_decr"><span class="id" title="lemma">split'_decr</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:100" class="idref" href="#l1:100"><span class="id" title="binder">l1</span></a> <a id="l2:101" class="idref" href="#l2:101"><span class="id" title="binder">l2</span></a> <a id="a:102" class="idref" href="#a:102"><span class="id" title="binder">a</span></a> <a id="b:103" class="idref" href="#b:103"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:100"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a:102"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#b:103"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:101"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'"><span class="id" title="definition">split'</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:100"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:100"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'"><span class="id" title="definition">split'</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:100"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:100"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split'_aux_eq" class="idref" href="#split'_aux_eq"><span class="id" title="lemma">split'_aux_eq</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:104" class="idref" href="#l:104"><span class="id" title="binder">l</span></a> <a id="l':105" class="idref" href="#l':105"><span class="id" title="binder">l'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:104"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':105"><span class="id" title="variable">l'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)++</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_aux"><span class="id" title="definition">split'_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:104"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l':105"><span class="id" title="variable">l'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:104"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="split'_permutation" class="idref" href="#split'_permutation"><span class="id" title="lemma">split'_permutation</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:106" class="idref" href="#l:106"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'"><span class="id" title="definition">split'</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:106"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)++</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#split'"><span class="id" title="definition">split'</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:106"><span class="id" title="variable">l</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:106"><span class="id" title="variable">l</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Splitting"><span class="id" title="section">Splitting</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Generic.Merging" class="idref" href="#Generic.Merging"><span class="id" title="section">Merging</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="merge" class="idref" href="#merge"><span class="id" title="definition">merge</span></a>  (<a id="leb:107" class="idref" href="#leb:107"><span class="id" title="binder">leb</span></a> : <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <a id="l1:108" class="idref" href="#l1:108"><span class="id" title="binder">l1</span></a> <a id="l2:109" class="idref" href="#l2:109"><span class="id" title="binder">l2</span></a>:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <a id="merge_aux:115" class="idref" href="#merge_aux:115"><span class="id" title="binder">merge_aux</span></a> <a id="l2:111" class="idref" href="#l2:111"><span class="id" title="binder">l2</span></a> {<span class="id" title="keyword">struct</span> <span class="id" title="var">l2</span>}:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:108"><span class="id" title="variable">l1</span></a>,<a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:111"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>,<span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:111"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:108"><span class="id" title="variable">l1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l1'</span>,<span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l2'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#leb:107"><span class="id" title="variable">leb</span></a> <span class="id" title="var">a1</span>  <span class="id" title="var">a2</span> <span class="id" title="keyword">then</span> <span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge:110"><span class="id" title="definition">merge</span></a>  <a class="idref" href="hydras.Prelude.Merge_Sort.html#leb:107"><span class="id" title="variable">leb</span></a> <span class="id" title="var">l1'</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:111"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">else</span> <span class="id" title="var">a2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_aux:112"><span class="id" title="variable">merge_aux</span></a> <span class="id" title="var">l2'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_aux:115"><span class="id" title="variable">merge_aux</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:109"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Generic.Merging.le" class="idref" href="#Generic.Merging.le"><span class="id" title="variable">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="le_total:117" class="idref" href="#le_total:117"><span class="id" title="binder">le_total</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="le_dec:118" class="idref" href="#le_dec:118"><span class="id" title="binder">le_dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="93d511194bf8dfeef7681d9e29124228" class="idref" href="#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">&quot;</span></a>a &lt;= b" := (<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="merge_rect" class="idref" href="#merge_rect"><span class="id" title="lemma">merge_rect</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:119" class="idref" href="#P:119"><span class="id" title="binder">P</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l:120" class="idref" href="#l:120"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:120"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:120"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l:121" class="idref" href="#l:121"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:121"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:121"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a1:122" class="idref" href="#a1:122"><span class="id" title="binder">a1</span></a> <a id="l1:123" class="idref" href="#l1:123"><span class="id" title="binder">l1</span></a> <a id="a2:124" class="idref" href="#a2:124"><span class="id" title="binder">a2</span></a> <a id="l2:125" class="idref" href="#l2:125"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:122"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:124"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:123"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:124"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:125"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:126" class="idref" href="#x:126"><span class="id" title="binder">x</span></a> <a id="y:127" class="idref" href="#y:127"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:126"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:127"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:123"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:124"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:125"><span class="id" title="variable">l2</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:122"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:123"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:124"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:125"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:122"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:128" class="idref" href="#x:128"><span class="id" title="binder">x</span></a> <a id="y:129" class="idref" href="#y:129"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:128"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:129"><span class="id" title="variable">y</span></a>))<a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:123"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:124"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:125"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a1:130" class="idref" href="#a1:130"><span class="id" title="binder">a1</span></a> <a id="l1:131" class="idref" href="#l1:131"><span class="id" title="binder">l1</span></a> <a id="a2:132" class="idref" href="#a2:132"><span class="id" title="binder">a2</span></a> <a id="l2:133" class="idref" href="#l2:133"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:130"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:132"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:130"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:131"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:133"><span class="id" title="variable">l2</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:134" class="idref" href="#x:134"><span class="id" title="binder">x</span></a> <a id="y:135" class="idref" href="#y:135"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:134"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:135"><span class="id" title="variable">y</span></a>))(<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:130"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:131"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:133"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:130"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:131"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:132"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:133"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a2:132"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:136" class="idref" href="#x:136"><span class="id" title="binder">x</span></a> <a id="y:137" class="idref" href="#y:137"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:136"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:137"><span class="id" title="variable">y</span></a>)) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#a1:130"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:131"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:133"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:138" class="idref" href="#l1:138"><span class="id" title="binder">l1</span></a> <a id="l2:139" class="idref" href="#l2:139"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#P:119"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:138"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:139"><span class="id" title="variable">l2</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:140" class="idref" href="#x:140"><span class="id" title="binder">x</span></a> <a id="y:141" class="idref" href="#y:141"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:140"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:141"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:138"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:139"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="merge_ind" class="idref" href="#merge_ind"><span class="id" title="definition">merge_ind</span></a> ( <a id="P:142" class="idref" href="#P:142"><span class="id" title="binder">P</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_rect"><span class="id" title="lemma">merge_rect</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#P:142"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="merge_rec" class="idref" href="#merge_rec"><span class="id" title="definition">merge_rec</span></a>( <a id="P:143" class="idref" href="#P:143"><span class="id" title="binder">P</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_rect"><span class="id" title="lemma">merge_rect</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#P:143"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">induction_merge</span> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span>:= <span class="id" title="tactic">pattern</span> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:144" class="idref" href="#x:144"><span class="id" title="binder">x</span></a> <a id="y:145" class="idref" href="#y:145"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:144"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:145"><span class="id" title="variable">y</span></a>)) <span class="id" title="var">l1</span> <span class="id" title="var">l2</span> ); <span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_rect"><span class="id" title="lemma">merge_rect</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="merge_equation" class="idref" href="#merge_equation"><span class="id" title="lemma">merge_equation</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:146" class="idref" href="#l1:146"><span class="id" title="binder">l1</span></a> <a id="l2:147" class="idref" href="#l2:147"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:148" class="idref" href="#x:148"><span class="id" title="binder">x</span></a> <a id="y:149" class="idref" href="#y:149"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:148"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:149"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:146"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:147"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:146"><span class="id" title="variable">l1</span></a>,<a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:147"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>,<span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:147"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>,<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:146"><span class="id" title="variable">l1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l1'</span>,<span class="id" title="var">a2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l2'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide"><span class="id" title="definition">decide</span></a> (<span class="id" title="var">a1</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <span class="id" title="var">a2</span>) <span class="id" title="keyword">then</span> <span class="id" title="var">a1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:154" class="idref" href="#x:154"><span class="id" title="binder">x</span></a> <a id="y:155" class="idref" href="#y:155"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:154"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:155"><span class="id" title="variable">y</span></a>)) <span class="id" title="var">l1'</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:147"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="var">a2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:152" class="idref" href="#x:152"><span class="id" title="binder">x</span></a> <a id="y:153" class="idref" href="#y:153"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:152"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:153"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:146"><span class="id" title="variable">l1</span></a> <span class="id" title="var">l2'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Generic.Merging.Correctness" class="idref" href="#Generic.Merging.Correctness"><span class="id" title="section">Correctness</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="merge_Forall" class="idref" href="#merge_Forall"><span class="id" title="lemma">merge_Forall</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  (<a id="f:156" class="idref" href="#f:156"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>) <a id="l1:157" class="idref" href="#l1:157"><span class="id" title="binder">l1</span></a> <a id="l2:158" class="idref" href="#l2:158"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#f:156"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:157"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#f:156"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:158"><span class="id" title="variable">l2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#f:156"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:159" class="idref" href="#x:159"><span class="id" title="binder">x</span></a> <a id="y:160" class="idref" href="#y:160"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:159"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:160"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:157"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:158"><span class="id" title="variable">l2</span></a>) .<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="merge_LocallySorted" class="idref" href="#merge_LocallySorted"><span class="id" title="lemma">merge_LocallySorted</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:161" class="idref" href="#l1:161"><span class="id" title="binder">l1</span></a> <a id="l2:162" class="idref" href="#l2:162"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a>  <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:161"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a>  <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:162"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:163" class="idref" href="#x:163"><span class="id" title="binder">x</span></a> <a id="y:164" class="idref" href="#y:164"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:163"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:164"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:161"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:162"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="merge_permutation" class="idref" href="#merge_permutation"><span class="id" title="lemma">merge_permutation</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:165" class="idref" href="#l1:165"><span class="id" title="binder">l1</span></a> <a id="l2:166" class="idref" href="#l2:166"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:165"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:166"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a> (<span class="id" title="keyword">fun</span> <a id="x:167" class="idref" href="#x:167"><span class="id" title="binder">x</span></a> <a id="y:168" class="idref" href="#y:168"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:167"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:168"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:165"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:166"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Generic.Merging.Correctness.merge_sort" class="idref" href="#Generic.Merging.Correctness.merge_sort"><span class="id" title="section">merge_sort</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Generic.Merging.Correctness.merge_sort.split" class="idref" href="#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Generic.Merging.Correctness.merge_sort.split_decr" class="idref" href="#Generic.Merging.Correctness.merge_sort.split_decr"><span class="id" title="variable">split_decr</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:170" class="idref" href="#l1:170"><span class="id" title="binder">l1</span></a> <a id="l2:171" class="idref" href="#l2:171"><span class="id" title="binder">l2</span></a> <a id="a:172" class="idref" href="#a:172"><span class="id" title="binder">a</span></a> <a id="b:173" class="idref" href="#b:173"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:170"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#a:172"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#b:173"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:171"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:170"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:170"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:170"><span class="id" title="variable">l1</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:170"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Generic.Merging.Correctness.merge_sort.split_permutation" class="idref" href="#Generic.Merging.Correctness.merge_sort.split_permutation"><span class="id" title="variable">split_permutation</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:175" class="idref" href="#l:175"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:175"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)++</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>(<a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:175"><span class="id" title="variable">l</span></a>)) <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:175"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Function</span> <span class="id" title="var">merge_sort</span> (<a id="leb:195" class="idref" href="#leb:195"><span class="id" title="binder">leb</span></a> : <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="l:196" class="idref" href="#l:196"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">measure</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span>} :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:178"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> | <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:178"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> (<a id="l1:198" class="idref" href="#l1:198"><span class="id" title="binder">l1</span></a>,<a id="l2:199" class="idref" href="#l2:199"><span class="id" title="binder">l2</span></a>) := <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split"><span class="id" title="variable">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l:178"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge"><span class="id" title="definition">merge</span></a>  <a class="idref" href="hydras.Prelude.Merge_Sort.html#leb:177"><span class="id" title="variable">leb</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_sort:179"><span class="id" title="definition">merge_sort</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#leb:177"><span class="id" title="variable">leb</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l1:181"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_sort:179"><span class="id" title="definition">merge_sort</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#leb:177"><span class="id" title="variable">leb</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#l2:182"><span class="id" title="variable">l2</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="merge_sort_correct" class="idref" href="#merge_sort_correct"><span class="id" title="lemma">merge_sort_correct</span></a>: <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_correct"><span class="id" title="definition">sort_correct</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le"><span class="id" title="variable">le</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_sort"><span class="id" title="lemma">merge_sort</span></a> (<span class="id" title="keyword">fun</span> <a id="x:218" class="idref" href="#x:218"><span class="id" title="binder">x</span></a> <a id="y:219" class="idref" href="#y:219"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#x:218"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#y:219"><span class="id" title="variable">y</span></a>))).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort"><span class="id" title="section">merge_sort</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness"><span class="id" title="section">Correctness</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="sp_merge_sort" class="idref" href="#sp_merge_sort"><span class="id" title="definition">sp_merge_sort</span></a>:= <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_sort"><span class="id" title="lemma">merge_sort</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#split"><span class="id" title="definition">split</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#split_decr"><span class="id" title="lemma">split_decr</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="stable_merge_sort" class="idref" href="#stable_merge_sort"><span class="id" title="definition">stable_merge_sort</span></a> := <a class="idref" href="hydras.Prelude.Merge_Sort.html#merge_sort"><span class="id" title="lemma">merge_sort</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#split'"><span class="id" title="definition">split'</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#split'_decr"><span class="id" title="lemma">split'_decr</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic.Merging"><span class="id" title="section">Merging</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Merge_Sort.html#Generic"><span class="id" title="section">Generic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#sp_merge_sort"><span class="id" title="definition">sp_merge_sort</span></a>: <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>).<br/>
+<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Prelude.Merge_Sort.html#stable_merge_sort"><span class="id" title="definition">stable_merge_sort</span></a>: <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="sp_mergesort_OK" class="idref" href="#sp_mergesort_OK"><span class="id" title="lemma">sp_mergesort_OK</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_spec"><span class="id" title="definition">sort_spec</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#sp_merge_sort"><span class="id" title="definition">sp_merge_sort</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="stable_mergesort_OK" class="idref" href="#stable_mergesort_OK"><span class="id" title="lemma">stable_mergesort_OK</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_spec"><span class="id" title="definition">sort_spec</span></a> <a class="idref" href="hydras.Prelude.Merge_Sort.html#stable_merge_sort"><span class="id" title="definition">stable_merge_sort</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.MoreDecidable</title>
+</head>
+
+<body>
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+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.MoreDecidable</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Decidable.html#"><span class="id" title="library">Decidable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="boundedCheck" class="idref" href="#boundedCheck"><span class="id" title="lemma">boundedCheck</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="P:1" class="idref" href="#P:1"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:2" class="idref" href="#x:2"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Decidable.html#decidable"><span class="id" title="definition">decidable</span></a> (<a class="idref" href="hydras.Prelude.MoreDecidable.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#x:2"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="c:3" class="idref" href="#c:3"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="d:4" class="idref" href="#d:4"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.MoreDecidable.html#d:4"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#c:3"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#d:4"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="d:5" class="idref" href="#d:5"><span class="id" title="binder">d</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#d:5"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#c:3"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#d:5"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="smallestExists" class="idref" href="#smallestExists"><span class="id" title="lemma">smallestExists</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="P:6" class="idref" href="#P:6"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Decidable.html#decidable"><span class="id" title="definition">decidable</span></a> (<a class="idref" href="hydras.Prelude.MoreDecidable.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#x:7"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="c:8" class="idref" href="#c:8"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.Prelude.MoreDecidable.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#c:8"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:10" class="idref" href="#b:10"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.MoreDecidable.html#b:10"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#P:6"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreDecidable.html#b:10"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.MoreLibHyps.html b/theories/html/hydras.Prelude.MoreLibHyps.html
new file mode 100644
index 00000000..37e66ba5
--- /dev/null
+++ b/theories/html/hydras.Prelude.MoreLibHyps.html
@@ -0,0 +1,59 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.MoreLibHyps</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.MoreLibHyps</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">LibHyps</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">LibHyps</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 4) <span class="id" title="var">tactic4</span>(<span class="id" title="var">Tac</span>) "/" "dr" := <br/>
+&nbsp;&nbsp;<span class="id" title="var">Tac</span> ; {&lt; <span class="id" title="keyword">fun</span> <span class="id" title="var">h</span> ⇒ <span class="id" title="tactic">try</span> <span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">h</span> }.<br/>
+
+<br/>
+<span class="id" title="keyword">Tactic Notation</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 4) <span class="id" title="var">tactic4</span>(<span class="id" title="var">Tac</span>) "/" "r?" :=<br/>
+&nbsp;&nbsp;<span class="id" title="var">Tac</span> ; {&lt; <span class="id" title="keyword">fun</span> <span class="id" title="var">h</span>  ⇒ <span class="id" title="tactic">try</span> <span class="id" title="var">revert</span> <span class="id" title="var">h</span> }.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">autonaming_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">old_rename</span> := <span class="id" title="var">rename_hyp_default</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">rename_short</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">th</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| (?<span class="id" title="var">f</span> ?<span class="id" title="var">x</span> ?<span class="id" title="var">y</span>) ⇒ <span class="id" title="var">name</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><span class="id" title="var">f</span> <span class="id" title="notation">#</span> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <span class="id" title="notation">#</span> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>)                          <br/>
+&nbsp;&nbsp;| (?<span class="id" title="var">f</span> ?<span class="id" title="var">x</span>) ⇒ <span class="id" title="var">name</span> ((<span class="id" title="var">f</span> <span class="id" title="notation">#</span> 1))<br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <span class="id" title="var">old_rename</span> <span class="id" title="var">n</span> <span class="id" title="var">th</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.MoreLists.html b/theories/html/hydras.Prelude.MoreLists.html
new file mode 100644
index 00000000..e9108978
--- /dev/null
+++ b/theories/html/hydras.Prelude.MoreLists.html
@@ -0,0 +1,417 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.MoreLists</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.MoreLists</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Complements on lists
+
+<div class="paragraph"> </div>
+
+   Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span>  <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorting.Sorted</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorting.Sorted</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab13"></a><h1 class="section">Sets of natural numbers as lists</h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab14"></a><h2 class="section">Definitions</h2>
+
+<div class="paragraph"> </div>
+
+ numbers from i to i+n-1  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="iota_from" class="idref" href="#iota_from"><span class="id" title="definition">iota_from</span></a> <a id="i:1" class="idref" href="#i:1"><span class="id" title="binder">i</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreLists.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒  <a class="idref" href="hydras.Prelude.MoreLists.html#i:1"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from:3"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:1"><span class="id" title="variable">i</span></a>) <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="interval" class="idref" href="#interval"><span class="id" title="definition">interval</span></a> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a> <a id="j:6" class="idref" href="#j:6"><span class="id" title="binder">j</span></a> := <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:5"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:6"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:5"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bounded_by" class="idref" href="#bounded_by"><span class="id" title="definition">bounded_by</span></a> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="s:8" class="idref" href="#s:8"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="i:9" class="idref" href="#i:9"><span class="id" title="binder">i</span></a> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#i:9"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#n:7"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="hydras.Prelude.MoreLists.html#s:8"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="shift" class="idref" href="#shift"><span class="id" title="definition">shift</span></a> (<a id="l:10" class="idref" href="#l:10"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:10"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="unshift" class="idref" href="#unshift"><span class="id" title="definition">unshift</span></a> (<a id="l:11" class="idref" href="#l:11"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreLists.html#l:11"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#unshift:12"><span class="id" title="definition">unshift</span></a> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#unshift:12"><span class="id" title="definition">unshift</span></a> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+sorted list of elements greater or equal than n 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="sorted_ge" class="idref" href="#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> (<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="sorted_ge_nil" class="idref" href="#sorted_ge_nil"><span class="id" title="constructor">sorted_ge_nil</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge:15"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+| <a id="sorted_ge_one" class="idref" href="#sorted_ge_one"><span class="id" title="constructor">sorted_ge_one</span></a> : <span class="id" title="keyword">∀</span> <a id="p:17" class="idref" href="#p:17"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#n:14"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#p:17"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge:15"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:14"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#p:17"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)<br/>
+| <a id="sorted_ge_cons" class="idref" href="#sorted_ge_cons"><span class="id" title="constructor">sorted_ge_cons</span></a>: <span class="id" title="keyword">∀</span> <a id="p:18" class="idref" href="#p:18"><span class="id" title="binder">p</span></a> <a id="q:19" class="idref" href="#q:19"><span class="id" title="binder">q</span></a> <a id="s:20" class="idref" href="#s:20"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#n:14"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:18"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:18"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#q:19"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge:15"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:18"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#q:19"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:20"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge:15"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:14"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#p:18"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#q:19"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:20"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+simpler than StdLib's last 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="simple_last" class="idref" href="#simple_last"><span class="id" title="definition">simple_last</span></a> {<a id="A:21" class="idref" href="#A:21"><span class="id" title="binder">A</span></a>} (<a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:21"><span class="id" title="variable">A</span></a>) <a id="s:23" class="idref" href="#s:23"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreLists.html#s:23"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#x:22"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s'</span> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last:24"><span class="id" title="definition">simple_last</span></a>  <span class="id" title="var">i</span> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+the list (x::s) without its last item 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="but_last" class="idref" href="#but_last"><span class="id" title="definition">but_last</span></a> {<a id="A:26" class="idref" href="#A:26"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="x:27" class="idref" href="#x:27"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:26"><span class="id" title="variable">A</span></a>) (<a id="s:28" class="idref" href="#s:28"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:26"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreLists.html#s:28"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s'</span> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#x:27"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#but_last:29"><span class="id" title="definition">but_last</span></a>  <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="but_last_iota_from" class="idref" href="#but_last_iota_from"><span class="id" title="lemma">but_last_iota_from</span></a> : <span class="id" title="keyword">∀</span> <a id="l:31" class="idref" href="#l:31"><span class="id" title="binder">l</span></a> <a id="i:32" class="idref" href="#i:32"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:32"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:32"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:31"><span class="id" title="variable">l</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:32"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:32"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:31"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_length" class="idref" href="#interval_length"><span class="id" title="lemma">interval_length</span></a> <a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> <a id="j:34" class="idref" href="#j:34"><span class="id" title="binder">j</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:34"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:34"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:33"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="but_last_interval" class="idref" href="#but_last_interval"><span class="id" title="lemma">but_last_interval</span></a> <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a> <a id="j:36" class="idref" href="#j:36"><span class="id" title="binder">j</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#i:35"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:36"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:35"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:35"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:36"><span class="id" title="variable">j</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:35"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:35"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#j:36"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="but_last_shift'" class="idref" href="#but_last_shift'"><span class="id" title="lemma">but_last_shift'</span></a>   <a id="s:37" class="idref" href="#s:37"><span class="id" title="binder">s</span></a> : <span class="id" title="keyword">∀</span> <a id="x:38" class="idref" href="#x:38"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:38"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#s:37"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:38"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:37"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_but_last" class="idref" href="#unshift_but_last"><span class="id" title="lemma">unshift_but_last</span></a>   <a id="s:39" class="idref" href="#s:39"><span class="id" title="binder">s</span></a> : <span class="id" title="keyword">∀</span> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Prelude.MoreLists.html#x:40"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:39"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:39"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:40"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:39"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_app" class="idref" href="#unshift_app"><span class="id" title="lemma">unshift_app</span></a> : <span class="id" title="keyword">∀</span> <a id="s:41" class="idref" href="#s:41"><span class="id" title="binder">s</span></a> <a id="t:42" class="idref" href="#t:42"><span class="id" title="binder">t</span></a>,  <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#s:41"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#t:42"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:41"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#t:42"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_not_nil" class="idref" href="#unshift_not_nil"><span class="id" title="lemma">unshift_not_nil</span></a> : <span class="id" title="keyword">∀</span> <a id="s:43" class="idref" href="#s:43"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Prelude.MoreLists.html#s:43"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:43"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:43"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="but_last_app" class="idref" href="#but_last_app"><span class="id" title="lemma">but_last_app</span></a> {<a id="A:44" class="idref" href="#A:44"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="s:45" class="idref" href="#s:45"><span class="id" title="binder">s</span></a> (<a id="x:46" class="idref" href="#x:46"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:44"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:46"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:46"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:46"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:45"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="but_last_iota_from'" class="idref" href="#but_last_iota_from'"><span class="id" title="lemma">but_last_iota_from'</span></a> <a id="j:47" class="idref" href="#j:47"><span class="id" title="binder">j</span></a> : <span class="id" title="keyword">∀</span> <a id="i:48" class="idref" href="#i:48"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#but_last"><span class="id" title="definition">but_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:48"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:48"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:47"><span class="id" title="variable">j</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:48"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:47"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ptwise_le" class="idref" href="#ptwise_le"><span class="id" title="definition">ptwise_le</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab15"></a><h2 class="section">Lemmas</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="empty_interval" class="idref" href="#empty_interval"><span class="id" title="lemma">empty_interval</span></a> <a id="i:49" class="idref" href="#i:49"><span class="id" title="binder">i</span></a> <a id="j:50" class="idref" href="#j:50"><span class="id" title="binder">j</span></a> : (<a class="idref" href="hydras.Prelude.MoreLists.html#j:50"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:49"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:49"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:50"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="shift_iota_from" class="idref" href="#shift_iota_from"><span class="id" title="lemma">shift_iota_from</span></a> : <span class="id" title="keyword">∀</span> <a id="i:51" class="idref" href="#i:51"><span class="id" title="binder">i</span></a> <a id="l:52" class="idref" href="#l:52"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:51"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:52"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:51"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:52"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="shift_interval" class="idref" href="#shift_interval"><span class="id" title="lemma">shift_interval</span></a> (<a id="i:53" class="idref" href="#i:53"><span class="id" title="binder">i</span></a> <a id="j:54" class="idref" href="#j:54"><span class="id" title="binder">j</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:53"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:54"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:53"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:54"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_iota_from" class="idref" href="#unshift_iota_from"><span class="id" title="lemma">unshift_iota_from</span></a> : <span class="id" title="keyword">∀</span> <a id="i:55" class="idref" href="#i:55"><span class="id" title="binder">i</span></a> <a id="l:56" class="idref" href="#l:56"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:55"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:56"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:55"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:56"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_interval" class="idref" href="#unshift_interval"><span class="id" title="lemma">unshift_interval</span></a> (<a id="i:57" class="idref" href="#i:57"><span class="id" title="binder">i</span></a> <a id="j:58" class="idref" href="#j:58"><span class="id" title="binder">j</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:57"><span class="id" title="variable">i</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:58"><span class="id" title="variable">j</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#i:57"><span class="id" title="variable">i</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#j:58"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_interval_pred" class="idref" href="#unshift_interval_pred"><span class="id" title="lemma">unshift_interval_pred</span></a> (<a id="i:59" class="idref" href="#i:59"><span class="id" title="binder">i</span></a> <a id="j:60" class="idref" href="#j:60"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:60"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:59"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#j:60"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:59"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:60"><span class="id" title="variable">j</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="shift_no_zero" class="idref" href="#shift_no_zero"><span class="id" title="lemma">shift_no_zero</span></a> <a id="l:61" class="idref" href="#l:61"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:61"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="shift_unshift" class="idref" href="#shift_unshift"><span class="id" title="lemma">shift_unshift</span></a> <a id="l:62" class="idref" href="#l:62"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:62"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:62"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_shift" class="idref" href="#unshift_shift"><span class="id" title="lemma">unshift_shift</span></a> <a id="l:63" class="idref" href="#l:63"><span class="id" title="binder">l</span></a> (<a id="H:64" class="idref" href="#H:64"><span class="id" title="binder">H</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Prelude.MoreLists.html#l:63"><span class="id" title="variable">l</span></a>): <a class="idref" href="hydras.Prelude.MoreLists.html#shift"><span class="id" title="definition">shift</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:63"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:63"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="unshift_pred" class="idref" href="#unshift_pred"><span class="id" title="lemma">unshift_pred</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:65" class="idref" href="#s:65"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> 0 <a class="idref" href="hydras.Prelude.MoreLists.html#s:65"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#unshift"><span class="id" title="definition">unshift</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:65"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">Nat.pred</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:65"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_Forall" class="idref" href="#sorted_ge_Forall"><span class="id" title="lemma">sorted_ge_Forall</span></a> (<a id="n:66" class="idref" href="#n:66"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">∀</span> <a id="l:67" class="idref" href="#l:67"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:66"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:67"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="x:68" class="idref" href="#x:68"><span class="id" title="binder">x</span></a> ⇒  <a class="idref" href="hydras.Prelude.MoreLists.html#n:66"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:68"><span class="id" title="variable">x</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:67"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_trans" class="idref" href="#sorted_ge_trans"><span class="id" title="lemma">sorted_ge_trans</span></a> <a id="n:71" class="idref" href="#n:71"><span class="id" title="binder">n</span></a> <a id="p:72" class="idref" href="#p:72"><span class="id" title="binder">p</span></a> <a id="l:73" class="idref" href="#l:73"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#n:71"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:72"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:72"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:73"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:71"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:73"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_not_In" class="idref" href="#sorted_ge_not_In"><span class="id" title="lemma">sorted_ge_not_In</span></a> (<a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">∀</span> <a id="l:75" class="idref" href="#l:75"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:74"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:75"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:75"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> : <span class="id" title="var">lists</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_inv_gt" class="idref" href="#sorted_inv_gt"><span class="id" title="lemma">sorted_inv_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="n:76" class="idref" href="#n:76"><span class="id" title="binder">n</span></a> <a id="p:77" class="idref" href="#p:77"><span class="id" title="binder">p</span></a> <a id="s:78" class="idref" href="#s:78"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:76"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#p:77"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:78"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#p:77"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:76"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_from_app" class="idref" href="#iota_from_app"><span class="id" title="lemma">iota_from_app</span></a> <a id="n:79" class="idref" href="#n:79"><span class="id" title="binder">n</span></a> <a id="p:80" class="idref" href="#p:80"><span class="id" title="binder">p</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:79"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:80"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:79"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:80"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#n:79"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#p:80"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_from_plus" class="idref" href="#iota_from_plus"><span class="id" title="lemma">iota_from_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="k:81" class="idref" href="#k:81"><span class="id" title="binder">k</span></a> <a id="i:82" class="idref" href="#i:82"><span class="id" title="binder">i</span></a> <a id="j:83" class="idref" href="#j:83"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:82"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#k:81"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#j:83"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:82"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#k:81"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#k:81"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#i:82"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#j:83"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_not_empty" class="idref" href="#interval_not_empty"><span class="id" title="lemma">interval_not_empty</span></a> : <span class="id" title="keyword">∀</span> <a id="i:84" class="idref" href="#i:84"><span class="id" title="binder">i</span></a> <a id="j:85" class="idref" href="#j:85"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#i:84"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:85"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:84"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:85"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_not_empty_iff" class="idref" href="#interval_not_empty_iff"><span class="id" title="lemma">interval_not_empty_iff</span></a> (<a id="n:86" class="idref" href="#n:86"><span class="id" title="binder">n</span></a> <a id="p:87" class="idref" href="#p:87"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:86"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:87"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#n:86"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:87"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_singleton" class="idref" href="#interval_singleton"><span class="id" title="lemma">interval_singleton</span></a> (<a id="i:88" class="idref" href="#i:88"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:88"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:88"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:88"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_app" class="idref" href="#interval_app"><span class="id" title="lemma">interval_app</span></a> (<a id="i:89" class="idref" href="#i:89"><span class="id" title="binder">i</span></a> <a id="j:90" class="idref" href="#j:90"><span class="id" title="binder">j</span></a> <a id="k:91" class="idref" href="#k:91"><span class="id" title="binder">k</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#i:89"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:90"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#j:90"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#k:91"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:89"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#k:91"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:89"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:90"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:90"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#k:91"><span class="id" title="variable">k</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_from_unroll" class="idref" href="#iota_from_unroll"><span class="id" title="lemma">iota_from_unroll</span></a> <a id="i:92" class="idref" href="#i:92"><span class="id" title="binder">i</span></a> <a id="l:93" class="idref" href="#l:93"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:92"><span class="id" title="variable">i</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:93"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:92"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:92"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:93"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_unroll" class="idref" href="#interval_unroll"><span class="id" title="lemma">interval_unroll</span></a> : <span class="id" title="keyword">∀</span> <a id="i:94" class="idref" href="#i:94"><span class="id" title="binder">i</span></a> <a id="j:95" class="idref" href="#j:95"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> , (<a class="idref" href="hydras.Prelude.MoreLists.html#i:94"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:95"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:94"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:95"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:94"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:94"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#j:95"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_from_sorted_ge" class="idref" href="#iota_from_sorted_ge"><span class="id" title="lemma">iota_from_sorted_ge</span></a> : <span class="id" title="keyword">∀</span> <a id="p:96" class="idref" href="#p:96"><span class="id" title="binder">p</span></a> <a id="n:97" class="idref" href="#n:97"><span class="id" title="binder">n</span></a> <a id="q:98" class="idref" href="#q:98"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#q:98"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:97"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#q:98"><span class="id" title="variable">q</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:97"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:96"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_sorted_ge" class="idref" href="#interval_sorted_ge"><span class="id" title="lemma">interval_sorted_ge</span></a>: <span class="id" title="keyword">∀</span> <a id="p:99" class="idref" href="#p:99"><span class="id" title="binder">p</span></a> <a id="n:100" class="idref" href="#n:100"><span class="id" title="binder">n</span></a> <a id="q:101" class="idref" href="#q:101"><span class="id" title="binder">q</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  (<a class="idref" href="hydras.Prelude.MoreLists.html#q:101"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:100"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#q:101"><span class="id" title="variable">q</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:100"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#p:99"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_from_lt_not_In" class="idref" href="#iota_from_lt_not_In"><span class="id" title="lemma">iota_from_lt_not_In</span></a> <a id="i:102" class="idref" href="#i:102"><span class="id" title="binder">i</span></a> <a id="j:103" class="idref" href="#j:103"><span class="id" title="binder">j</span></a> <a id="l:104" class="idref" href="#l:104"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#i:102"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:103"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:102"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#iota_from"><span class="id" title="definition">iota_from</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:103"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:104"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="interval_lt_not_In" class="idref" href="#interval_lt_not_In"><span class="id" title="lemma">interval_lt_not_In</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:105" class="idref" href="#i:105"><span class="id" title="binder">i</span></a> <a id="j:106" class="idref" href="#j:106"><span class="id" title="binder">j</span></a> <a id="k:107" class="idref" href="#k:107"><span class="id" title="binder">k</span></a>,  <a class="idref" href="hydras.Prelude.MoreLists.html#i:105"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:106"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:105"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#interval"><span class="id" title="definition">interval</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:106"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#k:107"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Forall2_right_induction" class="idref" href="#Forall2_right_induction"><span class="id" title="section">Forall2_right_induction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Forall2R" class="idref" href="#Forall2R"><span class="id" title="inductive">Forall2R</span></a> {<a id="A:108" class="idref" href="#A:108"><span class="id" title="binder">A</span></a> <a id="B:109" class="idref" href="#B:109"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="R:110" class="idref" href="#R:110"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#A:108"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:109"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <span class="id" title="var">B</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="Forall2R_nil" class="idref" href="#Forall2R_nil"><span class="id" title="constructor">Forall2R_nil</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R:111"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:110"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+| <a id="Forall2R_last" class="idref" href="#Forall2R_last"><span class="id" title="constructor">Forall2R_last</span></a> : <span class="id" title="keyword">∀</span> <a id="l:113" class="idref" href="#l:113"><span class="id" title="binder">l</span></a> <a id="l':114" class="idref" href="#l':114"><span class="id" title="binder">l'</span></a> <a id="x:115" class="idref" href="#x:115"><span class="id" title="binder">x</span></a> <a id="y:116" class="idref" href="#y:116"><span class="id" title="binder">y</span></a> <a id="l1:117" class="idref" href="#l1:117"><span class="id" title="binder">l1</span></a> <a id="l'1:118" class="idref" href="#l'1:118"><span class="id" title="binder">l'1</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R:111"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:110"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:113"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':114"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#R:110"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:115"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:116"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#l1:117"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:113"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++(</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#x:115"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#l'1:118"><span class="id" title="variable">l'1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':114"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#y:116"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R:111"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:110"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:117"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l'1:118"><span class="id" title="variable">l'1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Forall2R_cons" class="idref" href="#Forall2R_cons"><span class="id" title="lemma">Forall2R_cons</span></a> {<a id="A:119" class="idref" href="#A:119"><span class="id" title="binder">A</span></a> <a id="B:120" class="idref" href="#B:120"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="R:121" class="idref" href="#R:121"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#A:119"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:120"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:122" class="idref" href="#l:122"><span class="id" title="binder">l</span></a> <a id="l':123" class="idref" href="#l':123"><span class="id" title="binder">l'</span></a>,  <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:121"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:122"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':123"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="x:124" class="idref" href="#x:124"><span class="id" title="binder">x</span></a> <a id="y:125" class="idref" href="#y:125"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#R:121"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:124"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:125"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:121"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#x:124"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l:122"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.Prelude.MoreLists.html#y:125"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':123"><span class="id" title="variable">l'</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Forall2_R" class="idref" href="#Forall2_R"><span class="id" title="lemma">Forall2_R</span></a>  {<a id="A:126" class="idref" href="#A:126"><span class="id" title="binder">A</span></a> <a id="B:127" class="idref" href="#B:127"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="R:128" class="idref" href="#R:128"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#A:126"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:127"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:129" class="idref" href="#l:129"><span class="id" title="binder">l</span></a> <a id="l':130" class="idref" href="#l':130"><span class="id" title="binder">l'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:128"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:129"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':130"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:128"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:129"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':130"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="Forall2_RR" class="idref" href="#Forall2_RR"><span class="id" title="lemma">Forall2_RR</span></a>  {<a id="A:131" class="idref" href="#A:131"><span class="id" title="binder">A</span></a> <a id="B:132" class="idref" href="#B:132"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="R:133" class="idref" href="#R:133"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#A:131"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:132"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:134" class="idref" href="#l:134"><span class="id" title="binder">l</span></a> <a id="l':135" class="idref" href="#l':135"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:133"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:134"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':135"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:133"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:134"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':135"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall2R_iff" class="idref" href="#Forall2R_iff"><span class="id" title="lemma">Forall2R_iff</span></a> {<a id="A:136" class="idref" href="#A:136"><span class="id" title="binder">A</span></a> <a id="B:137" class="idref" href="#B:137"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="R:138" class="idref" href="#R:138"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Prelude.MoreLists.html#A:136"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:137"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:139" class="idref" href="#l:139"><span class="id" title="binder">l</span></a> <a id="l':140" class="idref" href="#l':140"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2R"><span class="id" title="inductive">Forall2R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:138"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:139"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':140"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:138"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:139"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':140"><span class="id" title="variable">l'</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall2_indR" class="idref" href="#Forall2_indR"><span class="id" title="lemma">Forall2_indR</span></a> {<a id="A:141" class="idref" href="#A:141"><span class="id" title="binder">A</span></a> <a id="B:142" class="idref" href="#B:142"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="R:143" class="idref" href="#R:143"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="P:144" class="idref" href="#P:144"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+<a class="idref" href="hydras.Prelude.MoreLists.html#P:144"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="l:145" class="idref" href="#l:145"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a>) (<a id="l':146" class="idref" href="#l':146"><span class="id" title="binder">l'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a>) (<a id="x:147" class="idref" href="#x:147"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a>) (<a id="y:148" class="idref" href="#y:148"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;(<a id="l1:149" class="idref" href="#l1:149"><span class="id" title="binder">l1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a>) (<a id="l'1:150" class="idref" href="#l'1:150"><span class="id" title="binder">l'1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a>),<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:143"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:145"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':146"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#P:144"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:145"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':146"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#R:143"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:147"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:148"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:149"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:145"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:147"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l'1:150"><span class="id" title="variable">l'1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l':146"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:148"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#P:144"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:149"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l'1:150"><span class="id" title="variable">l'1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:151" class="idref" href="#l:151"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:141"><span class="id" title="variable">A</span></a>) (<a id="l0:152" class="idref" href="#l0:152"><span class="id" title="binder">l0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#B:142"><span class="id" title="variable">B</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall2"><span class="id" title="inductive">Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:143"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:151"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l0:152"><span class="id" title="variable">l0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#P:144"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:151"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l0:152"><span class="id" title="variable">l0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span>  <a class="idref" href="hydras.Prelude.MoreLists.html#Forall2_right_induction"><span class="id" title="section">Forall2_right_induction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_le" class="idref" href="#sorted_le"><span class="id" title="lemma">sorted_le</span></a> : <span class="id" title="keyword">∀</span> <a id="i:153" class="idref" href="#i:153"><span class="id" title="binder">i</span></a> <a id="j:154" class="idref" href="#j:154"><span class="id" title="binder">j</span></a> <a id="X:155" class="idref" href="#X:155"><span class="id" title="binder">X</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#i:153"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:154"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:154"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:155"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:153"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:155"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_tail" class="idref" href="#sorted_tail"><span class="id" title="lemma">sorted_tail</span></a> : <span class="id" title="keyword">∀</span> <a id="i:156" class="idref" href="#i:156"><span class="id" title="binder">i</span></a> <a id="j:157" class="idref" href="#j:157"><span class="id" title="binder">j</span></a> <a id="X:158" class="idref" href="#X:158"><span class="id" title="binder">X</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:156"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#j:157"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#X:158"><span class="id" title="variable">X</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:156"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:158"><span class="id" title="variable">X</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_tail'" class="idref" href="#sorted_tail'"><span class="id" title="lemma">sorted_tail'</span></a> : <span class="id" title="keyword">∀</span> <a id="i:159" class="idref" href="#i:159"><span class="id" title="binder">i</span></a> <a id="j:160" class="idref" href="#j:160"><span class="id" title="binder">j</span></a> <a id="X:161" class="idref" href="#X:161"><span class="id" title="binder">X</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:159"><span class="id" title="variable">i</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#j:160"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#X:161"><span class="id" title="variable">X</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:160"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:161"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_head" class="idref" href="#sorted_head"><span class="id" title="lemma">sorted_head</span></a> : <span class="id" title="keyword">∀</span> <a id="n:162" class="idref" href="#n:162"><span class="id" title="binder">n</span></a> <a id="m:163" class="idref" href="#m:163"><span class="id" title="binder">m</span></a> <a id="s:164" class="idref" href="#s:164"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:162"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#m:163"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:164"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:162"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#m:163"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Sorted_mono" class="idref" href="#Sorted_mono"><span class="id" title="lemma">Sorted_mono</span></a> {<a id="A:165" class="idref" href="#A:165"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="R:166" class="idref" href="#R:166"><span class="id" title="binder">R</span></a> <a id="S:167" class="idref" href="#S:167"><span class="id" title="binder">S</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#A:165"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hincl:170" class="idref" href="#Hincl:170"><span class="id" title="binder">Hincl</span></a> : <span class="id" title="keyword">∀</span> <a id="x:168" class="idref" href="#x:168"><span class="id" title="binder">x</span></a> <a id="y:169" class="idref" href="#y:169"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#R:166"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:168"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:169"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#S:167"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:168"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:169"><span class="id" title="variable">y</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:171" class="idref" href="#l:171"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#R:166"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:171"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#S:167"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:171"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_iff0" class="idref" href="#sorted_ge_iff0"><span class="id" title="lemma">sorted_ge_iff0</span></a> : <span class="id" title="keyword">∀</span> <a id="l:172" class="idref" href="#l:172"><span class="id" title="binder">l</span></a> <a id="n:173" class="idref" href="#n:173"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:172"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:172"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="i:174" class="idref" href="#i:174"><span class="id" title="binder">i</span></a> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#n:173"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:174"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:172"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_iff" class="idref" href="#sorted_ge_iff"><span class="id" title="lemma">sorted_ge_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="l:175" class="idref" href="#l:175"><span class="id" title="binder">l</span></a> <a id="n:176" class="idref" href="#n:176"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:176"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:175"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l:175"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="i:177" class="idref" href="#i:177"><span class="id" title="binder">i</span></a> ⇒ <a class="idref" href="hydras.Prelude.MoreLists.html#n:176"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:177"><span class="id" title="variable">i</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l:175"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_prefix" class="idref" href="#sorted_ge_prefix"><span class="id" title="lemma">sorted_ge_prefix</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="l1:178" class="idref" href="#l1:178"><span class="id" title="binder">l1</span></a> <a id="n:179" class="idref" href="#n:179"><span class="id" title="binder">n</span></a> <a id="l2:180" class="idref" href="#l2:180"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:179"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#l1:178"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l2:180"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:179"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:178"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_In" class="idref" href="#sorted_In"><span class="id" title="lemma">sorted_In</span></a> : <span class="id" title="keyword">∀</span> <a id="i:181" class="idref" href="#i:181"><span class="id" title="binder">i</span></a> <a id="X:182" class="idref" href="#X:182"><span class="id" title="binder">X</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#i:181"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#X:182"><span class="id" title="variable">X</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="j:183" class="idref" href="#j:183"><span class="id" title="binder">j</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:183"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:182"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#i:181"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:183"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_not_in_tail" class="idref" href="#sorted_not_in_tail"><span class="id" title="lemma">sorted_not_in_tail</span></a> : <span class="id" title="keyword">∀</span>  <a id="i:184" class="idref" href="#i:184"><span class="id" title="binder">i</span></a> <a id="j:185" class="idref" href="#j:185"><span class="id" title="binder">j</span></a> <a id="X:186" class="idref" href="#X:186"><span class="id" title="binder">X</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#i:184"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#X:186"><span class="id" title="variable">X</span></a>)   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:185"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#i:184"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#j:185"><span class="id" title="variable">j</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#X:186"><span class="id" title="variable">X</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="simple_last_correct" class="idref" href="#simple_last_correct"><span class="id" title="lemma">simple_last_correct</span></a> {<a id="A:187" class="idref" href="#A:187"><span class="id" title="binder">A</span></a>}: <span class="id" title="keyword">∀</span> <a id="s:188" class="idref" href="#s:188"><span class="id" title="binder">s</span></a> (<a id="x:189" class="idref" href="#x:189"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:187"><span class="id" title="variable">A</span></a>), <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:189"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:188"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:188"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:189"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_sorted_ge_inv" class="idref" href="#In_sorted_ge_inv"><span class="id" title="lemma">In_sorted_ge_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="x:190" class="idref" href="#x:190"><span class="id" title="binder">x</span></a> <a id="y:191" class="idref" href="#y:191"><span class="id" title="binder">y</span></a> <a id="s:192" class="idref" href="#s:192"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:190"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#y:191"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#s:192"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:191"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#y:191"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:190"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:190"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:191"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:190"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="incl_inv" class="idref" href="#incl_inv"><span class="id" title="lemma">incl_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="x:193" class="idref" href="#x:193"><span class="id" title="binder">x</span></a> <a id="y:194" class="idref" href="#y:194"><span class="id" title="binder">y</span></a> <a id="l1:195" class="idref" href="#l1:195"><span class="id" title="binder">l1</span></a> <a id="l2:196" class="idref" href="#l2:196"><span class="id" title="binder">l2</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#x:193"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l1:195"><span class="id" title="variable">l1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#y:194"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l2:196"><span class="id" title="variable">l2</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#incl"><span class="id" title="definition">incl</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#x:193"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l1:195"><span class="id" title="variable">l1</span></a>)(<a class="idref" href="hydras.Prelude.MoreLists.html#y:194"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l2:196"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#y:194"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:193"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#incl"><span class="id" title="definition">incl</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:195"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l2:196"><span class="id" title="variable">l2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="incl_decomp" class="idref" href="#incl_decomp"><span class="id" title="lemma">incl_decomp</span></a> : <span class="id" title="keyword">∀</span> <a id="l1:197" class="idref" href="#l1:197"><span class="id" title="binder">l1</span></a> <a id="l2:198" class="idref" href="#l2:198"><span class="id" title="binder">l2</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:197"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l2:198"><span class="id" title="variable">l2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#incl"><span class="id" title="definition">incl</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:197"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l2:198"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l3:199" class="idref" href="#l3:199"><span class="id" title="binder">l3</span></a> <a id="l4:200" class="idref" href="#l4:200"><span class="id" title="binder">l4</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Prelude.MoreLists.html#l2:198"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l3:199"><span class="id" title="variable">l3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l4:200"><span class="id" title="variable">l4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#ptwise_le"><span class="id" title="definition">ptwise_le</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l3:199"><span class="id" title="variable">l3</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:197"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="simple_last_app" class="idref" href="#simple_last_app"><span class="id" title="lemma">simple_last_app</span></a> {<a id="A:201" class="idref" href="#A:201"><span class="id" title="binder">A</span></a>}: <span class="id" title="keyword">∀</span> <a id="l:202" class="idref" href="#l:202"><span class="id" title="binder">l</span></a> <a id="l1:203" class="idref" href="#l1:203"><span class="id" title="binder">l1</span></a> (<a id="x:204" class="idref" href="#x:204"><span class="id" title="binder">x</span></a> <a id="y:205" class="idref" href="#y:205"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:201"><span class="id" title="variable">A</span></a>), <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:204"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#l:202"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++(</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#y:205"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l1:203"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:205"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:203"><span class="id" title="variable">l1</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="simple_last_app1" class="idref" href="#simple_last_app1"><span class="id" title="lemma">simple_last_app1</span></a> {<a id="A:206" class="idref" href="#A:206"><span class="id" title="binder">A</span></a>}: <span class="id" title="keyword">∀</span> <a id="l:207" class="idref" href="#l:207"><span class="id" title="binder">l</span></a> (<a id="x:208" class="idref" href="#x:208"><span class="id" title="binder">x</span></a> <a id="y:209" class="idref" href="#y:209"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Prelude.MoreLists.html#A:206"><span class="id" title="variable">A</span></a>), <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:208"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#l:207"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++(</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#y:209"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#y:209"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_max_1" class="idref" href="#sorted_max_1"><span class="id" title="lemma">sorted_max_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:210" class="idref" href="#s:210"><span class="id" title="binder">s</span></a> <a id="n:211" class="idref" href="#n:211"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:211"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:210"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#n:211"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:211"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:210"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_cut" class="idref" href="#sorted_cut"><span class="id" title="lemma">sorted_cut</span></a> :<span class="id" title="keyword">∀</span> <a id="l1:212" class="idref" href="#l1:212"><span class="id" title="binder">l1</span></a> <a id="n:213" class="idref" href="#n:213"><span class="id" title="binder">n</span></a> <a id="x:214" class="idref" href="#x:214"><span class="id" title="binder">x</span></a> <a id="l2:215" class="idref" href="#l2:215"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:213"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#l1:212"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++(</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#x:214"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreLists.html#l2:215"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:213"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:212"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#x:214"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_max_2" class="idref" href="#sorted_max_2"><span class="id" title="lemma">sorted_max_2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:216" class="idref" href="#s:216"><span class="id" title="binder">s</span></a> <a id="n:217" class="idref" href="#n:217"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:217"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:216"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="i:218" class="idref" href="#i:218"><span class="id" title="binder">i</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.MoreLists.html#i:218"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:217"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#s:216"><span class="id" title="variable">s</span></a>)%<span class="id" title="var">nat</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreLists.html#s:216"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_ge_suffix" class="idref" href="#sorted_ge_suffix"><span class="id" title="lemma">sorted_ge_suffix</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="l1:219" class="idref" href="#l1:219"><span class="id" title="binder">l1</span></a> <a id="n:220" class="idref" href="#n:220"><span class="id" title="binder">n</span></a> <a id="l2:221" class="idref" href="#l2:221"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:220"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#l1:219"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l2:221"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#sorted_ge"><span class="id" title="inductive">sorted_ge</span></a> (<a class="idref" href="hydras.Prelude.MoreLists.html#simple_last"><span class="id" title="definition">simple_last</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#n:220"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreLists.html#l1:219"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.Prelude.MoreLists.html#l2:221"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.MoreOrders.html b/theories/html/hydras.Prelude.MoreOrders.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Prelude.MoreOrders.html
@@ -0,0 +1,115 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.MoreOrders</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.MoreOrders</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Complements on strict orders 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#"><span class="id" title="library">Relation_Operators</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#"><span class="id" title="module">Relation_Definitions</span></a>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="leq" class="idref" href="#leq"><span class="id" title="definition">leq</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="lt:2" class="idref" href="#lt:2"><span class="id" title="binder">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:1"><span class="id" title="variable">A</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:1"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_refl"><span class="id" title="inductive">clos_refl</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:2"><span class="id" title="variable">lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Defs" class="idref" href="#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Defs.A" class="idref" href="#Defs.A"><span class="id" title="variable">A</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Defs.lt" class="idref" href="#Defs.lt"><span class="id" title="variable">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:3"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Infix</span> "&lt;" := <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Infix</span> "&lt;=" := (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Least" class="idref" href="#Least"><span class="id" title="definition">Least</span></a> {<a id="sto:5" class="idref" href="#sto:5"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:7" class="idref" href="#y:7"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.MoreOrders.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#56ea2d5e19744d3f0cba9664d875d7ec"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:7"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Successor" class="idref" href="#Successor"><span class="id" title="definition">Successor</span></a> {<a id="sto:8" class="idref" href="#sto:8"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="y:9" class="idref" href="#y:9"><span class="id" title="binder">y</span></a> <a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>):=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:9"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="z:11" class="idref" href="#z:11"><span class="id" title="binder">z</span></a>,  <a class="idref" href="hydras.Prelude.MoreOrders.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#z:11"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.MoreOrders.html#z:11"><span class="id" title="variable">z</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Prelude.MoreOrders.html#y:9"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Limit" class="idref" href="#Limit"><span class="id" title="definition">Limit</span></a> {<a id="sto:12" class="idref" href="#sto:12"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>}  (<a id="x:13" class="idref" href="#x:13"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>)  :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="w:14" class="idref" href="#w:14"><span class="id" title="binder">w</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Prelude.MoreOrders.html#w:14"><span class="id" title="variable">w</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:13"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:15" class="idref" href="#y:15"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#y:15"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:13"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="z:16" class="idref" href="#z:16"><span class="id" title="binder">z</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:15"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#z:16"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#z:16"><span class="id" title="variable">z</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:13"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span>  <a id="Omega_limit" class="idref" href="#Omega_limit"><span class="id" title="definition">Omega_limit</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="sto:17" class="idref" href="#sto:17"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="s:18" class="idref" href="#s:18"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) (<a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>)  :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:20" class="idref" href="#i:20"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#s:18"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#i:20"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:19"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:21" class="idref" href="#y:21"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#y:21"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:22" class="idref" href="#i:22"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:21"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#s:18"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#i:22"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span>  <a id="Omega_limit_type" class="idref" href="#Omega_limit_type"><span class="id" title="definition">Omega_limit_type</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;`{<a id="lt:23" class="idref" href="#lt:23"><span class="id" title="binder">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="sto:24" class="idref" href="#sto:24"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:23"><span class="id" title="variable">lt</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:25" class="idref" href="#s:25"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) (<a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Type</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:27" class="idref" href="#i:27"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#i:27"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:26"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:28" class="idref" href="#y:28"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#y:28"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="i:29" class="idref" href="#i:29"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:28"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#i:29"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Omega_limit_not_Succ" class="idref" href="#Omega_limit_not_Succ"><span class="id" title="lemma">Omega_limit_not_Succ</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="sto:30" class="idref" href="#sto:30"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="s:31" class="idref" href="#s:31"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) (<a id="x:32" class="idref" href="#x:32"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Omega_limit"><span class="id" title="definition">Omega_limit</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#s:31"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:32"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:33" class="idref" href="#y:33"><span class="id" title="binder">y</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:32"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#y:33"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Least_not_Succ" class="idref" href="#Least_not_Succ"><span class="id" title="lemma">Least_not_Succ</span></a>  {<a id="sto:34" class="idref" href="#sto:34"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="x:35" class="idref" href="#x:35"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:35"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="z:36" class="idref" href="#z:36"><span class="id" title="binder">z</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:35"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#z:36"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Omega_limit_Limit" class="idref" href="#Omega_limit_Limit"><span class="id" title="lemma">Omega_limit_Limit</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="sto:37" class="idref" href="#sto:37"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} (<a id="s:38" class="idref" href="#s:38"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) (<a id="x:39" class="idref" href="#x:39"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.A"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Omega_limit"><span class="id" title="definition">Omega_limit</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#s:38"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:39"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:39"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Least_not_Limit" class="idref" href="#Least_not_Limit"><span class="id" title="lemma">Least_not_Limit</span></a>  {<a id="sto:40" class="idref" href="#sto:40"><span class="id" title="binder">sto</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs.lt"><span class="id" title="variable">lt</span></a>} <a id="x:41" class="idref" href="#x:41"><span class="id" title="binder">x</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Least"><span class="id" title="definition">Least</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:41"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#Limit"><span class="id" title="definition">Limit</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#x:41"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.MoreOrders.html#Defs"><span class="id" title="section">Defs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_eq" class="idref" href="#le_lt_eq"><span class="id" title="lemma">le_lt_eq</span></a>  {<a id="A:42" class="idref" href="#A:42"><span class="id" title="binder">A</span></a>}{<a id="lt:43" class="idref" href="#lt:43"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:42"><span class="id" title="variable">A</span></a>}:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> <a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:43"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:43"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#b:45"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#a:44"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#b:45"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="leq_trans" class="idref" href="#leq_trans"><span class="id" title="instance">leq_trans</span></a> {<a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="lt:47" class="idref" href="#lt:47"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#A:46"><span class="id" title="variable">A</span></a>}{<a id="sto:48" class="idref" href="#sto:48"><span class="id" title="binder">sto</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a>  <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:47"><span class="id" title="variable">lt</span></a>}:<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> (<a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Prelude.MoreOrders.html#lt:47"><span class="id" title="variable">lt</span></a>).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.MoreVectors</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.MoreVectors</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Bool.Bool.html#"><span class="id" title="library">Bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="library">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#"><span class="id" title="module">Vector</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#VectorNotations"><span class="id" title="module">VectorNotations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+generalities on vectors 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> {<span class="id" title="var">n</span>}  <span class="id" title="var">_</span> .<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> {<span class="id" title="var">A</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Vid" class="idref" href="#Vid"><span class="id" title="definition">Vid</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} {<a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:2"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <span class="id" title="keyword">fun</span> <a id="v:4" class="idref" href="#v:4"><span class="id" title="binder">v</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#4591d6c2b36fedc12d5b1c142546a37e"><span class="id" title="notation">[]</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <span class="id" title="keyword">fun</span> <a id="v:5" class="idref" href="#v:5"><span class="id" title="binder">v</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:5"><span class="id" title="variable">v</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Vid_eq" class="idref" href="#Vid_eq"><span class="id" title="lemma">Vid_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="A:7" class="idref" href="#A:7"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="v:8" class="idref" href="#v:8"><span class="id" title="binder">v</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:7"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:6"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.Prelude.MoreVectors.html#v:8"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#Vid"><span class="id" title="definition">Vid</span></a>   <a class="idref" href="hydras.Prelude.MoreVectors.html#v:8"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="t_0_nil" class="idref" href="#t_0_nil"><span class="id" title="lemma">t_0_nil</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:9" class="idref" href="#A:9"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="v:10" class="idref" href="#v:10"><span class="id" title="binder">v</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:9"><span class="id" title="variable">A</span></a> 0), <a class="idref" href="hydras.Prelude.MoreVectors.html#v:10"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="decomp" class="idref" href="#decomp"><span class="id" title="lemma">decomp</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="v:13" class="idref" href="#v:13"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:11"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:12"><span class="id" title="variable">n</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#v:13"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:13"><span class="id" title="variable">v</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:13"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decomp1" class="idref" href="#decomp1"><span class="id" title="lemma">decomp1</span></a>  {<a id="A:14" class="idref" href="#A:14"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}  (<a id="v:15" class="idref" href="#v:15"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:14"><span class="id" title="variable">A</span></a> 1):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#v:15"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:15"><span class="id" title="variable">v</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="vector_double_rect" class="idref" href="#vector_double_rect"><span class="id" title="definition">vector_double_rect</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:16" class="idref" href="#A:16"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="X:18" class="idref" href="#X:18"><span class="id" title="binder">X</span></a>: <span class="id" title="keyword">∀</span> (<a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:16"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:17"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:16"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:17"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#X:18"><span class="id" title="variable">X</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> (<a id="v1:20" class="idref" href="#v1:20"><span class="id" title="binder">v1</span></a> <a id="v2:21" class="idref" href="#v2:21"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:16"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:19"><span class="id" title="variable">n</span></a>) <a id="a:22" class="idref" href="#a:22"><span class="id" title="binder">a</span></a> <a id="b:23" class="idref" href="#b:23"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Prelude.MoreVectors.html#X:18"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:20"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:21"><span class="id" title="variable">v2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#X:18"><span class="id" title="variable">X</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:19"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:22"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:20"><span class="id" title="variable">v1</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Prelude.MoreVectors.html#b:23"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:21"><span class="id" title="variable">v2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:24" class="idref" href="#n:24"><span class="id" title="binder">n</span></a> (<a id="v1:25" class="idref" href="#v1:25"><span class="id" title="binder">v1</span></a> <a id="v2:26" class="idref" href="#v2:26"><span class="id" title="binder">v2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:16"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:24"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.Prelude.MoreVectors.html#X:18"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:24"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:25"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:26"><span class="id" title="variable">v2</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="vector_triple_rect" class="idref" href="#vector_triple_rect"><span class="id" title="definition">vector_triple_rect</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:27" class="idref" href="#A:27"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="X:29" class="idref" href="#X:29"><span class="id" title="binder">X</span></a>: <span class="id" title="keyword">∀</span> (<a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Type</span>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#X:29"><span class="id" title="variable">X</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:30" class="idref" href="#n:30"><span class="id" title="binder">n</span></a> (<a id="v1:31" class="idref" href="#v1:31"><span class="id" title="binder">v1</span></a> <a id="v2:32" class="idref" href="#v2:32"><span class="id" title="binder">v2</span></a> <a id="v3:33" class="idref" href="#v3:33"><span class="id" title="binder">v3</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:30"><span class="id" title="variable">n</span></a>) <a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a> <a id="b:35" class="idref" href="#b:35"><span class="id" title="binder">b</span></a> <a id="c:36" class="idref" href="#c:36"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Prelude.MoreVectors.html#X:29"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:30"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:31"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:32"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v3:33"><span class="id" title="variable">v3</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#X:29"><span class="id" title="variable">X</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:30"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:31"><span class="id" title="variable">v1</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a>  <a class="idref" href="hydras.Prelude.MoreVectors.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:32"><span class="id" title="variable">v2</span></a>)(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#c:36"><span class="id" title="variable">c</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v3:33"><span class="id" title="variable">v3</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:37" class="idref" href="#n:37"><span class="id" title="binder">n</span></a> (<a id="v1:38" class="idref" href="#v1:38"><span class="id" title="binder">v1</span></a> <a id="v2:39" class="idref" href="#v2:39"><span class="id" title="binder">v2</span></a> <a id="v3:40" class="idref" href="#v3:40"><span class="id" title="binder">v3</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:37"><span class="id" title="variable">n</span></a>), <a class="idref" href="hydras.Prelude.MoreVectors.html#X:29"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:37"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v1:38"><span class="id" title="variable">v1</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v2:39"><span class="id" title="variable">v2</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v3:40"><span class="id" title="variable">v3</span></a>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="vector_nth" class="idref" href="#vector_nth"><span class="id" title="definition">vector_nth</span></a> (<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="p:43" class="idref" href="#p:43"><span class="id" title="binder">p</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)(<a id="v:44" class="idref" href="#v:44"><span class="id" title="binder">v</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:41"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#p:43"><span class="id" title="variable">p</span></a>){<span class="id" title="keyword">struct</span> <span class="id" title="var">v</span>}<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:41"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:42"><span class="id" title="variable">n</span></a>,<a class="idref" href="hydras.Prelude.MoreVectors.html#v:44"><span class="id" title="variable">v</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">_</span>   , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;| 0   , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a>  <span class="id" title="var">b</span>  <span class="id" title="var">_</span>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n'</span>, @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a>  <span class="id" title="var">_</span>  <span class="id" title="var">_</span> <span class="id" title="var">p'</span>  <span class="id" title="var">v'</span> ⇒ <a class="idref" href="hydras.Prelude.MoreVectors.html#vector_nth:45"><span class="id" title="definition">vector_nth</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:41"><span class="id" title="variable">A</span></a> <span class="id" title="var">n'</span>  <span class="id" title="var">p'</span> <span class="id" title="var">v'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.MoreVectors.html#vector_nth"><span class="id" title="definition">vector_nth</span></a> {<span class="id" title="var">A</span> } <span class="id" title="var">n</span> {<span class="id" title="var">p</span>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall_inv" class="idref" href="#Forall_inv"><span class="id" title="lemma">Forall_inv</span></a> {<a id="A:48" class="idref" href="#A:48"><span class="id" title="binder">A</span></a> :<span class="id" title="keyword">Type</span>}(<a id="P:49" class="idref" href="#P:49"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.MoreVectors.html#A:48"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)(<a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;<a id="a:51" class="idref" href="#a:51"><span class="id" title="binder">a</span></a>  <a id="v:52" class="idref" href="#v:52"><span class="id" title="binder">v</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Vector.Forall</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:49"><span class="id" title="variable">P</span></a> (<span class="id" title="var">n</span>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:50"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:51"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:52"><span class="id" title="variable">v</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#P:49"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:51"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Vector.Forall</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:49"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:52"><span class="id" title="variable">v</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall2_inv" class="idref" href="#Forall2_inv"><span class="id" title="lemma">Forall2_inv</span></a> {<a id="A:53" class="idref" href="#A:53"><span class="id" title="binder">A</span></a> <a id="B:54" class="idref" href="#B:54"><span class="id" title="binder">B</span></a>:<span class="id" title="keyword">Type</span>}(<a id="P:55" class="idref" href="#P:55"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.MoreVectors.html#A:53"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#B:54"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)(<a id="n:56" class="idref" href="#n:56"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;<a id="a:57" class="idref" href="#a:57"><span class="id" title="binder">a</span></a> <a id="b:58" class="idref" href="#b:58"><span class="id" title="binder">b</span></a> <a id="v:59" class="idref" href="#v:59"><span class="id" title="binder">v</span></a> <a id="w:60" class="idref" href="#w:60"><span class="id" title="binder">w</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall2"><span class="id" title="inductive">Vector.Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:55"><span class="id" title="variable">P</span></a> (<span class="id" title="var">n</span>:=<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:56"><span class="id" title="variable">n</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:59"><span class="id" title="variable">v</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">Vector.cons</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#b:58"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#w:60"><span class="id" title="variable">w</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#P:55"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:57"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#b:58"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall2"><span class="id" title="inductive">Vector.Forall2</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:55"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:59"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#w:60"><span class="id" title="variable">w</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Computes the vector  f(from), f(from+1), f(from+2),...,f (from+n-1)  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="vect_from_fun" class="idref" href="#vect_from_fun"><span class="id" title="definition">vect_from_fun</span></a> {<a id="A:61" class="idref" href="#A:61"><span class="id" title="binder">A</span></a>} (<a id="f:62" class="idref" href="#f:62"><span class="id" title="binder">f</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Prelude.MoreVectors.html#A:61"><span class="id" title="variable">A</span></a>) <a id="n:63" class="idref" href="#n:63"><span class="id" title="binder">n</span></a> <a id="from:64" class="idref" href="#from:64"><span class="id" title="binder">from</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:61"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:63"><span class="id" title="variable">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:63"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:61"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:66"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0%<span class="id" title="var">nat</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> (<a class="idref" href="hydras.Prelude.MoreVectors.html#f:62"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#from:64"><span class="id" title="variable">from</span></a>) (<a class="idref" href="hydras.Prelude.MoreVectors.html#vect_from_fun:65"><span class="id" title="definition">vect_from_fun</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#f:62"><span class="id" title="variable">f</span></a> <span class="id" title="var">p</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#from:64"><span class="id" title="variable">from</span></a>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+On vector decomposition  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="vfst" class="idref" href="#vfst"><span class="id" title="abbreviation">vfst</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="vsnd" class="idref" href="#vsnd"><span class="id" title="abbreviation">vsnd</span></a> <span class="id" title="var">v</span> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> <span class="id" title="var">v</span>)).<br/>
+<span class="id" title="keyword">Notation</span> <a id="vthird" class="idref" href="#vthird"><span class="id" title="abbreviation">vthird</span></a> <span class="id" title="var">v</span> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> <span class="id" title="var">v</span>))).<br/>
+<span class="id" title="keyword">Notation</span> <a id="vfourth" class="idref" href="#vfourth"><span class="id" title="abbreviation">vfourth</span></a> <span class="id" title="var">v</span> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> <span class="id" title="var">v</span>)))).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decomp2" class="idref" href="#decomp2"><span class="id" title="lemma">decomp2</span></a> {<a id="A:67" class="idref" href="#A:67"><span class="id" title="binder">A</span></a>}  :  <span class="id" title="keyword">∀</span> <a id="v:68" class="idref" href="#v:68"><span class="id" title="binder">v</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:67"><span class="id" title="variable">A</span></a> 2,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#v:68"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:68"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">Vector.hd</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:68"><span class="id" title="variable">v</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="decompos2" class="idref" href="#decompos2"><span class="id" title="lemma">decompos2</span></a> {<a id="A:69" class="idref" href="#A:69"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="v:70" class="idref" href="#v:70"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:69"><span class="id" title="variable">A</span></a> 2, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><a id="a:71" class="idref" href="#a:71"><span class="id" title="binder">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:69"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="b:72" class="idref" href="#b:72"><span class="id" title="binder">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:69"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:70"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Prelude.MoreVectors.html#a:71"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">;</span></a><a class="idref" href="hydras.Prelude.MoreVectors.html#b:72"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#48c8ce35afc5c902143c59f20156f923"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="match2" class="idref" href="#match2"><span class="id" title="definition">match2</span></a> {<a id="A:73" class="idref" href="#A:73"><span class="id" title="binder">A</span></a> <a id="B:74" class="idref" href="#B:74"><span class="id" title="binder">B</span></a>:<span class="id" title="keyword">Type</span>} (<a id="f:75" class="idref" href="#f:75"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.MoreVectors.html#A:73"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:73"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#B:74"><span class="id" title="variable">B</span></a>) (<a id="v:76" class="idref" href="#v:76"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:73"><span class="id" title="variable">A</span></a> 2): <a class="idref" href="hydras.Prelude.MoreVectors.html#B:74"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span>   (<a class="idref" href="hydras.Prelude.MoreVectors.html#decompos2"><span class="id" title="lemma">decompos2</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:76"><span class="id" title="variable">v</span></a> )<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#existT"><span class="id" title="constructor">existT</span></a> <span class="id" title="var">_</span> <span class="id" title="var">x</span> <span class="id" title="var">Hx</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">Hx</span> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> <span class="id" title="var">y</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.Prelude.MoreVectors.html#f:75"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="keyword">end</span> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Vec2_proj" class="idref" href="#Vec2_proj"><span class="id" title="definition">Vec2_proj</span></a> {<a id="A:78" class="idref" href="#A:78"><span class="id" title="binder">A</span></a>} (<a id="P2:79" class="idref" href="#P2:79"><span class="id" title="binder">P2</span></a> : <a class="idref" href="hydras.Prelude.MoreVectors.html#A:78"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:78"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:78"><span class="id" title="variable">A</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">vdec2</span> <span class="id" title="var">v</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">x</span> := <span class="id" title="tactic">fresh</span> <span class="id" title="var">a</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">y</span> :=<span class="id" title="tactic">fresh</span> <span class="id" title="var">b</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">tmp</span> := <span class="id" title="tactic">fresh</span> "tmp" <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> :=<span class="id" title="tactic">fresh</span> "e" <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;<span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.Prelude.MoreVectors.html#decompos2"><span class="id" title="lemma">decompos2</span></a> <span class="id" title="var">v</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">x</span> <span class="id" title="var">tmp</span>]; <span class="id" title="tactic">destruct</span> <span class="id" title="var">tmp</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span> <span class="id" title="var">e</span>]; <span class="id" title="tactic">subst</span> <span class="id" title="var">v</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="In_cases" class="idref" href="#In_cases"><span class="id" title="lemma">In_cases</span></a> {<a id="A:80" class="idref" href="#A:80"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}{<a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="v:82" class="idref" href="#v:82"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:80"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:81"><span class="id" title="variable">n</span></a>)) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:83" class="idref" href="#x:83"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#In"><span class="id" title="inductive">In</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#x:83"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:82"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#x:83"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:82"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#In"><span class="id" title="inductive">In</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#x:83"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:82"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall_and" class="idref" href="#Forall_and"><span class="id" title="lemma">Forall_and</span></a> {<a id="A:84" class="idref" href="#A:84"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="P:85" class="idref" href="#P:85"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.MoreVectors.html#A:84"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) {<a id="n:86" class="idref" href="#n:86"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="v:87" class="idref" href="#v:87"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:84"><span class="id" title="variable">A</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:86"><span class="id" title="variable">n</span></a>)) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Forall</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:85"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:87"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:85"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#hd"><span class="id" title="definition">hd</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:87"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Forall</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:85"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:87"><span class="id" title="variable">v</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For V8.11 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Forall_forall" class="idref" href="#Forall_forall"><span class="id" title="lemma">Forall_forall</span></a> {<a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="P:89" class="idref" href="#P:89"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.MoreVectors.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> {<a id="n:90" class="idref" href="#n:90"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="v:91" class="idref" href="#v:91"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:90"><span class="id" title="variable">n</span></a>) ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Forall</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:91"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:92" class="idref" href="#a:92"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#In"><span class="id" title="inductive">In</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:91"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#a:92"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Vectors of natural numbers 
+<div class="paragraph"> </div>
+
+<a id="lab16"></a><h1 class="section">Maximum of a vector of nat</h1>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="max_v" class="idref" href="#max_v"><span class="id" title="definition">max_v</span></a> {<a id="n:93" class="idref" href="#n:93"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>} (<a id="v:94" class="idref" href="#v:94"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:93"><span class="id" title="variable">n</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:94"><span class="id" title="variable">v</span></a>  <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a> ⇒  0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a> <span class="id" title="var">x</span> (<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v:95"><span class="id" title="definition">max_v</span></a> <span class="id" title="var">t</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_v_2" class="idref" href="#max_v_2"><span class="id" title="lemma">max_v_2</span></a> : <span class="id" title="keyword">∀</span> <a id="x:97" class="idref" href="#x:97"><span class="id" title="binder">x</span></a> <a id="y:98" class="idref" href="#y:98"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> (<a class="idref" href="hydras.Prelude.MoreVectors.html#x:97"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.MoreVectors.html#y:98"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.VectorDef.html#VectorNotations.::vector_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#x:97"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#y:98"><span class="id" title="variable">y</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_v_lub" class="idref" href="#max_v_lub"><span class="id" title="lemma">max_v_lub</span></a> : <span class="id" title="keyword">∀</span> <a id="n:99" class="idref" href="#n:99"><span class="id" title="binder">n</span></a> (<a id="v:100" class="idref" href="#v:100"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:99"><span class="id" title="variable">n</span></a>) <a id="y:101" class="idref" href="#y:101"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="x:102" class="idref" href="#x:102"><span class="id" title="binder">x</span></a> ⇒  <a class="idref" href="hydras.Prelude.MoreVectors.html#x:102"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#y:101"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.Prelude.MoreVectors.html#v:100"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:100"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#y:101"><span class="id" title="variable">y</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_v_ge" class="idref" href="#max_v_ge"><span class="id" title="lemma">max_v_ge</span></a> : <span class="id" title="keyword">∀</span> <a id="n:103" class="idref" href="#n:103"><span class="id" title="binder">n</span></a> (<a id="v:104" class="idref" href="#v:104"><span class="id" title="binder">v</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:103"><span class="id" title="variable">n</span></a>) <a id="y:105" class="idref" href="#y:105"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#In"><span class="id" title="inductive">In</span></a>  <a class="idref" href="hydras.Prelude.MoreVectors.html#y:105"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Prelude.MoreVectors.html#v:104"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#y:105"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:104"><span class="id" title="variable">v</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_v_tl" class="idref" href="#max_v_tl"><span class="id" title="lemma">max_v_tl</span></a> {<a id="n:106" class="idref" href="#n:106"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>}(<a id="v:107" class="idref" href="#v:107"><span class="id" title="binder">v</span></a>:  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#t"><span class="id" title="inductive">Vector.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#n:106"><span class="id" title="variable">n</span></a>)) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Vectors.Vector.html#tl"><span class="id" title="definition">Vector.tl</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:107"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#max_v"><span class="id" title="definition">max_v</span></a> <a class="idref" href="hydras.Prelude.MoreVectors.html#v:107"><span class="id" title="variable">v</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
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+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.More_Arith</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.More_Arith</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Note by Pierre:
+
+<div class="paragraph"> </div>
+
+  Some lemmas of this file are possibly in Standard Library 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> .<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Arith_lemmas" class="idref" href="#Arith_lemmas"><span class="id" title="section">Arith_lemmas</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nat_double_or_s_double" class="idref" href="#nat_double_or_s_double"><span class="id" title="lemma">nat_double_or_s_double</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p:2" class="idref" href="#p:2"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:2"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p:3" class="idref" href="#p:3"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:3"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div2_double_is_id" class="idref" href="#div2_double_is_id"><span class="id" title="lemma">div2_double_is_id</span></a> :<span class="id" title="keyword">∀</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:4"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:4"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="double_S" class="idref" href="#double_S"><span class="id" title="lemma">double_S</span></a> (<a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:5"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:5"><span class="id" title="variable">n</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="double_plus" class="idref" href="#double_plus"><span class="id" title="lemma">double_plus</span></a> (<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a> <a id="y:7" class="idref" href="#y:7"><span class="id" title="binder">y</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> (<a class="idref" href="hydras.Prelude.More_Arith.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#y:7"><span class="id" title="variable">y</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#y:7"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="double_inj" class="idref" href="#double_inj"><span class="id" title="lemma">double_inj</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m:8" class="idref" href="#m:8"><span class="id" title="binder">m</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:8"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:8"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:9"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="double_is_even" class="idref" href="#double_is_even"><span class="id" title="lemma">double_is_even</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.Even"><span class="id" title="definition">Nat.Even</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:10"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_double_is_s_double" class="idref" href="#not_double_is_s_double"><span class="id" title="lemma">not_double_is_s_double</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="m:11" class="idref" href="#m:11"><span class="id" title="binder">m</span></a> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:11"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:12"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div2_of_Even" class="idref" href="#div2_of_Even"><span class="id" title="lemma">div2_of_Even</span></a> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.Even"><span class="id" title="definition">Nat.Even</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:13"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:13"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:13"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="even_prod" class="idref" href="#even_prod"><span class="id" title="lemma">even_prod</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:14" class="idref" href="#p:14"><span class="id" title="binder">p</span></a> <a id="q:15" class="idref" href="#q:15"><span class="id" title="binder">q</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.Even"><span class="id" title="definition">Nat.Even</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#p:14"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:15"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#p:14"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:15"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_2" class="idref" href="#plus_2"><span class="id" title="lemma">plus_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:16"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:16"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 2.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div2_incr" class="idref" href="#div2_incr"><span class="id" title="lemma">div2_incr</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> <a id="m:18" class="idref" href="#m:18"><span class="id" title="binder">m</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:18"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:18"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div2_even_plus" class="idref" href="#div2_even_plus"><span class="id" title="lemma">div2_even_plus</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> <a id="m:20" class="idref" href="#m:20"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.Even"><span class="id" title="definition">Even</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:20"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.div2"><span class="id" title="definition">div2</span></a> (<a class="idref" href="hydras.Prelude.More_Arith.html#n:19"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.double"><span class="id" title="definition">double</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#m:20"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_lt_lt" class="idref" href="#mult_lt_lt"><span class="id" title="lemma">mult_lt_lt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:21" class="idref" href="#p:21"><span class="id" title="binder">p</span></a> <a id="p':22" class="idref" href="#p':22"><span class="id" title="binder">p'</span></a> <a id="k:23" class="idref" href="#k:23"><span class="id" title="binder">k</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#k:23"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p':22"><span class="id" title="variable">p'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#k:23"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p':22"><span class="id" title="variable">p'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minus_semi_assoc" class="idref" href="#minus_semi_assoc"><span class="id" title="lemma">minus_semi_assoc</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a> <a id="c:26" class="idref" href="#c:26"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#b:25"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#c:26"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#b:25"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#c:26"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:25"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#c:26"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div_not_qlt" class="idref" href="#div_not_qlt"><span class="id" title="lemma">div_not_qlt</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="b:28" class="idref" href="#b:28"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="q:29" class="idref" href="#q:29"><span class="id" title="binder">q</span></a> <a id="q':30" class="idref" href="#q':30"><span class="id" title="binder">q'</span></a> <a id="r:31" class="idref" href="#r:31"><span class="id" title="binder">r</span></a> <a id="r':32" class="idref" href="#r':32"><span class="id" title="binder">r'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.More_Arith.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:29"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:31"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':30"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':32"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:31"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':32"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:29"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':30"><span class="id" title="variable">q'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="div_eucl_unique" class="idref" href="#div_eucl_unique"><span class="id" title="lemma">div_eucl_unique</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a> <a id="b:34" class="idref" href="#b:34"><span class="id" title="binder">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="q:35" class="idref" href="#q:35"><span class="id" title="binder">q</span></a> <a id="q':36" class="idref" href="#q':36"><span class="id" title="binder">q'</span></a> <a id="r:37" class="idref" href="#r:37"><span class="id" title="binder">r</span></a> <a id="r':38" class="idref" href="#r':38"><span class="id" title="binder">r'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.More_Arith.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:35"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:37"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':36"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':38"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:37"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:34"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&gt;'_x"><span class="id" title="notation">&gt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':38"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:35"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':36"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:37"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':38"><span class="id" title="variable">r'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_le_plus" class="idref" href="#max_le_plus"><span class="id" title="lemma">max_le_plus</span></a> (<a id="n:39" class="idref" href="#n:39"><span class="id" title="binder">n</span></a> <a id="p:40" class="idref" href="#p:40"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">Nat.max</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:40"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:39"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:40"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_le_regR" class="idref" href="#max_le_regR"><span class="id" title="lemma">max_le_regR</span></a> : <span class="id" title="keyword">∀</span> <a id="n:41" class="idref" href="#n:41"><span class="id" title="binder">n</span></a> <a id="p:42" class="idref" href="#p:42"><span class="id" title="binder">p</span></a> <a id="q:43" class="idref" href="#q:43"><span class="id" title="binder">q</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#p:42"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:43"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:42"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:41"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:43"><span class="id" title="variable">q</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:41"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="max_le_regL" class="idref" href="#max_le_regL"><span class="id" title="lemma">max_le_regL</span></a> :  <span class="id" title="keyword">∀</span> <a id="n:44" class="idref" href="#n:44"><span class="id" title="binder">n</span></a> <a id="p:45" class="idref" href="#p:45"><span class="id" title="binder">p</span></a> <a id="q:46" class="idref" href="#q:46"><span class="id" title="binder">q</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#p:45"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:46"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:44"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:45"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.max"><span class="id" title="definition">max</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:44"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:46"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_lt_Sn" class="idref" href="#lt_lt_Sn"><span class="id" title="lemma">lt_lt_Sn</span></a> : <span class="id" title="keyword">∀</span> <a id="a:47" class="idref" href="#a:47"><span class="id" title="binder">a</span></a> <a id="b:48" class="idref" href="#b:48"><span class="id" title="binder">b</span></a> <a id="c:49" class="idref" href="#c:49"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#a:47"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:48"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:48"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#c:49"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#a:47"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#c:49"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.More_Arith.html#Arith_lemmas"><span class="id" title="section">Arith_lemmas</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+From Cantor contrib 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="power" class="idref" href="#power"><span class="id" title="abbreviation">power</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pow"><span class="id" title="definition">Nat.pow</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="power_of_1" class="idref" href="#power_of_1"><span class="id" title="lemma">power_of_1</span></a> : <span class="id" title="keyword">∀</span> <a id="p:50" class="idref" href="#p:50"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Prelude.More_Arith.html#power"><span class="id" title="abbreviation">power</span></a> 1 <a class="idref" href="hydras.Prelude.More_Arith.html#p:50"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">∀</span></a> <a id="a:51" class="idref" href="#a:51"><span class="id" title="binder">a</span></a> <a id="b:52" class="idref" href="#b:52"><span class="id" title="binder">b</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">,</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">0</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">&lt;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">power</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">(</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">S</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">a</span></a><a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="pred_of_power" class="idref" href="#pred_of_power"><span class="id" title="lemma">pred_of_power</span></a> : <span class="id" title="keyword">∀</span> <a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a> <a id="e:54" class="idref" href="#e:54"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Prelude.More_Arith.html#power"><span class="id" title="abbreviation">power</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:53"><span class="id" title="variable">b</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#e:54"><span class="id" title="variable">e</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#power"><span class="id" title="abbreviation">power</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:53"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Prelude.More_Arith.html#e:54"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)*</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#b:53"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="hydras.Prelude.More_Arith.html#power"><span class="id" title="abbreviation">power</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:53"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Prelude.More_Arith.html#e:54"><span class="id" title="variable">e</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="get_predecessor" class="idref" href="#get_predecessor"><span class="id" title="lemma">get_predecessor</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:55" class="idref" href="#n:55"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:55"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><a id="p:56" class="idref" href="#p:56"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#n:55"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#p:56"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">pred_exhib</span> <span class="id" title="var">H</span> <span class="id" title="var">name</span> := <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">type</span> <span class="id" title="keyword">of</span> <span class="id" title="var">H</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> ?<span class="id" title="var">n</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="hydras.Prelude.More_Arith.html#get_predecessor"><span class="id" title="lemma">get_predecessor</span></a>  <span class="id" title="var">H</span>); <span class="id" title="tactic">intro</span> <span class="id" title="var">name</span>; <span class="id" title="tactic">intro</span> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Euc1" class="idref" href="#Euc1"><span class="id" title="lemma">Euc1</span></a> : <span class="id" title="keyword">∀</span> <a id="b:57" class="idref" href="#b:57"><span class="id" title="binder">b</span></a> <a id="q:58" class="idref" href="#q:58"><span class="id" title="binder">q</span></a>  <a id="q':59" class="idref" href="#q':59"><span class="id" title="binder">q'</span></a> <a id="r:60" class="idref" href="#r:60"><span class="id" title="binder">r</span></a> <a id="r':61" class="idref" href="#r':61"><span class="id" title="binder">r'</span></a>, 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.More_Arith.html#q:58"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r:60"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':59"><span class="id" title="variable">q'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a><a class="idref" href="hydras.Prelude.More_Arith.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':61"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.More_Arith.html#r:60"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#r':61"><span class="id" title="variable">r'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q:58"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.More_Arith.html#q':59"><span class="id" title="variable">q'</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.OrdNotations.html b/theories/html/hydras.Prelude.OrdNotations.html
new file mode 100644
index 00000000..cc89b016
--- /dev/null
+++ b/theories/html/hydras.Prelude.OrdNotations.html
@@ -0,0 +1,46 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.OrdNotations</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.OrdNotations</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&gt;=' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&gt;' y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;=' y 'o&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;=' y 'o&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;' y 'o&lt;' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;x 'o&lt;' y 'o&lt;=' z" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
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\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Restriction.html b/theories/html/hydras.Prelude.Restriction.html
new file mode 100644
index 00000000..685fcd9b
--- /dev/null
+++ b/theories/html/hydras.Prelude.Restriction.html
@@ -0,0 +1,130 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Restriction</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Restriction</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Pierre Casteran 
+    LaBRI, University of Bordeaux and LaBRI
+
+<div class="paragraph"> </div>
+
+    Restriction of a binary relation 
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="restrict" class="idref" href="#restrict"><span class="id" title="definition">restrict</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="E:2" class="idref" href="#E:2"><span class="id" title="binder">E</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#A:1"><span class="id" title="variable">A</span></a>)(<a id="R:3" class="idref" href="#R:3"><span class="id" title="binder">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="a:4" class="idref" href="#a:4"><span class="id" title="binder">a</span></a> <a id="b:5" class="idref" href="#b:5"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="hydras.Prelude.Restriction.html#E:2"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:4"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#R:3"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:4"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#b:5"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#E:2"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#b:5"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="restricted_recursion" class="idref" href="#restricted_recursion"><span class="id" title="section">restricted_recursion</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="restricted_recursion.A" class="idref" href="#restricted_recursion.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="restricted_recursion.E" class="idref" href="#restricted_recursion.E"><span class="id" title="variable">E</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#A:6"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>)(<a id="restricted_recursion.R" class="idref" href="#restricted_recursion.R"><span class="id" title="variable">R</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#A:6"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Prelude.Restriction.html#A:6"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="well_founded_restriction" class="idref" href="#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>), <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.Prelude.Restriction.html#a:9"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Induction principle for the restriction of R to E 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="well_founded_restriction_rect" class="idref" href="#well_founded_restriction_rect"><span class="id" title="definition">well_founded_restriction_rect</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="X:10" class="idref" href="#X:10"><span class="id" title="binder">X</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:11"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:11"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:12"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:11"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:13"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Restriction.html#restricted_recursion"><span class="id" title="section">restricted_recursion</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Fix" class="idref" href="#Fix"><span class="id" title="section">Fix</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Fix.A" class="idref" href="#Fix.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="Fix.E" class="idref" href="#Fix.E"><span class="id" title="variable">E</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#A:14"><span class="id" title="variable">A</span></a>)(<a id="Fix.R" class="idref" href="#Fix.R"><span class="id" title="variable">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#A:14"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Fix.Hwf" class="idref" href="#Fix.Hwf"><span class="id" title="variable">Hwf</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Fix.P" class="idref" href="#Fix.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Fix.F" class="idref" href="#Fix.F"><span class="id" title="variable">F</span></a>  : (<span class="id" title="keyword">∀</span> <a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:20" class="idref" href="#y:20"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:20"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:20"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:20"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:19"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="restriction_fwd" class="idref" href="#restriction_fwd"><span class="id" title="lemma">restriction_fwd</span></a> : <span class="id" title="keyword">∀</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> <a id="y:23" class="idref" href="#y:23"><span class="id" title="binder">y</span></a> (<a id="H:24" class="idref" href="#H:24"><span class="id" title="binder">H</span></a>: <a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:23"><span class="id" title="variable">y</span></a>), <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:22"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="restrict_build" class="idref" href="#restrict_build"><span class="id" title="definition">restrict_build</span></a> <a id="x:25" class="idref" href="#x:25"><span class="id" title="binder">x</span></a> <a id="y:26" class="idref" href="#y:26"><span class="id" title="binder">y</span></a> (<a id="Hx:27" class="idref" href="#Hx:27"><span class="id" title="binder">Hx</span></a>: <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:25"><span class="id" title="variable">x</span></a>)(<a id="Hy:28" class="idref" href="#Hy:28"><span class="id" title="binder">Hy</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:26"><span class="id" title="variable">y</span></a>)(<a id="H:29" class="idref" href="#H:29"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:26"><span class="id" title="variable">y</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:27"><span class="id" title="variable">Hx</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#H:29"><span class="id" title="variable">H</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hy:28"><span class="id" title="variable">Hy</span></a>) .<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="FixR_F" class="idref" href="#FixR_F"><span class="id" title="definition">FixR_F</span></a> (<a id="x:30" class="idref" href="#x:30"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>)(<a id="Hx:31" class="idref" href="#Hx:31"><span class="id" title="binder">Hx</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a>)(<a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a>  (<a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>)<a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a> ) : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#Fix.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:31"><span class="id" title="variable">Hx</span></a> (<span class="id" title="keyword">fun</span> (<a id="y:34" class="idref" href="#y:34"><span class="id" title="binder">y</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>) (<a id="h0:35" class="idref" href="#h0:35"><span class="id" title="binder">h0</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a>) (<a id="h:36" class="idref" href="#h:36"><span class="id" title="binder">h</span></a> :  <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>  <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#FixR_F:33"><span class="id" title="definition">FixR_F</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restriction_fwd"><span class="id" title="lemma">restriction_fwd</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Prelude.Restriction.html#restrict_build"><span class="id" title="definition">restrict_build</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h0:35"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:31"><span class="id" title="variable">Hx</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h:36"><span class="id" title="variable">h</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc_inv"><span class="id" title="lemma">Acc_inv</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#a:32"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restrict_build"><span class="id" title="definition">restrict_build</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:34"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:30"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h0:35"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:31"><span class="id" title="variable">Hx</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h:36"><span class="id" title="variable">h</span></a>))).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="FixR_F_eq" class="idref" href="#FixR_F_eq"><span class="id" title="lemma">FixR_F_eq</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="x:37" class="idref" href="#x:37"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>)(<a id="Hx:38" class="idref" href="#Hx:38"><span class="id" title="binder">Hx</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a>) (<a id="r:39" class="idref" href="#r:39"><span class="id" title="binder">r</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a>) ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#Fix.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span>  <a class="idref" href="hydras.Prelude.Restriction.html#Hx:38"><span class="id" title="variable">Hx</span></a> (<span class="id" title="keyword">fun</span> (<a id="y:40" class="idref" href="#y:40"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>)( <a id="h0:41" class="idref" href="#h0:41"><span class="id" title="binder">h0</span></a> : <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a>) (<a id="h:42" class="idref" href="#h:42"><span class="id" title="binder">h</span></a> :  <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>  <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#FixR_F"><span class="id" title="definition">FixR_F</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restriction_fwd"><span class="id" title="lemma">restriction_fwd</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a>                                                                               (<a class="idref" href="hydras.Prelude.Restriction.html#restrict_build"><span class="id" title="definition">restrict_build</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h0:41"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:38"><span class="id" title="variable">Hx</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h:42"><span class="id" title="variable">h</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc_inv"><span class="id" title="lemma">Acc_inv</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#r:39"><span class="id" title="variable">r</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#restrict_build"><span class="id" title="definition">restrict_build</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#y:40"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h0:41"><span class="id" title="variable">h0</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:38"><span class="id" title="variable">Hx</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#h:42"><span class="id" title="variable">h</span></a>)))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#FixR_F"><span class="id" title="definition">FixR_F</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Hx:38"><span class="id" title="variable">Hx</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#r:39"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Fix.Rwf" class="idref" href="#Fix.Rwf"><span class="id" title="variable">Rwf</span></a> :  <a class="idref" href="hydras.Prelude.Restriction.html#well_founded_restriction"><span class="id" title="definition">well_founded_restriction</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#Fix.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="FixR" class="idref" href="#FixR"><span class="id" title="definition">FixR</span></a> (<a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#Fix.A"><span class="id" title="variable">A</span></a>)(<a id="H:45" class="idref" href="#H:45"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Prelude.Restriction.html#Fix.E"><span class="id" title="variable">E</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:44"><span class="id" title="variable">x</span></a>) := <a class="idref" href="hydras.Prelude.Restriction.html#FixR_F"><span class="id" title="definition">FixR_F</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#H:45"><span class="id" title="variable">H</span></a> (<a class="idref" href="hydras.Prelude.Restriction.html#Fix.Rwf"><span class="id" title="variable">Rwf</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Restriction.html#H:45"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Restriction.html#Fix"><span class="id" title="section">Fix</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.Restriction.html#FixR"><span class="id" title="definition">FixR</span></a> [<span class="id" title="var">A</span> <span class="id" title="var">E</span> <span class="id" title="var">R</span> <span class="id" title="var">P</span>] <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.STDPP_compat.html b/theories/html/hydras.Prelude.STDPP_compat.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Prelude.STDPP_compat.html
@@ -0,0 +1,75 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.STDPP_compat</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.STDPP_compat</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Assoc" class="idref" href="#Assoc"><span class="id" title="record">Assoc</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>} (<a id="R:2" class="idref" href="#R:2"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="f:3" class="idref" href="#f:3"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#A:1"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="assoc" class="idref" href="#assoc"><span class="id" title="projection">assoc</span></a> <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> <a id="z:7" class="idref" href="#z:7"><span class="id" title="binder">z</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#R:2"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.STDPP_compat.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#x:5"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Prelude.STDPP_compat.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#z:7"><span class="id" title="variable">z</span></a>)) (<a class="idref" href="hydras.Prelude.STDPP_compat.html#f:3"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Prelude.STDPP_compat.html#f:3"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#y:6"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.Prelude.STDPP_compat.html#z:7"><span class="id" title="variable">z</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Decision typeclasses following Spitters and van der Weegen 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="Decision" class="idref" href="#Decision"><span class="id" title="record">Decision</span></a> (<a id="P:9" class="idref" href="#P:9"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>) := <a id="decide" class="idref" href="#decide"><span class="id" title="projection">decide</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.Prelude.STDPP_compat.html#P:9"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Prelude.STDPP_compat.html#P:9"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="var">Mode</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#Decision"><span class="id" title="class">Decision</span></a> ! : <span class="id" title="var">typeclass_instances</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide"><span class="id" title="definition">decide</span></a> <span class="id" title="var">_</span> {<span class="id" title="var">_</span>} : <span class="id" title="tactic">simpl</span> <span class="id" title="var">never</span>, <span class="id" title="tactic">assert</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="RelDecision" class="idref" href="#RelDecision"><span class="id" title="record">RelDecision</span></a> {<a id="A:12" class="idref" href="#A:12"><span class="id" title="binder">A</span></a> <a id="B:13" class="idref" href="#B:13"><span class="id" title="binder">B</span></a>} (<a id="R:14" class="idref" href="#R:14"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#B:13"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :=<br/>
+&nbsp;&nbsp;<a id="decide_rel" class="idref" href="#decide_rel"><span class="id" title="projection">decide_rel</span></a> <a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a> <a id="y:17" class="idref" href="#y:17"><span class="id" title="binder">y</span></a> :&gt; <a class="idref" href="hydras.Prelude.STDPP_compat.html#Decision"><span class="id" title="class">Decision</span></a> (<a class="idref" href="hydras.Prelude.STDPP_compat.html#R:14"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#y:17"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="var">Mode</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> ! ! ! : <span class="id" title="var">typeclass_instances</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#decide_rel"><span class="id" title="definition">decide_rel</span></a> {<span class="id" title="var">_</span> <span class="id" title="var">_</span>} <span class="id" title="var">_</span> {<span class="id" title="var">_</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> : <span class="id" title="tactic">simpl</span> <span class="id" title="var">never</span>, <span class="id" title="tactic">assert</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="EqDecision" class="idref" href="#EqDecision"><span class="id" title="abbreviation">EqDecision</span></a> <span class="id" title="var">A</span> := (<a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <span class="id" title="var">A</span>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="bool_decide" class="idref" href="#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a id="P:19" class="idref" href="#P:19"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>) {<a id="dec:20" class="idref" href="#dec:20"><span class="id" title="binder">dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#Decision"><span class="id" title="class">Decision</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#P:19"><span class="id" title="variable">P</span></a>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#dec:20"><span class="id" title="variable">dec</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bool_decide_eq_true" class="idref" href="#bool_decide_eq_true"><span class="id" title="lemma">bool_decide_eq_true</span></a> (<a id="P:21" class="idref" href="#P:21"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>) `{<a id="H:22" class="idref" href="#H:22"><span class="id" title="binder">Decision</span></a> <a id="H:22" class="idref" href="#H:22"><span class="id" title="binder">P</span></a>} : <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#P:21"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#P:21"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="bool_decide_eq_false" class="idref" href="#bool_decide_eq_false"><span class="id" title="lemma">bool_decide_eq_false</span></a> (<a id="P:23" class="idref" href="#P:23"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>) `{<a id="H:24" class="idref" href="#H:24"><span class="id" title="binder">Decision</span></a> <a id="H:24" class="idref" href="#H:24"><span class="id" title="binder">P</span></a>} : <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> <a class="idref" href="hydras.Prelude.STDPP_compat.html#P:23"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.Prelude.STDPP_compat.html#P:23"><span class="id" title="variable">P</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Simple_LexProd.html b/theories/html/hydras.Prelude.Simple_LexProd.html
new file mode 100644
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+++ b/theories/html/hydras.Prelude.Simple_LexProd.html
@@ -0,0 +1,97 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Simple_LexProd</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Simple_LexProd</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Non dependent lexicographic product  Pierre Castéran, Univ. Bordeaux and LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Lexicographic_Product.html#"><span class="id" title="library">Wellfounded.Lexicographic_Product</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Definitions" class="idref" href="#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Definitions.A" class="idref" href="#Definitions.A"><span class="id" title="variable">A</span></a> <a id="Definitions.B" class="idref" href="#Definitions.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Definitions.ltA" class="idref" href="#Definitions.ltA"><span class="id" title="variable">ltA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#A:2"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Definitions.ltB" class="idref" href="#Definitions.ltB"><span class="id" title="variable">ltB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#B:1"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Definitions.wfA" class="idref" href="#Definitions.wfA"><span class="id" title="variable">wfA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltA"><span class="id" title="variable">ltA</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Definitions.wfB" class="idref" href="#Definitions.wfB"><span class="id" title="variable">wfB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltB"><span class="id" title="variable">ltB</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Definitions.pair2sig" class="idref" href="#Definitions.pair2sig"><span class="id" title="variable">pair2sig</span></a> (<a id="p:7" class="idref" href="#p:7"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.B"><span class="id" title="variable">B</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#existT"><span class="id" title="constructor">existT</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.A"><span class="id" title="variable">A</span></a> ⇒ <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#p:7"><span class="id" title="variable">p</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#p:7"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="lexico" class="idref" href="#lexico"><span class="id" title="inductive">lexico</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.B"><span class="id" title="variable">B</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="lex_1" class="idref" href="#lex_1"><span class="id" title="constructor">lex_1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:10" class="idref" href="#a:10"><span class="id" title="binder">a</span></a> <a id="a':11" class="idref" href="#a':11"><span class="id" title="binder">a'</span></a> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a> <a id="b':13" class="idref" href="#b':13"><span class="id" title="binder">b'</span></a>, <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltA"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#a':11"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico:8"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#a:10"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#b:12"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#a':11"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#b':13"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;|  <a id="lex_2" class="idref" href="#lex_2"><span class="id" title="constructor">lex_2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a>  <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a> <a id="b':16" class="idref" href="#b':16"><span class="id" title="binder">b'</span></a>, <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltB"><span class="id" title="variable">ltB</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#b':16"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico:8"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#a:14"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#b:15"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#a:14"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Simple_LexProd.html#b':16"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> .<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Trans_lex" class="idref" href="#Trans_lex"><span class="id" title="instance">Trans_lex</span></a>  {<a id="SA:17" class="idref" href="#SA:17"><span class="id" title="binder">SA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltA"><span class="id" title="variable">ltA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="SB:18" class="idref" href="#SB:18"><span class="id" title="binder">SB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltB"><span class="id" title="variable">ltB</span></a>}  :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="Strict_lex" class="idref" href="#Strict_lex"><span class="id" title="instance">Strict_lex</span></a> {<a id="SA:19" class="idref" href="#SA:19"><span class="id" title="binder">SA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltA"><span class="id" title="variable">ltA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="SB:20" class="idref" href="#SB:20"><span class="id" title="binder">SB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions.ltB"><span class="id" title="variable">ltB</span></a>}  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="wf_lexico" class="idref" href="#wf_lexico"><span class="id" title="lemma">wf_lexico</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.Simple_LexProd.html#wf_lexico"><span class="id" title="lemma">wf_lexico</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">ltA</span> <span class="id" title="var">ltB</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>5<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>4<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> : <a class="idref" href="hydras.Prelude.Simple_LexProd.html#lexico"><span class="id" title="inductive">lexico</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>5<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>6<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.Sort_spec.html b/theories/html/hydras.Prelude.Sort_spec.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Prelude.Sort_spec.html
@@ -0,0 +1,214 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.Sort_spec</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.Sort_spec</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Formal specification of list sorting functions 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#"><span class="id" title="library">Sorting.Permutation</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorting.Sorted</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.DecPreOrder.html#"><span class="id" title="library">DecPreOrder</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#"><span class="id" title="library">DecPreOrder_Instances</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sort_fun_t" class="idref" href="#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a> := <span class="id" title="keyword">∀</span> <a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:1"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="R_given" class="idref" href="#R_given"><span class="id" title="section">R_given</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="R_given.A" class="idref" href="#R_given.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)(<a id="R_given.R" class="idref" href="#R_given.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:2"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LocallySorted_cons" class="idref" href="#LocallySorted_cons"><span class="id" title="lemma">LocallySorted_cons</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>  <a id="l:4" class="idref" href="#l:4"><span class="id" title="binder">l</span></a> <a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a> <a id="b:6" class="idref" href="#b:6"><span class="id" title="binder">b</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#b:6"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:4"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#b:6"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#a:5"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#b:6"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:4"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sort_rel" class="idref" href="#sort_rel"><span class="id" title="definition">sort_rel</span></a> (<a id="l:7" class="idref" href="#l:7"><span class="id" title="binder">l</span></a> <a id="l':8" class="idref" href="#l':8"><span class="id" title="binder">l'</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>) := <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l':8"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a>   <a class="idref" href="hydras.Prelude.Sort_spec.html#l:7"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l':8"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sort_correct" class="idref" href="#sort_correct"><span class="id" title="definition">sort_correct</span></a>  (<a id="f:9" class="idref" href="#f:9"><span class="id" title="binder">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:10" class="idref" href="#l:10"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_rel"><span class="id" title="definition">sort_rel</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:10"><span class="id" title="variable">l</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#f:9"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:10"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+already defined in DecPreOrder ? 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="equiv" class="idref" href="#equiv"><span class="id" title="definition">equiv</span></a> (<a id="x:11" class="idref" href="#x:11"><span class="id" title="binder">x</span></a> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#x:11"><span class="id" title="variable">x</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#y:12"><span class="id" title="variable">y</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#x:11"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="equiv_equiv" class="idref" href="#equiv_equiv"><span class="id" title="instance">equiv_equiv</span></a> (<a id="P:13" class="idref" href="#P:13"><span class="id" title="binder">P</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#equiv"><span class="id" title="definition">equiv</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+Abstract properties  TODO: look into StdLib's Order whether some lemmas are already
+    proved 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="forall_weak" class="idref" href="#forall_weak"><span class="id" title="lemma">forall_weak</span></a> (<a id="H:14" class="idref" href="#H:14"><span class="id" title="binder">H</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:15" class="idref" href="#l:15"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>) (<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a> <a id="b:17" class="idref" href="#b:17"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.A"><span class="id" title="variable">A</span></a>),  <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a:16"><span class="id" title="variable">a</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#b:17"><span class="id" title="variable">b</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#b:17"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Prelude.Sort_spec.html#l:15"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a:16"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Prelude.Sort_spec.html#l:15"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+To remove when it compiles again 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LocallySorted_cons'" class="idref" href="#LocallySorted_cons'"><span class="id" title="lemma">LocallySorted_cons'</span></a> (<a id="Htrans:18" class="idref" href="#Htrans:18"><span class="id" title="binder">Htrans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a> <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">List.Forall</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a:20"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.Prelude.Sort_spec.html#l:19"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:19"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#a:20"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:19"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LocallySorted_trans" class="idref" href="#LocallySorted_trans"><span class="id" title="lemma">LocallySorted_trans</span></a> (<a id="Htrans:21" class="idref" href="#Htrans:21"><span class="id" title="binder">Htrans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:22" class="idref" href="#l:22"><span class="id" title="binder">l</span></a> <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="x:24" class="idref" href="#x:24"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#a:23"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:22"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#x:24"><span class="id" title="variable">x</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#x:24"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:22"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LocallySorted_inv_In" class="idref" href="#LocallySorted_inv_In"><span class="id" title="lemma">LocallySorted_inv_In</span></a> (<a id="Htrans:25" class="idref" href="#Htrans:25"><span class="id" title="binder">Htrans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:26" class="idref" href="#l:26"><span class="id" title="binder">l</span></a> <a id="x:27" class="idref" href="#x:27"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#x:27"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:26"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="keyword">∀</span> <a id="y:28" class="idref" href="#y:28"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#y:28"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:26"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#x:27"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#y:28"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.Sort_spec.html#R_given"><span class="id" title="section">R_given</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Prelude.Sort_spec.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> : <span class="id" title="var">lists</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A sort must work on any decidable total pre-order 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sort_spec" class="idref" href="#sort_spec"><span class="id" title="definition">sort_spec</span></a> (<a id="f:29" class="idref" href="#f:29"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:30" class="idref" href="#A:30"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) (<a id="le:31" class="idref" href="#le:31"><span class="id" title="binder">le</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:30"><span class="id" title="variable">A</span></a>) (<a id="P:32" class="idref" href="#P:32"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:31"><span class="id" title="variable">le</span></a>) (<a id="dec:33" class="idref" href="#dec:33"><span class="id" title="binder">dec</span></a>:<a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:31"><span class="id" title="variable">le</span></a>) (<a id="l:34" class="idref" href="#l:34"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:30"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;<span class="id" title="keyword">let</span> <a id="l':37" class="idref" href="#l':37"><span class="id" title="binder">l'</span></a> := <a class="idref" href="hydras.Prelude.Sort_spec.html#f:29"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:30"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="x:35" class="idref" href="#x:35"><span class="id" title="binder">x</span></a> <a id="y:36" class="idref" href="#y:36"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#le:31"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#x:35"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#y:36"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Sort_spec.html#l:34"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Permutation.html#Permutation"><span class="id" title="inductive">Permutation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:34"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l':37"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:31"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l':37"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A prototype for using TotalDecPreOrder type class 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sort" class="idref" href="#sort"><span class="id" title="definition">sort</span></a> (<a id="f:38" class="idref" href="#f:38"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>)<br/>
+&nbsp;{<a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a>} {<a id="le:40" class="idref" href="#le:40"><span class="id" title="binder">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:39"><span class="id" title="variable">A</span></a>} {<a id="P:41" class="idref" href="#P:41"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:40"><span class="id" title="variable">le</span></a>} {<a id="dec:42" class="idref" href="#dec:42"><span class="id" title="binder">dec</span></a>:<a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:40"><span class="id" title="variable">le</span></a>}<br/>
+&nbsp;(<a id="l:43" class="idref" href="#l:43"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:39"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:39"><span class="id" title="variable">A</span></a> (<span class="id" title="keyword">fun</span> <a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a> <a id="y:45" class="idref" href="#y:45"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.Prelude.STDPP_compat.html#bool_decide"><span class="id" title="definition">bool_decide</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#le:40"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#x:44"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#y:45"><span class="id" title="variable">y</span></a>)) <a class="idref" href="hydras.Prelude.Sort_spec.html#l:43"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+stability 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="marked" class="idref" href="#marked"><span class="id" title="inductive">marked</span></a> {<a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a> <a id="B:47" class="idref" href="#B:47"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>}(<a id="leA:48" class="idref" href="#leA:48"><span class="id" title="binder">leA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:46"><span class="id" title="variable">A</span></a>)(<a id="leB:49" class="idref" href="#leB:49"><span class="id" title="binder">leB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#B:47"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="pA:50" class="idref" href="#pA:50"><span class="id" title="binder">pA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:48"><span class="id" title="variable">leA</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="pB:51" class="idref" href="#pB:51"><span class="id" title="binder">pB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#PreOrder"><span class="id" title="class">PreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:49"><span class="id" title="variable">leB</span></a>}:<br/>
+<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<span class="id" title="var">A</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <span class="id" title="var">B</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+| <a id="marked0" class="idref" href="#marked0"><span class="id" title="constructor">marked0</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#marked:52"><span class="id" title="inductive">marked</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:48"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:49"><span class="id" title="variable">leB</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+| <a id="marked1" class="idref" href="#marked1"><span class="id" title="constructor">marked1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:54" class="idref" href="#a:54"><span class="id" title="binder">a</span></a> <a id="b:55" class="idref" href="#b:55"><span class="id" title="binder">b</span></a> <a id="l:56" class="idref" href="#l:56"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.Prelude.Sort_spec.html#marked:52"><span class="id" title="inductive">marked</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:48"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:49"><span class="id" title="variable">leB</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#l:56"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a':57" class="idref" href="#a':57"><span class="id" title="binder">a'</span></a> <a id="b':58" class="idref" href="#b':58"><span class="id" title="binder">b'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#a':57"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#b':58"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:56"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#leA:48"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a:54"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#a':57"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.DecPreOrder.html#lt"><span class="id" title="definition">lt</span></a>  <a class="idref" href="hydras.Prelude.Sort_spec.html#b:55"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#b':58"><span class="id" title="variable">b'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#marked:52"><span class="id" title="inductive">marked</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:48"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:49"><span class="id" title="variable">leB</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#a:54"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#b:55"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#l:56"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="stable" class="idref" href="#stable"><span class="id" title="definition">stable</span></a> (<a id="f:59" class="idref" href="#f:59"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:60" class="idref" href="#A:60"><span class="id" title="binder">A</span></a> <a id="B:61" class="idref" href="#B:61"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>) <a id="leA:62" class="idref" href="#leA:62"><span class="id" title="binder">leA</span></a> <a id="leB:63" class="idref" href="#leB:63"><span class="id" title="binder">leB</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="PA:64" class="idref" href="#PA:64"><span class="id" title="binder">PA</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:62"><span class="id" title="variable">leA</span></a>) (<a id="PB:65" class="idref" href="#PB:65"><span class="id" title="binder">PB</span></a> : <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:63"><span class="id" title="variable">leB</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="dec:66" class="idref" href="#dec:66"><span class="id" title="binder">dec</span></a> : <a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:62"><span class="id" title="variable">leA</span></a>) (<a id="l:67" class="idref" href="#l:67"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#A:60"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#B:61"><span class="id" title="variable">B</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#marked"><span class="id" title="inductive">marked</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:62"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:63"><span class="id" title="variable">leB</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:67"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#marked"><span class="id" title="inductive">marked</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leA:62"><span class="id" title="variable">leA</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#leB:63"><span class="id" title="variable">leB</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#f:59"><span class="id" title="variable">f</span></a>  (<span class="id" title="var">P</span>:= <a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#Total_Inverse_fun"><span class="id" title="instance">Total_Inverse_fun</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span>:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#A:60"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#B:61"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:60"><span class="id" title="variable">A</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#l:67"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+for testing only 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="mark" class="idref" href="#mark"><span class="id" title="definition">mark</span></a> {<a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a>}(<a id="l:69" class="idref" href="#l:69"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:68"><span class="id" title="variable">A</span></a>)(<a id="from:70" class="idref" href="#from:70"><span class="id" title="binder">from</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#A:68"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=<br/>
+<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:69"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+| (<span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l'</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#from:70"><span class="id" title="variable">from</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Prelude.Sort_spec.html#mark:71"><span class="id" title="definition">mark</span></a> <span class="id" title="var">l'</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#from:70"><span class="id" title="variable">from</span></a>)<br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="stable_test" class="idref" href="#stable_test"><span class="id" title="definition">stable_test</span></a> (<a id="f:73" class="idref" href="#f:73"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Prelude.Sort_spec.html#sort_fun_t"><span class="id" title="definition">sort_fun_t</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="A:74" class="idref" href="#A:74"><span class="id" title="binder">A</span></a>} {<a id="le:75" class="idref" href="#le:75"><span class="id" title="binder">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:74"><span class="id" title="variable">A</span></a>}{<a id="P:76" class="idref" href="#P:76"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Prelude.DecPreOrder.html#TotalPreOrder"><span class="id" title="class">TotalPreOrder</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:75"><span class="id" title="variable">le</span></a>} {<a id="dec:77" class="idref" href="#dec:77"><span class="id" title="binder">dec</span></a>:<a class="idref" href="hydras.Prelude.STDPP_compat.html#RelDecision"><span class="id" title="class">RelDecision</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:75"><span class="id" title="variable">le</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="l:78" class="idref" href="#l:78"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#A:74"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#A:74"><span class="id" title="variable">A</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)                                                               :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="m:79" class="idref" href="#m:79"><span class="id" title="binder">m</span></a> := <a class="idref" href="hydras.Prelude.Sort_spec.html#mark"><span class="id" title="definition">mark</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#l:78"><span class="id" title="variable">l</span></a> 0 <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Sort_spec.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#f:73"><span class="id" title="variable">f</span></a>  (<span class="id" title="var">P</span>:= @<a class="idref" href="hydras.Prelude.DecPreOrder_Instances.html#Total_Inverse_fun"><span class="id" title="instance">Total_Inverse_fun</span></a> (<a class="idref" href="hydras.Prelude.Sort_spec.html#A:74"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="hydras.Prelude.Sort_spec.html#A:74"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#le:75"><span class="id" title="variable">le</span></a> <a class="idref" href="hydras.Prelude.Sort_spec.html#P:76"><span class="id" title="variable">P</span></a>) <a class="idref" href="hydras.Prelude.Sort_spec.html#m:79"><span class="id" title="variable">m</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.WfVariant.html b/theories/html/hydras.Prelude.WfVariant.html
new file mode 100644
index 00000000..7fa98694
--- /dev/null
+++ b/theories/html/hydras.Prelude.WfVariant.html
@@ -0,0 +1,79 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.WfVariant</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.WfVariant</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Pierre Casteran, LaBRI, University of Bordeaux  Definition of termination variants 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Basics.html#"><span class="id" title="library">Basics</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inverse_Image.html#"><span class="id" title="library">Wellfounded.Inverse_Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Inclusion.html#"><span class="id" title="library">Wellfounded.Inclusion</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Variants" class="idref" href="#Variants"><span class="id" title="section">Variants</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Variants.E" class="idref" href="#Variants.E"><span class="id" title="variable">E</span></a>: <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Variants.tr" class="idref" href="#Variants.tr"><span class="id" title="variable">tr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.E"><span class="id" title="variable">E</span></a>. 
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="terminates" class="idref" href="#terminates"><span class="id" title="definition">terminates</span></a>  :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.tr"><span class="id" title="variable">tr</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Variants.T" class="idref" href="#Variants.T"><span class="id" title="variable">T</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Variants.lt" class="idref" href="#Variants.lt"><span class="id" title="variable">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#T:3"><span class="id" title="variable">T</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Variants.m" class="idref" href="#Variants.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.E"><span class="id" title="variable">E</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#T:3"><span class="id" title="variable">T</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Infix</span> <a id="Variants.:::x_'&lt;'_x" class="idref" href="#Variants.:::x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>&lt;" := <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.lt"><span class="id" title="variable">lt</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Class</span> <a id="WfVariant" class="idref" href="#WfVariant"><span class="id" title="record">WfVariant</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="wf" class="idref" href="#wf"><span class="id" title="projection">wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.lt"><span class="id" title="variable">lt</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="decr" class="idref" href="#decr"><span class="id" title="projection">decr</span></a> : <span class="id" title="keyword">∀</span> <a id="x:8" class="idref" href="#x:8"><span class="id" title="binder">x</span></a> <a id="y:9" class="idref" href="#y:9"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.tr"><span class="id" title="variable">tr</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#y:9"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#y:9"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.:::x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.Prelude.WfVariant.html#x:8"><span class="id" title="variable">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Variant_termination" class="idref" href="#Variant_termination"><span class="id" title="lemma">Variant_termination</span></a> (<a id="Var:11" class="idref" href="#Var:11"><span class="id" title="binder">Var</span></a> : <a class="idref" href="hydras.Prelude.WfVariant.html#WfVariant"><span class="id" title="class">WfVariant</span></a> ) :  <a class="idref" href="hydras.Prelude.WfVariant.html#terminates"><span class="id" title="definition">terminates</span></a> .<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Prelude.WfVariant.html#Variants"><span class="id" title="section">Variants</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.WfVariant.html#decr"><span class="id" title="method">decr</span></a> {<span class="id" title="var">E</span> <span class="id" title="var">tr</span> <span class="id" title="var">T</span> <span class="id" title="var">lt</span> <span class="id" title="var">m</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Prelude.WfVariant.html#wf"><span class="id" title="method">wf</span></a> {<span class="id" title="var">E</span> <span class="id" title="var">tr</span> <span class="id" title="var">T</span> <span class="id" title="var">lt</span> <span class="id" title="var">m</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>  .<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Prelude.ssrnat_extracts.html b/theories/html/hydras.Prelude.ssrnat_extracts.html
new file mode 100644
index 00000000..11160b9a
--- /dev/null
+++ b/theories/html/hydras.Prelude.ssrnat_extracts.html
@@ -0,0 +1,61 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Prelude.ssrnat_extracts</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Prelude.ssrnat_extracts</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+<a id="lab8"></a><h1 class="section">Notations adapted from Mathcomp's ssrnat</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="bda89d73ec4a8f23ae92b565ffb5aaa6" class="idref" href="#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">&quot;</span></a>n .+1" := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .+1") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="bc63483175fac06bcd5541a5e9093b18" class="idref" href="#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">&quot;</span></a>n .+2" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1.+1</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .+2") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="331a87f8f0f3c9336601c27f1e458dc3" class="idref" href="#331a87f8f0f3c9336601c27f1e458dc3"><span class="id" title="notation">&quot;</span></a>n .+3" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .+3") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="854d640e6c55155f8eda90636a67ef35" class="idref" href="#854d640e6c55155f8eda90636a67ef35"><span class="id" title="notation">&quot;</span></a>n .+4" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2.+2</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .+4") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="f78f3853e8d6bffffe9b6bc1ce727fd3" class="idref" href="#f78f3853e8d6bffffe9b6bc1ce727fd3"><span class="id" title="notation">&quot;</span></a>n .+5" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#331a87f8f0f3c9336601c27f1e458dc3"><span class="id" title="notation">.+3</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .+5") : <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nat_scope:x_'.-1'" class="idref" href="#::nat_scope:x_'.-1'"><span class="id" title="notation">&quot;</span></a>n .-1" := (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#pred"><span class="id" title="abbreviation">pred</span></a> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .-1") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nat_scope:x_'.-2'" class="idref" href="#::nat_scope:x_'.-2'"><span class="id" title="notation">&quot;</span></a>n .-2" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-1'"><span class="id" title="notation">.-1.-1</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .-2") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nat_scope:x_'.-3'" class="idref" href="#::nat_scope:x_'.-3'"><span class="id" title="notation">&quot;</span></a>n .-3" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-2'"><span class="id" title="notation">.-2</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-1'"><span class="id" title="notation">.-1</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .-3") : <span class="id" title="var">nat_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a id="::nat_scope:x_'.-4'" class="idref" href="#::nat_scope:x_'.-4'"><span class="id" title="notation">&quot;</span></a>n .-4" := <span class="id" title="var">n</span><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-3'"><span class="id" title="notation">.-3</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-1'"><span class="id" title="notation">.-1</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>,<br/>
+&nbsp;&nbsp;<span class="id" title="var">format</span> "n .-4") : <span class="id" title="var">nat_scope</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.AP</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.AP</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab178"></a><h1 class="section">Additive principal ordinals</h1>
+
+<div class="paragraph"> </div>
+
+ Pierre Casteran, LaBRI, Universite de Bordeaux 
+<div class="paragraph"> </div>
+
+
+<div class="paragraph"> </div>
+
+In this library, we define the exponential of basis omega, also called <span class="inlinecode"><span class="id" title="var">phi0</span></span>.
+
+<div class="paragraph"> </div>
+
+In fact,  <math> &omega; <sup> &alpha; </sup> </math> , written  <span class="inlinecode"><span class="id" title="var">phi0</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> in Coq,
+ is defined as the <span class="inlinecode"><span class="id" title="var">alpha</span></span>-th <i>additive principal</i> ordinal.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab179"></a><h2 class="section">Main Definitions</h2>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab180"></a><h3 class="section">Additive principal ordinals</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="AP" class="idref" href="#AP"><span class="id" title="definition">AP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:2" class="idref" href="#beta:2"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Schutte.AP.html#beta:2"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:2"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:1"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab181"></a><h3 class="section">Exponential of basis omega</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="_phi0" class="idref" href="#_phi0"><span class="id" title="definition">_phi0</span></a> := <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="phi0" class="idref" href="#phi0"><span class="id" title="abbreviation">phi0</span></a> := <a class="idref" href="hydras.Schutte.AP.html#_phi0"><span class="id" title="definition">_phi0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="6b5cc3fabae1095df9f5452df6b2c0df" class="idref" href="#6b5cc3fabae1095df9f5452df6b2c0df"><span class="id" title="notation">&quot;</span></a>'omega^'" := <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab182"></a><h3 class="section">Omega-towers</h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="omega_tower" class="idref" href="#omega_tower"><span class="id" title="definition">omega_tower</span></a> (<a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.AP.html#i:3"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒  1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Schutte.AP.html#omega_tower:4"><span class="id" title="definition">omega_tower</span></a> <span class="id" title="var">j</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab183"></a><h3 class="section">The limit ordinal <span class="inlinecode"><span class="id" title="var">epsilon0</span></span></h3>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="epsilon0" class="idref" href="#epsilon0"><span class="id" title="definition">epsilon0</span></a> := <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.AP.html#omega_tower"><span class="id" title="definition">omega_tower</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab184"></a><h2 class="section">Proofs, proofs, proofs ...</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab185"></a><h2 class="section">About additive principals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_one" class="idref" href="#AP_one"><span class="id" title="lemma">AP_one</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="least_AP" class="idref" href="#least_AP"><span class="id" title="lemma">least_AP</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_omega" class="idref" href="#AP_omega"><span class="id" title="lemma">AP_omega</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_lt_omega</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_finite_eq_one" class="idref" href="#AP_finite_eq_one"><span class="id" title="lemma">AP_finite_eq_one</span></a> : <span class="id" title="keyword">∀</span> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Thus, omega is the second additive principal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_second_AP" class="idref" href="#omega_second_AP"><span class="id" title="lemma">omega_second_AP</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> ⇒ 1 <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:7"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_plus_closed" class="idref" href="#AP_plus_closed"><span class="id" title="lemma">AP_plus_closed</span></a> (<a id="alpha:8" class="idref" href="#alpha:8"><span class="id" title="binder">alpha</span></a> <a id="beta:9" class="idref" href="#beta:9"><span class="id" title="binder">beta</span></a> <a id="gamma:10" class="idref" href="#gamma:10"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:9"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#gamma:10"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:8"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#beta:9"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#gamma:10"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:8"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_mult_Sn_closed" class="idref" href="#AP_mult_Sn_closed"><span class="id" title="lemma">AP_mult_Sn_closed</span></a> (<a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> <a id="beta:12" class="idref" href="#beta:12"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:12"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Schutte.Addition.html#mult_Sn"><span class="id" title="definition">mult_Sn</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:12"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.AP.html#n:13"><span class="id" title="variable">n</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:11"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_mult_fin_r_closed" class="idref" href="#AP_mult_fin_r_closed"><span class="id" title="lemma">AP_mult_fin_r_closed</span></a>  (<a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> <a id="beta:15" class="idref" href="#beta:15"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:15"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:14"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Schutte.AP.html#beta:15"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Schutte.AP.html#n:16"><span class="id" title="variable">n</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:14"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="AP_unbounded" class="idref" href="#AP_unbounded"><span class="id" title="lemma">AP_unbounded</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>.  
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="AP_closed" class="idref" href="#AP_closed"><span class="id" title="lemma">AP_closed</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>. 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="AP_o_segment" class="idref" href="#AP_o_segment"><span class="id" title="lemma">AP_o_segment</span></a> :  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab186"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">phi0</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="normal_phi0" class="idref" href="#normal_phi0"><span class="id" title="lemma">normal_phi0</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_ordering" class="idref" href="#phi0_ordering"><span class="id" title="lemma">phi0_ordering</span></a> :  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_elim" class="idref" href="#phi0_elim"><span class="id" title="lemma">phi0_elim</span></a> : <span class="id" title="keyword">∀</span> <a id="P:44" class="idref" href="#P:44"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.AP.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#P:44"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.AP.html#f:45"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#P:44"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_phi0" class="idref" href="#AP_phi0"><span class="id" title="lemma">AP_phi0</span></a> (<a id="alpha:46" class="idref" href="#alpha:46"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> (<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:46"><span class="id" title="variable">alpha</span></a>).  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_zero" class="idref" href="#phi0_zero"><span class="id" title="lemma">phi0_zero</span></a> : <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  1. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono" class="idref" href="#phi0_mono"><span class="id" title="lemma">phi0_mono</span></a> (<a id="alpha:47" class="idref" href="#alpha:47"><span class="id" title="binder">alpha</span></a> <a id="beta:48" class="idref" href="#beta:48"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:48"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:47"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:48"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono_weak" class="idref" href="#phi0_mono_weak"><span class="id" title="lemma">phi0_mono_weak</span></a> (<a id="alpha:49" class="idref" href="#alpha:49"><span class="id" title="binder">alpha</span></a> <a id="beta:50" class="idref" href="#beta:50"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:50"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:49"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:50"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono_R" class="idref" href="#phi0_mono_R"><span class="id" title="lemma">phi0_mono_R</span></a> (<a id="alpha:51" class="idref" href="#alpha:51"><span class="id" title="binder">alpha</span></a> <a id="beta:52" class="idref" href="#beta:52"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:52"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_mono_R_weak" class="idref" href="#phi0_mono_R_weak"><span class="id" title="lemma">phi0_mono_R_weak</span></a> (<a id="alpha:53" class="idref" href="#alpha:53"><span class="id" title="binder">alpha</span></a> <a id="beta:54" class="idref" href="#beta:54"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:53"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:54"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:53"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:54"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_inj" class="idref" href="#phi0_inj"><span class="id" title="lemma">phi0_inj</span></a> (<a id="alpha:55" class="idref" href="#alpha:55"><span class="id" title="binder">alpha</span></a> <a id="beta:56" class="idref" href="#beta:56"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:56"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:56"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_positive" class="idref" href="#phi0_positive"><span class="id" title="lemma">phi0_positive</span></a> (<a id="alpha:57" class="idref" href="#alpha:57"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):  <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:57"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_lt_phi0" class="idref" href="#plus_lt_phi0"><span class="id" title="lemma">plus_lt_phi0</span></a> (<a id="ksi:58" class="idref" href="#ksi:58"><span class="id" title="binder">ksi</span></a> <a id="alpha:59" class="idref" href="#alpha:59"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#ksi:58"><span class="id" title="variable">ksi</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.AP.html#ksi:58"><span class="id" title="variable">ksi</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:59"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_alpha_phi0_beta" class="idref" href="#phi0_alpha_phi0_beta"><span class="id" title="lemma">phi0_alpha_phi0_beta</span></a> (<a id="alpha:60" class="idref" href="#alpha:60"><span class="id" title="binder">alpha</span></a> <a id="beta:61" class="idref" href="#beta:61"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:60"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:61"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:60"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:61"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:61"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_sup" class="idref" href="#phi0_sup"><span class="id" title="lemma">phi0_sup</span></a> : <span class="id" title="keyword">∀</span> <a id="U:62" class="idref" href="#U:62"><span class="id" title="binder">U</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.AP.html#U:62"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.AP.html#U:62"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.AP.html#U:62"><span class="id" title="variable">U</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.AP.html#U:62"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_of_limit" class="idref" href="#phi0_of_limit"><span class="id" title="lemma">phi0_of_limit</span></a> (<a id="alpha:63" class="idref" href="#alpha:63"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:63"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_to_phi0" class="idref" href="#AP_to_phi0"><span class="id" title="lemma">AP_to_phi0</span></a> (<a id="alpha:64" class="idref" href="#alpha:64"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:64"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:65" class="idref" href="#beta:65"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.AP.html#alpha:64"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:65"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="AP_plus_AP" class="idref" href="#AP_plus_AP"><span class="id" title="lemma">AP_plus_AP</span></a> (<a id="alpha:66" class="idref" href="#alpha:66"><span class="id" title="binder">alpha</span></a> <a id="beta:67" class="idref" href="#beta:67"><span class="id" title="binder">beta</span></a> <a id="gamma:68" class="idref" href="#gamma:68"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:67"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:66"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#beta:67"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#gamma:68"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:66"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#gamma:68"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="hydras.Schutte.AP.html#beta:67"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#gamma:68"><span class="id" title="variable">gamma</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_phi0" class="idref" href="#is_limit_phi0"><span class="id" title="lemma">is_limit_phi0</span></a> (<a id="alpha:69" class="idref" href="#alpha:69"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:69"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> (<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:69"><span class="id" title="variable">alpha</span></a>). 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_eqn" class="idref" href="#omega_eqn"><span class="id" title="lemma">omega_eqn</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> 1. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_phi0" class="idref" href="#le_phi0"><span class="id" title="lemma">le_phi0</span></a> (<a id="alpha:70" class="idref" href="#alpha:70"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.AP.html#alpha:70"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:70"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab187"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">epsilon0</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="epsilon0_fxp" class="idref" href="#epsilon0_fxp"><span class="id" title="lemma">epsilon0_fxp</span></a> : <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="epsilon0_AP" class="idref" href="#epsilon0_AP"><span class="id" title="lemma">epsilon0_AP</span></a> : <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_tower_mono" class="idref" href="#omega_tower_mono"><span class="id" title="lemma">omega_tower_mono</span></a> (<a id="i:71" class="idref" href="#i:71"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Schutte.AP.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> <a class="idref" href="hydras.Schutte.AP.html#i:71"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#omega_tower"><span class="id" title="definition">omega_tower</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.AP.html#i:71"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_phi0" class="idref" href="#lt_phi0"><span class="id" title="lemma">lt_phi0</span></a> (<a id="alpha:72" class="idref" href="#alpha:72"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:72"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:72"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:72"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="epsilon0_lfp" class="idref" href="#epsilon0_lfp"><span class="id" title="lemma">epsilon0_lfp</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#least_fixpoint"><span class="id" title="definition">least_fixpoint</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_lt_epsilon0" class="idref" href="#phi0_lt_epsilon0"><span class="id" title="lemma">phi0_lt_epsilon0</span></a> (<a id="alpha:73" class="idref" href="#alpha:73"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:73"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_lt_epsilon0_R" class="idref" href="#phi0_lt_epsilon0_R"><span class="id" title="lemma">phi0_lt_epsilon0_R</span></a> (<a id="alpha:74" class="idref" href="#alpha:74"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:74"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#alpha:74"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Addition</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Addition</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="library">Logic.Epsilon</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">CountableTypes</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#"><span class="id" title="library">Ordering_Functions</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="library">PartialFun</span></a>  <a class="idref" href="hydras.Schutte.Countable.html#"><span class="id" title="library">Countable</span></a>  <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="library">MoreEpsilonIota</span></a>.<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Prelude.STDPP_compat.html#"><span class="id" title="library">STDPP_compat</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab211"></a><h1 class="section">Definitions</h1>
+
+<div class="paragraph"> </div>
+
+<a id="lab212"></a><h2 class="section">addition, multiplication by a positive integer</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="plus" class="idref" href="#plus"><span class="id" title="definition">plus</span></a> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> := <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:1"><span class="id" title="variable">alpha</span></a>).<br/>
+<span class="id" title="keyword">Infix</span> <a id="ef17d4fde9f4f9a265266bf8f6c56f86" class="idref" href="#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">&quot;</span></a>+"  := <a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+returns <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="mult_Sn" class="idref" href="#mult_Sn"><span class="id" title="definition">mult_Sn</span></a> (<a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>){<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.Addition.html#n:3"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.Schutte.Addition.html#alpha:2"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Schutte.Addition.html#mult_Sn:4"><span class="id" title="definition">mult_Sn</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:2"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">p</span> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:2"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="mult_fin_r" class="idref" href="#mult_fin_r"><span class="id" title="definition">mult_fin_r</span></a> <a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.Addition.html#n:7"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.Schutte.Addition.html#mult_Sn"><span class="id" title="definition">mult_Sn</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:6"><span class="id" title="variable">alpha</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="c07b3f4290a4e84a9a38136928201701" class="idref" href="#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">&quot;</span></a>×" := <a class="idref" href="hydras.Schutte.Addition.html#mult_fin_r"><span class="id" title="definition">mult_fin_r</span></a> : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab213"></a><h1 class="section">Proofs, proofs, proofs</h1>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Unbounded_ge" class="idref" href="#Unbounded_ge"><span class="id" title="lemma">Unbounded_ge</span></a> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)  :   <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:9"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ge_o_segment" class="idref" href="#ge_o_segment"><span class="id" title="lemma">ge_o_segment</span></a> (<a id="alpha:10" class="idref" href="#alpha:10"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:10"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_ordering" class="idref" href="#plus_ordering"><span class="id" title="lemma">plus_ordering</span></a> (<a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:11"><span class="id" title="variable">alpha</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:11"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_elim" class="idref" href="#plus_elim"><span class="id" title="lemma">plus_elim</span></a> (<a id="alpha:12" class="idref" href="#alpha:12"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:13" class="idref" href="#P:13"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)-&gt;</span></a><span class="id" title="keyword">Prop</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:14" class="idref" href="#f:14"><span class="id" title="binder">f</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Addition.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:12"><span class="id" title="variable">alpha</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#P:13"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Addition.html#f:14"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#P:13"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:12"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="normal_plus_alpha" class="idref" href="#normal_plus_alpha"><span class="id" title="lemma">normal_plus_alpha</span></a> (<a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:15"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:15"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab214"></a><h2 class="section">Basic properties of addition</h2>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="alpha_plus_zero" class="idref" href="#alpha_plus_zero"><span class="id" title="lemma">alpha_plus_zero</span></a> (<a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):   <a class="idref" href="hydras.Schutte.Addition.html#alpha:16"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:16"><span class="id" title="variable">alpha</span></a>.  
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="ge_zero" class="idref" href="#ge_zero"><span class="id" title="lemma">ge_zero</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_plus_alpha" class="idref" href="#zero_plus_alpha"><span class="id" title="lemma">zero_plus_alpha</span></a> (<a id="alpha:17" class="idref" href="#alpha:17"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:17"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:17"><span class="id" title="variable">alpha</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_plus_l" class="idref" href="#le_plus_l"><span class="id" title="lemma">le_plus_l</span></a> (<a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a> <a id="beta:21" class="idref" href="#beta:21"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Addition.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:21"><span class="id" title="variable">beta</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_plus_r" class="idref" href="#le_plus_r"><span class="id" title="lemma">le_plus_r</span></a> (<a id="alpha:22" class="idref" href="#alpha:22"><span class="id" title="binder">alpha</span></a> <a id="beta:23" class="idref" href="#beta:23"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :  <a class="idref" href="hydras.Schutte.Addition.html#beta:23"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:22"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:23"><span class="id" title="variable">beta</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_r" class="idref" href="#plus_mono_r"><span class="id" title="lemma">plus_mono_r</span></a> (<a id="alpha:24" class="idref" href="#alpha:24"><span class="id" title="binder">alpha</span></a> <a id="beta:25" class="idref" href="#beta:25"><span class="id" title="binder">beta</span></a> <a id="gamma:26" class="idref" href="#gamma:26"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#beta:25"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:26"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:25"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:26"><span class="id" title="variable">gamma</span></a>.  
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_of_succ" class="idref" href="#plus_of_succ"><span class="id" title="lemma">plus_of_succ</span></a> (<a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a> <a id="beta:28" class="idref" href="#beta:28"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:28"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:28"><span class="id" title="variable">beta</span></a>).  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_r_weak" class="idref" href="#plus_mono_r_weak"><span class="id" title="lemma">plus_mono_r_weak</span></a> (<a id="alpha:29" class="idref" href="#alpha:29"><span class="id" title="binder">alpha</span></a> <a id="beta:30" class="idref" href="#beta:30"><span class="id" title="binder">beta</span></a> <a id="gamma:31" class="idref" href="#gamma:31"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#beta:30"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:31"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:30"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:29"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:31"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_reg_r" class="idref" href="#plus_reg_r"><span class="id" title="lemma">plus_reg_r</span></a> (<a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a> <a id="beta:33" class="idref" href="#beta:33"><span class="id" title="binder">beta</span></a> <a id="gamma:34" class="idref" href="#gamma:34"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:33"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:34"><span class="id" title="variable">gamma</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#beta:33"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:34"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_is_plus_1" class="idref" href="#succ_is_plus_1"><span class="id" title="lemma">succ_is_plus_1</span></a> <a id="alpha:35" class="idref" href="#alpha:35"><span class="id" title="binder">alpha</span></a> :  <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="alpha_plus_sup" class="idref" href="#alpha_plus_sup"><span class="id" title="lemma">alpha_plus_sup</span></a> (<a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="A:37" class="idref" href="#A:37"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Addition.html#A:37"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Addition.html#A:37"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Addition.html#A:37"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Addition.html#A:37"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:36"><span class="id" title="variable">alpha</span></a>)<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_limit" class="idref" href="#plus_limit"><span class="id" title="lemma">plus_limit</span></a> (<a id="alpha:38" class="idref" href="#alpha:38"><span class="id" title="binder">alpha</span></a> <a id="beta:39" class="idref" href="#beta:39"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:39"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:38"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:39"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:39"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:38"><span class="id" title="variable">alpha</span></a>)<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_FF" class="idref" href="#plus_FF"><span class="id" title="lemma">plus_FF</span></a> : <span class="id" title="keyword">∀</span> <a id="i:40" class="idref" href="#i:40"><span class="id" title="binder">i</span></a> <a id="j:41" class="idref" href="#j:41"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#i:40"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#j:41"><span class="id" title="variable">j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Addition.html#i:40"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Addition.html#j:41"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="one_plus_omega" class="idref" href="#one_plus_omega"><span class="id" title="lemma">one_plus_omega</span></a> :  1 <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="minus_exists" class="idref" href="#minus_exists"><span class="id" title="lemma">minus_exists</span></a> (<a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a> <a id="beta:43" class="idref" href="#beta:43"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:43"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="gamma:44" class="idref" href="#gamma:44"><span class="id" title="binder">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:44"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:43"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="proof_of_associativity" class="idref" href="#proof_of_associativity"><span class="id" title="section">proof_of_associativity</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="proof_of_associativity.alpha" class="idref" href="#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a id="proof_of_associativity.beta" class="idref" href="#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="plus_assoc1" class="idref" href="#plus_assoc1"><span class="id" title="lemma">plus_assoc1</span></a> (<a id="gamma:47" class="idref" href="#gamma:47"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:47"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a> .<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="plus_assoc2" class="idref" href="#plus_assoc2"><span class="id" title="lemma">plus_assoc2</span></a> (<a id="gamma:48" class="idref" href="#gamma:48"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:48"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="khi:49" class="idref" href="#khi:49"><span class="id" title="binder">khi</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#gamma:48"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#khi:49"><span class="id" title="variable">khi</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="proof_of_associativity.f_alpha_beta" class="idref" href="#proof_of_associativity.f_alpha_beta"><span class="id" title="variable">f_alpha_beta</span></a> := <a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="proof_of_associativity.g_alpha_beta" class="idref" href="#proof_of_associativity.g_alpha_beta"><span class="id" title="variable">g_alpha_beta</span></a> <a id="gamma:50" class="idref" href="#gamma:50"><span class="id" title="binder">gamma</span></a> := <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:50"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="of_g" class="idref" href="#of_g"><span class="id" title="lemma">of_g</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.g_alpha_beta"><span class="id" title="variable">g_alpha_beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#ge"><span class="id" title="definition">ge</span></a> (<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="of_u" class="idref" href="#of_u"><span class="id" title="lemma">of_u</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#fun_equiv"><span class="id" title="definition">fun_equiv</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.f_alpha_beta"><span class="id" title="variable">f_alpha_beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.g_alpha_beta"><span class="id" title="variable">g_alpha_beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="plus_assoc3" class="idref" href="#plus_assoc3"><span class="id" title="lemma">plus_assoc3</span></a> (<a id="gamma:51" class="idref" href="#gamma:51"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.f_alpha_beta"><span class="id" title="variable">f_alpha_beta</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#gamma:51"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.g_alpha_beta"><span class="id" title="variable">g_alpha_beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:51"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="plus_assoc'" class="idref" href="#plus_assoc'"><span class="id" title="lemma">plus_assoc'</span></a> (<a id="gamma:52" class="idref" href="#gamma:52"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:52"><span class="id" title="variable">gamma</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity.beta"><span class="id" title="variable">beta</span></a><a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:52"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Addition.html#proof_of_associativity"><span class="id" title="section">proof_of_associativity</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="plus_assoc" class="idref" href="#plus_assoc"><span class="id" title="instance">plus_assoc</span></a>: <a class="idref" href="hydras.Prelude.STDPP_compat.html#Assoc"><span class="id" title="class">Assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="one_plus_infinite" class="idref" href="#one_plus_infinite"><span class="id" title="lemma">one_plus_infinite</span></a> (<a id="alpha:53" class="idref" href="#alpha:53"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:53"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  1 <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:53"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:53"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_plus_infinite" class="idref" href="#finite_plus_infinite"><span class="id" title="lemma">finite_plus_infinite</span></a> (<a id="n:54" class="idref" href="#n:54"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="alpha:55" class="idref" href="#alpha:55"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:54"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:55"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:55"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+<span class="id" title="keyword">Example</span> <a id="L_3_plus_omega" class="idref" href="#L_3_plus_omega"><span class="id" title="definition">L_3_plus_omega</span></a> : 3 <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_weak_l" class="idref" href="#plus_mono_weak_l"><span class="id" title="lemma">plus_mono_weak_l</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:56" class="idref" href="#alpha:56"><span class="id" title="binder">alpha</span></a> <a id="beta:57" class="idref" href="#beta:57"><span class="id" title="binder">beta</span></a> <a id="gamma:58" class="idref" href="#gamma:58"><span class="id" title="binder">gamma</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:57"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:58"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:57"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:58"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="plus_mono_bi" class="idref" href="#plus_mono_bi"><span class="id" title="lemma">plus_mono_bi</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:59" class="idref" href="#alpha:59"><span class="id" title="binder">alpha</span></a> <a id="beta:60" class="idref" href="#beta:60"><span class="id" title="binder">beta</span></a> <a id="gamma:61" class="idref" href="#gamma:61"><span class="id" title="binder">gamma</span></a> <a id="delta:62" class="idref" href="#delta:62"><span class="id" title="binder">delta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:61"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#beta:60"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#delta:62"><span class="id" title="variable">delta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:59"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:60"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#gamma:61"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#delta:62"><span class="id" title="variable">delta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_fin_r_one" class="idref" href="#mult_fin_r_one"><span class="id" title="lemma">mult_fin_r_one</span></a> : <span class="id" title="keyword">∀</span> <a id="n:63" class="idref" href="#n:63"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> 1<a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:63"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:63"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_fin_r_mono" class="idref" href="#mult_fin_r_mono"><span class="id" title="lemma">mult_fin_r_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:64" class="idref" href="#alpha:64"><span class="id" title="binder">alpha</span></a> <a id="beta:65" class="idref" href="#beta:65"><span class="id" title="binder">beta</span></a> , <a class="idref" href="hydras.Schutte.Addition.html#alpha:64"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#beta:65"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:66" class="idref" href="#n:66"><span class="id" title="binder">n</span></a>,  <a class="idref" href="hydras.Schutte.Addition.html#alpha:64"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:66"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#beta:65"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:66"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_a_mult_Sn_a" class="idref" href="#le_a_mult_Sn_a"><span class="id" title="lemma">le_a_mult_Sn_a</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:67" class="idref" href="#alpha:67"><span class="id" title="binder">alpha</span></a> <a id="n:68" class="idref" href="#n:68"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:68"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_Sn_mono2" class="idref" href="#mult_Sn_mono2"><span class="id" title="lemma">mult_Sn_mono2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a> <a id="p:71" class="idref" href="#p:71"><span class="id" title="binder">p</span></a>, (<a class="idref" href="hydras.Schutte.Addition.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#p:71"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#p:71"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="mult_Sn_mono3" class="idref" href="#mult_Sn_mono3"><span class="id" title="lemma">mult_Sn_mono3</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:72" class="idref" href="#alpha:72"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:72"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:73" class="idref" href="#n:73"><span class="id" title="binder">n</span></a> <a id="p:74" class="idref" href="#p:74"><span class="id" title="binder">p</span></a>, (<a class="idref" href="hydras.Schutte.Addition.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Addition.html#p:74"><span class="id" title="variable">p</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:72"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#n:73"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Addition.html#alpha:72"><span class="id" title="variable">alpha</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#alpha:72"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Addition.html#p:74"><span class="id" title="variable">p</span></a>.<br/>
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.CNF.html b/theories/html/hydras.Schutte.CNF.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Schutte.CNF.html
@@ -0,0 +1,258 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+<h1 class="libtitle">Library hydras.Schutte.CNF</h1>
+
+<div class="code">
+
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+</div>
+
+<div class="doc">
+ Cantor normal form 
+<div class="paragraph"> </div>
+
+ Pierre Casteran, Labri and Univ Bordeaux. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorting.Sorted</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="library">Logic.Epsilon</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#"><span class="id" title="library">Ordering_Functions</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="library">PartialFun</span></a>  <a class="idref" href="hydras.Schutte.Countable.html#"><span class="id" title="library">Countable</span></a> <a class="idref" href="hydras.Schutte.Addition.html#"><span class="id" title="library">Schutte.Addition</span></a> <a class="idref" href="hydras.Schutte.AP.html#"><span class="id" title="library">AP</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="library">Classical</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ListNotations"><span class="id" title="module">ListNotations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A Cantor normal form for a countable ordinal <span class="inlinecode"><span class="id" title="var">alpha</span></span> is just a sorted
+list <span class="inlinecode"><span class="id" title="var">l</span></span> (in decreasing order) such that  <span class="inlinecode"><span class="id" title="var">alpha</span></span>  is equal to the sum of the terms <span class="inlinecode"><span class="id" title="var">phi0</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span>, for every term <span class="inlinecode"><span class="id" title="keyword">beta</span></span> in <span class="inlinecode"><span class="id" title="var">l</span></span>.
+
+<div class="paragraph"> </div>
+
+Note that, if <span class="inlinecode"><span class="id" title="var">alpha</span></span> is greater or equal than <span class="inlinecode"><span class="id" title="var">epsilon0</span></span>, the members of <span class="inlinecode"><span class="id" title="var">l</span></span>
+are less <i>or equal</i> than <span class="inlinecode"><span class="id" title="var">alpha</span></span>.
+
+<div class="paragraph"> </div>
+
+For instance, the Cantor Normal Form of <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> is just <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> <span class="inlinecode">::</span> <span class="inlinecode"><span class="id" title="var">nil</span></span>.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="cnf_t" class="idref" href="#cnf_t"><span class="id" title="definition">cnf_t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="eval" class="idref" href="#eval"><span class="id" title="definition">eval</span></a> (<a id="l:1" class="idref" href="#l:1"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.CNF.html#l:1"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="keyword">beta</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval:2"><span class="id" title="definition">eval</span></a> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sorted" class="idref" href="#sorted"><span class="id" title="definition">sorted</span></a> (<a id="l:4" class="idref" href="#l:4"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> (<span class="id" title="keyword">fun</span> <a id="alpha:5" class="idref" href="#alpha:5"><span class="id" title="binder">alpha</span></a> <a id="beta:6" class="idref" href="#beta:6"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Schutte.CNF.html#beta:6"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:5"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.CNF.html#l:4"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="is_cnf_of" class="idref" href="#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> (<a id="alpha:7" class="idref" href="#alpha:7"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="l:8" class="idref" href="#l:8"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:8"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:7"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:8"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="exponents_lt" class="idref" href="#exponents_lt"><span class="id" title="definition">exponents_lt</span></a> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:10" class="idref" href="#beta:10"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.Schutte.CNF.html#beta:10"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:9"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="exponents_le" class="idref" href="#exponents_le"><span class="id" title="definition">exponents_le</span></a> (<a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:12" class="idref" href="#beta:12"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Schutte.CNF.html#beta:12"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:11"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exponents_lt_eval" class="idref" href="#exponents_lt_eval"><span class="id" title="lemma">exponents_lt_eval</span></a> <a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="l:14" class="idref" href="#l:14"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#exponents_lt"><span class="id" title="definition">exponents_lt</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:14"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:14"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:13"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_tail" class="idref" href="#sorted_tail"><span class="id" title="lemma">sorted_tail</span></a>  <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> <a id="l:16" class="idref" href="#l:16"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:16"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:16"><span class="id" title="variable">l</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nf_bounded" class="idref" href="#nf_bounded"><span class="id" title="lemma">nf_bounded</span></a> <a id="beta:17" class="idref" href="#beta:17"><span class="id" title="binder">beta</span></a>  <a id="l:18" class="idref" href="#l:18"><span class="id" title="binder">l</span></a> <a id="alpha:19" class="idref" href="#alpha:19"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:17"><span class="id" title="variable">beta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:19"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:18"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#exponents_le"><span class="id" title="definition">exponents_le</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:18"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_of_ap" class="idref" href="#cnf_of_ap"><span class="id" title="lemma">cnf_of_ap</span></a> (<a id="alpha:20" class="idref" href="#alpha:20"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l:21" class="idref" href="#l:21"><span class="id" title="binder">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:20"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:21"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_lt_lt" class="idref" href="#sorted_lt_lt"><span class="id" title="lemma">sorted_lt_lt</span></a> (<a id="beta:22" class="idref" href="#beta:22"><span class="id" title="binder">beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <span class="id" title="keyword">∀</span> <a id="l:23" class="idref" href="#l:23"><span class="id" title="binder">l</span></a> <a id="alpha:24" class="idref" href="#alpha:24"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:23"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.CNF.html#beta:22"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:24"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:23"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:22"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_lt_lt_2" class="idref" href="#sorted_lt_lt_2"><span class="id" title="lemma">sorted_lt_lt_2</span></a> <a id="l:25" class="idref" href="#l:25"><span class="id" title="binder">l</span></a> <a id="alpha:26" class="idref" href="#alpha:26"><span class="id" title="binder">alpha</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:25"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:25"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:26"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_head_eq" class="idref" href="#cnf_head_eq"><span class="id" title="lemma">cnf_head_eq</span></a> <a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a> <a id="beta:28" class="idref" href="#beta:28"><span class="id" title="binder">beta</span></a> <a id="ol:29" class="idref" href="#ol:29"><span class="id" title="binder">ol</span></a> <a id="ol':30" class="idref" href="#ol':30"><span class="id" title="binder">ol'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:29"><span class="id" title="variable">ol</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:28"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':30"><span class="id" title="variable">ol'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:29"><span class="id" title="variable">ol</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:28"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':30"><span class="id" title="variable">ol'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:28"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_eq" class="idref" href="#cnf_eq"><span class="id" title="lemma">cnf_eq</span></a>  <a id="alpha:31" class="idref" href="#alpha:31"><span class="id" title="binder">alpha</span></a> <a id="beta:32" class="idref" href="#beta:32"><span class="id" title="binder">beta</span></a> <a id="ol:33" class="idref" href="#ol:33"><span class="id" title="binder">ol</span></a> <a id="ol':34" class="idref" href="#ol':34"><span class="id" title="binder">ol'</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:33"><span class="id" title="variable">ol</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:32"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':34"><span class="id" title="variable">ol'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:33"><span class="id" title="variable">ol</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:32"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':34"><span class="id" title="variable">ol'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#alpha:31"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:32"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:33"><span class="id" title="variable">ol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':34"><span class="id" title="variable">ol'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_plus1" class="idref" href="#cnf_plus1"><span class="id" title="lemma">cnf_plus1</span></a> (<a id="ol:35" class="idref" href="#ol:35"><span class="id" title="binder">ol</span></a> : <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:35"><span class="id" title="variable">ol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <span class="id" title="keyword">∀</span> <a id="alpha:36" class="idref" href="#alpha:36"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="ol':37" class="idref" href="#ol':37"><span class="id" title="binder">ol'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> (<a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:36"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:35"><span class="id" title="variable">ol</span></a>) <a class="idref" href="hydras.Schutte.CNF.html#ol':37"><span class="id" title="variable">ol'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_plus2" class="idref" href="#cnf_plus2"><span class="id" title="lemma">cnf_plus2</span></a> : <span class="id" title="keyword">∀</span> <a id="ol:38" class="idref" href="#ol:38"><span class="id" title="binder">ol</span></a>, <a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:38"><span class="id" title="variable">ol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="ol':39" class="idref" href="#ol':39"><span class="id" title="binder">ol'</span></a>, <a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':39"><span class="id" title="variable">ol'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="ol'':40" class="idref" href="#ol'':40"><span class="id" title="binder">ol''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> (<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:38"><span class="id" title="variable">ol</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':39"><span class="id" title="variable">ol'</span></a>) <a class="idref" href="hydras.Schutte.CNF.html#ol'':40"><span class="id" title="variable">ol''</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_plus" class="idref" href="#cnf_plus"><span class="id" title="lemma">cnf_plus</span></a> : <span class="id" title="keyword">∀</span> <a id="ol:41" class="idref" href="#ol:41"><span class="id" title="binder">ol</span></a> <a id="alpha:42" class="idref" href="#alpha:42"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol:41"><span class="id" title="variable">ol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="ol':43" class="idref" href="#ol':43"><span class="id" title="binder">ol'</span></a> <a id="beta:44" class="idref" href="#beta:44"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.CNF.html#ol':43"><span class="id" title="variable">ol'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="ol'':45" class="idref" href="#ol'':45"><span class="id" title="binder">ol''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> (<a class="idref" href="hydras.Schutte.CNF.html#alpha:42"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.CNF.html#beta:44"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.CNF.html#ol'':45"><span class="id" title="variable">ol''</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab174"></a><h3 class="section">Every countable ordinal has (at least) a Cantor normal form</h3>
+
+
+<div class="paragraph"> </div>
+
+  (Proof by transfinite induction)
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="cnf_exists" class="idref" href="#cnf_exists"><span class="id" title="lemma">cnf_exists</span></a> (<a id="alpha:51" class="idref" href="#alpha:51"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l:52" class="idref" href="#l:52"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:52"><span class="id" title="variable">l</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sorted_lt" class="idref" href="#sorted_lt"><span class="id" title="lemma">sorted_lt</span></a>  <a id="l:53" class="idref" href="#l:53"><span class="id" title="binder">l</span></a> <a id="alpha:54" class="idref" href="#alpha:54"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="hydras.Schutte.CNF.html#alpha:54"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.Schutte.CNF.html#l:53"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:53"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:54"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:53"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab175"></a><h3 class="section">Unicity of cnf</h3>
+
+
+<div class="paragraph"> </div>
+
+(Proof by induction on lists)
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_unicity" class="idref" href="#cnf_unicity"><span class="id" title="lemma">cnf_unicity</span></a> <a id="l:55" class="idref" href="#l:55"><span class="id" title="binder">l</span></a> <a id="alpha:56" class="idref" href="#alpha:56"><span class="id" title="binder">alpha</span></a>:<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:55"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l':57" class="idref" href="#l':57"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:56"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l':57"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:55"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l':57"><span class="id" title="variable">l'</span></a>. 
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab176"></a><h3 class="section">The main result</h3>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="cnf_exists_unique" class="idref" href="#cnf_exists_unique"><span class="id" title="lemma">cnf_exists_unique</span></a> (<a id="alpha:58" class="idref" href="#alpha:58"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="l:59" class="idref" href="#l:59"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.Schutte.CNF.html#cnf_t"><span class="id" title="definition">cnf_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:58"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:59"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab177"></a><h2 class="section">Cantor Normal Form and the ordinal epsilon0</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_lt_epsilon0" class="idref" href="#cnf_lt_epsilon0"><span class="id" title="lemma">cnf_lt_epsilon0</span></a> : <span class="id" title="keyword">∀</span> <a id="l:60" class="idref" href="#l:60"><span class="id" title="binder">l</span></a> <a id="alpha:61" class="idref" href="#alpha:61"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.CNF.html#l:60"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:62" class="idref" href="#beta:62"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.Schutte.CNF.html#beta:62"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.CNF.html#alpha:61"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.CNF.html#l:60"><span class="id" title="variable">l</span></a>. 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The normal form of <span class="inlinecode"><span class="id" title="var">epsilon0</span></span> is just <span class="inlinecode"><span class="id" title="var">phi0</span></span> <span class="inlinecode"><span class="id" title="var">epsilon0</span></span>
+ 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="cnf_of_epsilon0" class="idref" href="#cnf_of_epsilon0"><span class="id" title="lemma">cnf_of_epsilon0</span></a> : <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">[</span></a><a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#ddd65c2f7ee73ecec433744948d846bb"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.Correctness_E0.html b/theories/html/hydras.Schutte.Correctness_E0.html
new file mode 100644
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+++ b/theories/html/hydras.Schutte.Correctness_E0.html
@@ -0,0 +1,175 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Correctness_E0</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Correctness_E0</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Injection from the set of  ordinal terms in Cantor normal form
+  into the set of Schutte's countable ordinal numbers stricly less than
+  epsilon0.
+
+<div class="paragraph"> </div>
+
+  Pierre Castéran, Univ. Bordeaux and LaBRI
+
+<div class="paragraph"> </div>
+
+   This is intented to be a validation of main constructions and functions
+   designed for the type <span class="inlinecode"><span class="id" title="var">T1</span></span>.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0.Epsilon0</span></a> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#"><span class="id" title="library">Schutte.Addition</span></a>  <a class="idref" href="hydras.Schutte.AP.html#"><span class="id" title="library">AP</span></a> <a class="idref" href="hydras.Schutte.CNF.html#"><span class="id" title="library">CNF</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="module">List</span></a>  <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="module">PartialFun</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="module">Ensembles</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="inject" class="idref" href="#inject"><span class="id" title="definition">inject</span></a> (<a id="t:1" class="idref" href="#t:1"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:1"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">n</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="hydras.Schutte.AP.html#_phi0"><span class="id" title="definition">AP._phi0</span></a> (<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject:2"><span class="id" title="definition">inject</span></a> <span class="id" title="var">a</span>) <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject:2"><span class="id" title="definition">inject</span></a> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inject_of_finite_pos" class="idref" href="#inject_of_finite_pos"><span class="id" title="lemma">inject_of_finite_pos</span></a> : <span class="id" title="keyword">∀</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:4"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:4"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_of_zero" class="idref" href="#inject_of_zero"><span class="id" title="lemma">inject_of_zero</span></a> : <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">T1.zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_of_finite" class="idref" href="#inject_of_finite"><span class="id" title="lemma">inject_of_finite</span></a> (<a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:5"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:5"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_of_omega" class="idref" href="#inject_of_omega"><span class="id" title="lemma">inject_of_omega</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1omega"><span class="id" title="abbreviation">T1omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#_omega"><span class="id" title="definition">Schutte_basics._omega</span></a>. 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_of_phi0" class="idref" href="#inject_of_phi0"><span class="id" title="lemma">inject_of_phi0</span></a> (<a id="alpha:6" class="idref" href="#alpha:6"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">T1.phi0</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:6"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#_phi0"><span class="id" title="definition">AP._phi0</span></a> (<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:6"><span class="id" title="variable">alpha</span></a>). 
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_mono" class="idref" href="#inject_mono"><span class="id" title="lemma">inject_mono</span></a> (<a id="beta:35" class="idref" href="#beta:35"><span class="id" title="binder">beta</span></a> <a id="gamma:36" class="idref" href="#gamma:36"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">T1.lt</span></a>  <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:35"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:36"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:35"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:36"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:35"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:36"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_injective" class="idref" href="#inject_injective"><span class="id" title="lemma">inject_injective</span></a> (<a id="beta:37" class="idref" href="#beta:37"><span class="id" title="binder">beta</span></a> <a id="gamma:38" class="idref" href="#gamma:38"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:38"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:38"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:38"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_monoR" class="idref" href="#inject_monoR"><span class="id" title="lemma">inject_monoR</span></a> (<a id="beta:39" class="idref" href="#beta:39"><span class="id" title="binder">beta</span></a> <a id="gamma:40" class="idref" href="#gamma:40"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:39"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">T1.nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:40"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:39"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:40"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:39"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#gamma:40"><span class="id" title="variable">gamma</span></a>)%<span class="id" title="var">t1</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_lt_epsilon0" class="idref" href="#inject_lt_epsilon0"><span class="id" title="lemma">inject_lt_epsilon0</span></a> (<a id="alpha:41" class="idref" href="#alpha:41"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:41"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inject_rw" class="idref" href="#inject_rw"><span class="id" title="lemma">inject_rw</span></a> (<a id="a:60" class="idref" href="#a:60"><span class="id" title="binder">a</span></a> <a id="b:61" class="idref" href="#b:61"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>) <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">T1.cons</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#b:61"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#mult_Sn"><span class="id" title="definition">mult_Sn</span></a> (<a class="idref" href="hydras.Schutte.AP.html#_phi0"><span class="id" title="definition">AP._phi0</span></a> (<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#a:60"><span class="id" title="variable">a</span></a>)) <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#b:61"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_plus" class="idref" href="#inject_plus"><span class="id" title="lemma">inject_plus</span></a> (<a id="alpha:63" class="idref" href="#alpha:63"><span class="id" title="binder">alpha</span></a> <a id="beta:64" class="idref" href="#beta:64"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:64"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:64"><span class="id" title="variable">beta</span></a>)%<span class="id" title="var">t1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:63"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#beta:64"><span class="id" title="variable">beta</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_mult_fin_r" class="idref" href="#inject_mult_fin_r"><span class="id" title="lemma">inject_mult_fin_r</span></a> (<a id="alpha:65" class="idref" href="#alpha:65"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:66" class="idref" href="#n:66"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866"><span class="id" title="notation">×</span></a>  <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:66"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">t1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:65"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#n:66"><span class="id" title="variable">n</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inject_lt_epsilon0_ex_cnf" class="idref" href="#inject_lt_epsilon0_ex_cnf"><span class="id" title="lemma">inject_lt_epsilon0_ex_cnf</span></a>  (<a id="alpha:67" class="idref" href="#alpha:67"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="H:68" class="idref" href="#H:68"><span class="id" title="binder">H</span></a>  : <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="l:69" class="idref" href="#l:69"><span class="id" title="binder">l</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>),  <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:67"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#l:69"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="t:70" class="idref" href="#t:70"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:70"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:70"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.CNF.html#eval"><span class="id" title="definition">eval</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#l:69"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_lt_epsilon0_ex" class="idref" href="#inject_lt_epsilon0_ex"><span class="id" title="lemma">inject_lt_epsilon0_ex</span></a>  (<a id="alpha:71" class="idref" href="#alpha:71"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="H:72" class="idref" href="#H:72"><span class="id" title="binder">H</span></a>  : <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:71"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="t:73" class="idref" href="#t:73"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:73"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:73"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:71"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="inject_lt_epsilon0_ex_unique" class="idref" href="#inject_lt_epsilon0_ex_unique"><span class="id" title="lemma">inject_lt_epsilon0_ex_unique</span></a>  (<a id="alpha:74" class="idref" href="#alpha:74"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="H:75" class="idref" href="#H:75"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:74"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="t:76" class="idref" href="#t:76"><span class="id" title="binder">t</span></a>: <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:74"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="embedding" class="idref" href="#embedding"><span class="id" title="lemma">embedding</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> (<a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a>. 
+<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Epsilon0_correct" class="idref" href="#Epsilon0_correct"><span class="id" title="instance">Epsilon0_correct</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_correct"><span class="id" title="class">ON_correct</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#Epsilon0"><span class="id" title="instance">Epsilon0</span></a>  (<span class="id" title="keyword">fun</span> <a id="alpha:77" class="idref" href="#alpha:77"><span class="id" title="binder">alpha</span></a> ⇒ <a class="idref" href="hydras.Schutte.Correctness_E0.html#inject"><span class="id" title="definition">inject</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#cnf"><span class="id" title="method">cnf</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#alpha:77"><span class="id" title="variable">alpha</span></a>)). 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Correctness of E0.plus 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span>  <a id="E0_plus_correct" class="idref" href="#E0_plus_correct"><span class="id" title="lemma">E0_plus_correct</span></a> :  <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON_op_ok"><span class="id" title="definition">ON_op_ok</span></a>  <a class="idref" href="hydras.Epsilon0.E0.html#E0add"><span class="id" title="instance">E0add</span></a> <a class="idref" href="hydras.Schutte.Addition.html#plus"><span class="id" title="definition">plus</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Countable</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Countable</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+A formalization of denumerable sets.  by Florian Hatat and Stéphane Desarzens  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Finite_sets.html#"><span class="id" title="library">Finite_sets</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="library">Logic.Epsilon</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#"><span class="id" title="library">Sets.Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Classical_Wf</span> <span class="id" title="library">CountableTypes</span> <span class="id" title="library">Families</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="library">MoreEpsilonIota</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="library">PartialFun</span></a>  <a class="idref" href="hydras.Schutte.GRelations.html#"><span class="id" title="library">GRelations</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.More_Arith.html#"><span class="id" title="library">Prelude.More_Arith</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A is countable if there exists an injection from <span class="inlinecode"><span class="id" title="var">A</span></span> to
+  <span class="inlinecode"><span class="id" title="var">Full_set</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Countable" class="idref" href="#Countable"><span class="id" title="section">Countable</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Countable.Definitions" class="idref" href="#Countable.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Definitions.U" class="idref" href="#Countable.Definitions.U"><span class="id" title="variable">U</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Definitions.A" class="idref" href="#Countable.Definitions.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.U"><span class="id" title="variable">U</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Predicate for relations which number the elements of A.
+
+<div class="paragraph"> </div>
+
+  These relations map each element of A to at least one integer, but they
+  are not required to be functional (injectivity is only needed to ensure that
+  A is countable). 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_numbers" class="idref" href="#rel_numbers"><span class="id" title="definition">rel_numbers</span></a> (<a id="R:3" class="idref" href="#R:3"><span class="id" title="binder">R</span></a>: <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.Schutte.GRelations.html#rel_injection"><span class="id" title="inductive">rel_injection</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Full_set"><span class="id" title="inductive">Full_set</span></a> <a class="idref" href="hydras.Schutte.Countable.html#R:3"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Predicate for relations which enumerate A. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_enumerates" class="idref" href="#rel_enumerates"><span class="id" title="definition">rel_enumerates</span></a> (<a id="R:4" class="idref" href="#R:4"><span class="id" title="binder">R</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.U"><span class="id" title="variable">U</span></a>) := <a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Full_set"><span class="id" title="inductive">Full_set</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Countable.html#R:4"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="countable_surj" class="idref" href="#countable_surj"><span class="id" title="lemma">countable_surj</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="R:5" class="idref" href="#R:5"><span class="id" title="binder">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Countable.html#rel_enumerates"><span class="id" title="definition">rel_enumerates</span></a> <a class="idref" href="hydras.Schutte.Countable.html#R:5"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.U" class="idref" href="#Countable.U"><span class="id" title="variable">U</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Countable.Countable_seq_range" class="idref" href="#Countable.Countable_seq_range"><span class="id" title="section">Countable_seq_range</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="seq_range" class="idref" href="#seq_range"><span class="id" title="definition">seq_range</span></a> (<a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.U"><span class="id" title="variable">U</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.U"><span class="id" title="variable">U</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Full_set"><span class="id" title="inductive">Full_set</span></a> <a class="idref" href="hydras.Schutte.Countable.html#f:27"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="seq_range_countable" class="idref" href="#seq_range_countable"><span class="id" title="lemma">seq_range_countable</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:28" class="idref" href="#f:28"><span class="id" title="binder">f</span></a>, <span class="id" title="definition">Countable</span> (<a class="idref" href="hydras.Schutte.Countable.html#seq_range"><span class="id" title="definition">seq_range</span></a> <a class="idref" href="hydras.Schutte.Countable.html#f:28"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_seq_range"><span class="id" title="section">Countable_seq_range</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Countable.Countable_bijection" class="idref" href="#Countable.Countable_bijection"><span class="id" title="section">Countable_bijection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Countable_bijection.V" class="idref" href="#Countable.Countable_bijection.V"><span class="id" title="variable">V</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Countable_bijection.A" class="idref" href="#Countable.Countable_bijection.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.U"><span class="id" title="variable">U</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Countable_bijection.B" class="idref" href="#Countable.Countable_bijection.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.V"><span class="id" title="variable">V</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Countable.Countable_bijection.g" class="idref" href="#Countable.Countable_bijection.g"><span class="id" title="variable">g</span></a> : <a class="idref" href="hydras.Schutte.Countable.html#Countable.U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.V"><span class="id" title="variable">V</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Countable.Countable_bijection.g_bij" class="idref" href="#Countable.Countable_bijection.g_bij"><span class="id" title="variable">g_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.g"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="countable_bij_fun" class="idref" href="#countable_bij_fun"><span class="id" title="lemma">countable_bij_fun</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="countable_bij_funR" class="idref" href="#countable_bij_funR"><span class="id" title="lemma">countable_bij_funR</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable.Countable_bijection"><span class="id" title="section">Countable_bijection</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Countable.html#Countable"><span class="id" title="section">Countable</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="countable_image" class="idref" href="#countable_image"><span class="id" title="lemma">countable_image</span></a> : <span class="id" title="keyword">∀</span> (<a id="U:34" class="idref" href="#U:34"><span class="id" title="binder">U</span></a> <a id="V:35" class="idref" href="#V:35"><span class="id" title="binder">V</span></a>:<span class="id" title="keyword">Type</span>)(<a id="DA:36" class="idref" href="#DA:36"><span class="id" title="binder">DA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Countable.html#U:34"><span class="id" title="variable">U</span></a>)(<a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Schutte.Countable.html#U:34"><span class="id" title="variable">U</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Countable.html#V:35"><span class="id" title="variable">V</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Countable.html#DA:36"><span class="id" title="variable">DA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Countable.html#DA:36"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.Countable.html#f:37"><span class="id" title="variable">f</span></a>).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Critical</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Critical</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+ We  adapt Schutte's definition of critical ordinals :
+
+<div class="paragraph"> </div>
+
+<ul class="doclist">
+<li> Cr(zero) = AP   (the set of additive principal ordinals )
+
+<div class="paragraph"> </div>
+
+
+</li>
+<li> if zero &lt; alpha, then
+       Cr(alpha) is the   intersection of all the sets of fixpoints
+       of the ordering functions of Cr(beta) for beta &lt; alpha.
+
+</li>
+</ul>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="library">Logic.Epsilon</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="library">Classical</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">CountableTypes</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#"><span class="id" title="library">Ordering_Functions</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Countable.html#"><span class="id" title="library">Countable</span></a>  <a class="idref" href="hydras.Schutte.Addition.html#"><span class="id" title="library">Schutte.Addition</span></a> <a class="idref" href="hydras.Schutte.AP.html#"><span class="id" title="library">AP</span></a> <a class="idref" href="hydras.Schutte.CNF.html#"><span class="id" title="library">CNF</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#"><span class="id" title="library">Well_Orders</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="library">MoreEpsilonIota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Let us define a functional, the fixpoint of which we shall consider 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Cr_fun" class="idref" href="#Cr_fun"><span class="id" title="definition">Cr_fun</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:2" class="idref" href="#beta:2"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#beta:2"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> (<a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Cr:5" class="idref" href="#Cr:5"><span class="id" title="binder">Cr</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:4" class="idref" href="#beta:4"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#beta:4"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) ⇒ (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Critical.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:6"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:3"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:7" class="idref" href="#beta:7"><span class="id" title="binder">beta</span></a> (<a id="H:8" class="idref" href="#H:8"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Schutte.Critical.html#beta:7"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:3"><span class="id" title="variable">alpha</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr:5"><span class="id" title="variable">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:7"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Critical.html#H:8"><span class="id" title="variable">H</span></a>)) <a class="idref" href="hydras.Schutte.Critical.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr:5"><span class="id" title="variable">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:7"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Critical.html#H:8"><span class="id" title="variable">H</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:6"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Cr" class="idref" href="#Cr"><span class="id" title="definition">Cr</span></a> (<a id="alpha:9" class="idref" href="#alpha:9"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Fix"><span class="id" title="definition">Fix</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#all_ord_acc"><span class="id" title="lemma">all_ord_acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<span class="id" title="var">_</span>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#Cr_fun"><span class="id" title="definition">Cr_fun</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#alpha:9"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="strongly_critical" class="idref" href="#strongly_critical"><span class="id" title="definition">strongly_critical</span></a> <a id="alpha:10" class="idref" href="#alpha:10"><span class="id" title="binder">alpha</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:10"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#alpha:10"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="maximal_critical" class="idref" href="#maximal_critical"><span class="id" title="definition">maximal_critical</span></a> <a id="alpha:11" class="idref" href="#alpha:11"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="gamma:12" class="idref" href="#gamma:12"><span class="id" title="binder">gamma</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:11"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#gamma:12"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="xi:13" class="idref" href="#xi:13"><span class="id" title="binder">xi</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#alpha:11"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:13"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:13"><span class="id" title="variable">xi</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#gamma:12"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Gamma0" class="idref" href="#Gamma0"><span class="id" title="definition">Gamma0</span></a> := <a class="idref" href="hydras.Schutte.Well_Orders.html#the_least"><span class="id" title="definition">the_least</span></a> <a class="idref" href="hydras.Schutte.Critical.html#strongly_critical"><span class="id" title="definition">strongly_critical</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ See Gamma0.Gamma0.phi 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="phi" class="idref" href="#phi"><span class="id" title="definition">phi</span></a> (<a id="alpha:14" class="idref" href="#alpha:14"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:14"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="A_" class="idref" href="#A_"><span class="id" title="definition">A_</span></a> (<a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:15"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_extensional" class="idref" href="#Cr_extensional"><span class="id" title="lemma">Cr_extensional</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="x:16" class="idref" href="#x:16"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:18" class="idref" href="#f:18"><span class="id" title="binder">f</span></a> <a id="g:19" class="idref" href="#g:19"><span class="id" title="binder">g</span></a> : <span class="id" title="keyword">∀</span> <a id="y:17" class="idref" href="#y:17"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#y:17"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:16"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#y:17"><span class="id" title="variable">y</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="y:20" class="idref" href="#y:20"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="p:21" class="idref" href="#p:21"><span class="id" title="binder">p</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#y:20"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:16"><span class="id" title="variable">x</span></a>), <a class="idref" href="hydras.Schutte.Critical.html#f:18"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Critical.html#y:20"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Critical.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#g:19"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Schutte.Critical.html#y:20"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Critical.html#p:21"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Critical.html#Cr_fun"><span class="id" title="definition">Cr_fun</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#f:18"><span class="id" title="variable">f</span></a>  :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Critical.html#Cr_fun"><span class="id" title="definition">Cr_fun</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#g:19"><span class="id" title="variable">g</span></a> :<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_equation" class="idref" href="#Cr_equation"><span class="id" title="lemma">Cr_equation</span></a> (<a id="alpha:22" class="idref" href="#alpha:22"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#alpha:22"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr_fun"><span class="id" title="definition">Cr_fun</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="y:23" class="idref" href="#y:23"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="h:24" class="idref" href="#h:24"><span class="id" title="binder">h</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#y:23"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:22"><span class="id" title="variable">alpha</span></a>) ⇒  <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#y:23"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_inv" class="idref" href="#Cr_inv"><span class="id" title="lemma">Cr_inv</span></a> (<a id="alpha:25" class="idref" href="#alpha:25"><span class="id" title="binder">alpha</span></a> <a id="x:26" class="idref" href="#x:26"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Critical.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:25"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:27" class="idref" href="#beta:27"><span class="id" title="binder">beta</span></a> (<a id="H:28" class="idref" href="#H:28"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Schutte.Critical.html#beta:27"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:25"><span class="id" title="variable">alpha</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:27"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#beta:27"><span class="id" title="variable">beta</span></a> ) <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:26"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_zero" class="idref" href="#Cr_zero"><span class="id" title="lemma">Cr_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="x:29" class="idref" href="#x:29"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:29"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#x:29"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_pos" class="idref" href="#Cr_pos"><span class="id" title="lemma">Cr_pos</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:30" class="idref" href="#alpha:30"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:30"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:31" class="idref" href="#x:31"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:32" class="idref" href="#beta:32"><span class="id" title="binder">beta</span></a> (<a id="H:33" class="idref" href="#H:33"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Schutte.Critical.html#beta:32"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:30"><span class="id" title="variable">alpha</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#beta:32"><span class="id" title="variable">beta</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>   <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#beta:32"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:31"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:30"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:31"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_zero_inv" class="idref" href="#Cr_zero_inv"><span class="id" title="lemma">Cr_zero_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="x:34" class="idref" href="#x:34"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:34"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:34"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_zero_AP" class="idref" href="#Cr_zero_AP"><span class="id" title="lemma">Cr_zero_AP</span></a> :  <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_pos_inv" class="idref" href="#Cr_pos_inv"><span class="id" title="lemma">Cr_pos_inv</span></a> (<a id="alpha:35" class="idref" href="#alpha:35"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:36" class="idref" href="#x:36"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:35"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:36"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:37" class="idref" href="#beta:37"><span class="id" title="binder">beta</span></a> (<a id="H:38" class="idref" href="#H:38"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Schutte.Critical.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:35"><span class="id" title="variable">alpha</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:37"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#x:36"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:37"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:36"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:36"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_pos_iff" class="idref" href="#Cr_pos_iff"><span class="id" title="lemma">Cr_pos_iff</span></a> (<a id="alpha:39" class="idref" href="#alpha:39"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:39"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:41" class="idref" href="#beta:41"><span class="id" title="binder">beta</span></a> (<a id="H:42" class="idref" href="#H:42"><span class="id" title="binder">H</span></a>:<a class="idref" href="hydras.Schutte.Critical.html#beta:41"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:39"><span class="id" title="variable">alpha</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:41"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:41"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:40"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="A_Cr" class="idref" href="#A_Cr"><span class="id" title="lemma">A_Cr</span></a> (<a id="alpha:43" class="idref" href="#alpha:43"><span class="id" title="binder">alpha</span></a> <a id="beta:44" class="idref" href="#beta:44"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:43"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:43"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:44"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:43"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#beta:44"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_lt" class="idref" href="#Cr_lt"><span class="id" title="lemma">Cr_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:45" class="idref" href="#alpha:45"><span class="id" title="binder">alpha</span></a> <a id="beta:46" class="idref" href="#beta:46"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#beta:46"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:45"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="x:47" class="idref" href="#x:47"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:45"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:47"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:46"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Critical.html#x:47"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_incl" class="idref" href="#Cr_incl"><span class="id" title="lemma">Cr_incl</span></a> (<a id="alpha:48" class="idref" href="#alpha:48"><span class="id" title="binder">alpha</span></a> <a id="beta:49" class="idref" href="#beta:49"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="H:50" class="idref" href="#H:50"><span class="id" title="binder">H</span></a> :<a class="idref" href="hydras.Schutte.Critical.html#beta:49"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:48"><span class="id" title="variable">alpha</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:48"><span class="id" title="variable">alpha</span></a>)  (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#beta:49"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="phi0_well_named" class="idref" href="#phi0_well_named"><span class="id" title="lemma">phi0_well_named</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:51" class="idref" href="#alpha:51"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> 0 <a class="idref" href="hydras.Schutte.Critical.html#alpha:51"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Cr_1_iff" class="idref" href="#Cr_1_iff"><span class="id" title="lemma">Cr_1_iff</span></a> (<a id="alpha:52" class="idref" href="#alpha:52"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> 1 <a class="idref" href="hydras.Schutte.Critical.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="hydras.Schutte.AP.html#AP"><span class="id" title="definition">AP</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:52"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:52"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="epsilon0_Cr1" class="idref" href="#epsilon0_Cr1"><span class="id" title="lemma">epsilon0_Cr1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> 1) <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>. 
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Lemma 5, p 82 of Schutte's book 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_Lemma5" class="idref" href="#Proof_of_Lemma5"><span class="id" title="section">Proof_of_Lemma5</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_Lemma5.P" class="idref" href="#Proof_of_Lemma5.P"><span class="id" title="variable">P</span></a> (<a id="alpha:53" class="idref" href="#alpha:53"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) := <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:53"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:53"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_0" class="idref" href="#Lemma5_0"><span class="id" title="lemma">Lemma5_0</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Lemma5.Alpha_positive" class="idref" href="#Proof_of_Lemma5.Alpha_positive"><span class="id" title="section">Alpha_positive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Lemma5.Alpha_positive.alpha" class="idref" href="#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Lemma5.Alpha_positive.alpha_pos" class="idref" href="#Proof_of_Lemma5.Alpha_positive.alpha_pos"><span class="id" title="variable">alpha_pos</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Lemma5.Alpha_positive.IHalpha" class="idref" href="#Proof_of_Lemma5.Alpha_positive.IHalpha"><span class="id" title="variable">IHalpha</span></a> : <span class="id" title="keyword">∀</span> <a id="xi:56" class="idref" href="#xi:56"><span class="id" title="binder">xi</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#xi:56"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:56"><span class="id" title="variable">xi</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Lemma5.Alpha_positive.Proof_unbounded" class="idref" href="#Proof_of_Lemma5.Alpha_positive.Proof_unbounded"><span class="id" title="section">Proof_unbounded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta" class="idref" href="#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="gamma_" class="idref" href="#gamma_"><span class="id" title="definition">gamma_</span></a> (<a id="n:59" class="idref" href="#n:59"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.Critical.html#n:59"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta"><span class="id" title="variable">beta</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="y:62" class="idref" href="#y:62"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="xi:63" class="idref" href="#xi:63"><span class="id" title="binder">xi</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:63"><span class="id" title="variable">xi</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#y:62"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:63"><span class="id" title="variable">xi</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#gamma_:60"><span class="id" title="definition">gamma_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#n:59"><span class="id" title="variable">n</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma" class="idref" href="#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma"><span class="id" title="variable">gamma</span></a> := <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Critical.html#gamma_"><span class="id" title="definition">gamma_</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_01" class="idref" href="#Lemma5_01"><span class="id" title="lemma">Lemma5_01</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_02" class="idref" href="#Lemma5_02"><span class="id" title="lemma">Lemma5_02</span></a> : <span class="id" title="keyword">∀</span> <a id="xi:64" class="idref" href="#xi:64"><span class="id" title="binder">xi</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#xi:64"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:64"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_03" class="idref" href="#Lemma5_03"><span class="id" title="lemma">Lemma5_03</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="A_full" class="idref" href="#A_full"><span class="id" title="lemma">A_full</span></a> : <span class="id" title="keyword">∀</span> <a id="xi:73" class="idref" href="#xi:73"><span class="id" title="binder">xi</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#xi:73"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:73"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_04" class="idref" href="#Lemma5_04"><span class="id" title="lemma">Lemma5_04</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="gamma:74" class="idref" href="#gamma:74"><span class="id" title="binder">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Critical.html#gamma:74"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#gamma:74"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded"><span class="id" title="section">Proof_unbounded</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_1" class="idref" href="#Lemma5_1"><span class="id" title="lemma">Lemma5_1</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_Lemma5.Alpha_positive.closedness" class="idref" href="#Proof_of_Lemma5.Alpha_positive.closedness"><span class="id" title="section">closedness</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_Lemma5.Alpha_positive.closedness.M" class="idref" href="#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Lemma5.Alpha_positive.closedness.HM" class="idref" href="#Proof_of_Lemma5.Alpha_positive.closedness.HM"><span class="id" title="variable">HM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Lemma5.Alpha_positive.closedness.CM" class="idref" href="#Proof_of_Lemma5.Alpha_positive.closedness.CM"><span class="id" title="variable">CM</span></a> : <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_Lemma5.Alpha_positive.closedness.IM" class="idref" href="#Proof_of_Lemma5.Alpha_positive.closedness.IM"><span class="id" title="variable">IM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a>  <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_2" class="idref" href="#Lemma5_2"><span class="id" title="lemma">Lemma5_2</span></a> : <span class="id" title="keyword">∀</span> <a id="xi:79" class="idref" href="#xi:79"><span class="id" title="binder">xi</span></a> <a id="eta:80" class="idref" href="#eta:80"><span class="id" title="binder">eta</span></a>,  <a class="idref" href="hydras.Schutte.Critical.html#xi:79"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a> <a class="idref" href="hydras.Schutte.Critical.html#eta:80"><span class="id" title="variable">eta</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#xi:79"><span class="id" title="variable">xi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#eta:80"><span class="id" title="variable">eta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Critical.html#eta:80"><span class="id" title="variable">eta</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Check</span> (<a class="idref" href="hydras.Schutte.Critical.html#Cr_pos_iff"><span class="id" title="lemma">Cr_pos_iff</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha_pos"><span class="id" title="variable">alpha_pos</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5_7" class="idref" href="#Lemma5_7"><span class="id" title="lemma">Lemma5_7</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M"><span class="id" title="variable">M</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness"><span class="id" title="section">closedness</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="induct_step" class="idref" href="#induct_step"><span class="id" title="lemma">induct_step</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive"><span class="id" title="section">Alpha_positive</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lemma5" class="idref" href="#Lemma5"><span class="id" title="lemma">Lemma5</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:81" class="idref" href="#alpha:81"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:81"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Critical.html#Proof_of_Lemma5"><span class="id" title="section">Proof_of_Lemma5</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Unbounded_Cr" class="idref" href="#Unbounded_Cr"><span class="id" title="lemma">Unbounded_Cr</span></a> <a id="alpha:82" class="idref" href="#alpha:82"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:82"><span class="id" title="variable">alpha</span></a>). 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Closed_Cr" class="idref" href="#Closed_Cr"><span class="id" title="lemma">Closed_Cr</span></a> <a id="alpha:83" class="idref" href="#alpha:83"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:83"><span class="id" title="variable">alpha</span></a>). 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Th13_8" class="idref" href="#Th13_8"><span class="id" title="lemma">Th13_8</span></a> <a id="alpha:84" class="idref" href="#alpha:84"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#normal"><span class="id" title="definition">normal</span></a> (<a class="idref" href="hydras.Schutte.Critical.html#phi"><span class="id" title="definition">phi</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:84"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="hydras.Schutte.Critical.html#Cr"><span class="id" title="definition">Cr</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:84"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Corollary</span> <a id="Th13_8_1" class="idref" href="#Th13_8_1"><span class="id" title="lemma">Th13_8_1</span></a> <a id="alpha:85" class="idref" href="#alpha:85"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Critical.html#A_"><span class="id" title="definition">A_</span></a> <a class="idref" href="hydras.Schutte.Critical.html#alpha:85"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.GRelations.html b/theories/html/hydras.Schutte.GRelations.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Schutte.GRelations.html
@@ -0,0 +1,155 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.GRelations</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.GRelations</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+General relations  
+<div class="paragraph"> </div>
+
+  by  Florian Hatat, ENS-Lyon 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="library">Classical</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="library">PartialFun</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="General_Relations" class="idref" href="#General_Relations"><span class="id" title="section">General_Relations</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="General_Relations.Definitions" class="idref" href="#General_Relations.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="General_Relations.Definitions.A" class="idref" href="#General_Relations.Definitions.A"><span class="id" title="variable">A</span></a> <a id="General_Relations.Definitions.B" class="idref" href="#General_Relations.Definitions.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+domain 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.Definitions.DA" class="idref" href="#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+codomain 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.Definitions.DB" class="idref" href="#General_Relations.Definitions.DB"><span class="id" title="variable">DB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="GRelation" class="idref" href="#GRelation"><span class="id" title="definition">GRelation</span></a> := <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.Definitions.R" class="idref" href="#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="rel_injection" class="idref" href="#rel_injection"><span class="id" title="inductive">rel_injection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="rel_inj_i" class="idref" href="#rel_inj_i"><span class="id" title="constructor">rel_inj_i</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#rel_domain"><span class="id" title="definition">rel_domain</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_codomain"><span class="id" title="definition">rel_codomain</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_inj"><span class="id" title="definition">rel_inj</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.GRelations.html#rel_injection:6"><span class="id" title="inductive">rel_injection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span>  <a id="rel_surjection" class="idref" href="#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="rel_surj_i" class="idref" href="#rel_surj_i"><span class="id" title="constructor">rel_surj_i</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#rel_codomain"><span class="id" title="definition">rel_codomain</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_functional"><span class="id" title="definition">rel_functional</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_onto"><span class="id" title="definition">rel_onto</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection:8"><span class="id" title="inductive">rel_surjection</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.GRelations.html#rel_injection"><span class="id" title="inductive">rel_injection</span></a> [<span class="id" title="var">A</span> <span class="id" title="var">B</span>].<br/>
+&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> [<span class="id" title="var">A</span> <span class="id" title="var">B</span>].<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="General_Relations.A" class="idref" href="#General_Relations.A"><span class="id" title="variable">A</span></a> <a id="General_Relations.B" class="idref" href="#General_Relations.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.DA" class="idref" href="#General_Relations.DA"><span class="id" title="variable">DA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.DB" class="idref" href="#General_Relations.DB"><span class="id" title="variable">DB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="General_Relations.injection2surjection" class="idref" href="#General_Relations.injection2surjection"><span class="id" title="section">injection2surjection</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.injection2surjection.R" class="idref" href="#General_Relations.injection2surjection.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.B"><span class="id" title="variable">B</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="General_Relations.injection2surjection.R_inj" class="idref" href="#General_Relations.injection2surjection.R_inj"><span class="id" title="variable">R_inj</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#rel_injection"><span class="id" title="inductive">rel_injection</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R_inv_surj" class="idref" href="#R_inv_surj"><span class="id" title="lemma">R_inv_surj</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#rel_inv"><span class="id" title="definition">rel_inv</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection"><span class="id" title="section">injection2surjection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="General_Relations.surjection2injection" class="idref" href="#General_Relations.surjection2injection"><span class="id" title="section">surjection2injection</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.surjection2injection.R" class="idref" href="#General_Relations.surjection2injection.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.B"><span class="id" title="variable">B</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="General_Relations.surjection2injection.R_surj" class="idref" href="#General_Relations.surjection2injection.R_surj"><span class="id" title="variable">R_surj</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="R_inv" class="idref" href="#R_inv"><span class="id" title="definition">R_inv</span></a> := <a class="idref" href="hydras.Schutte.PartialFun.html#rel_inv"><span class="id" title="definition">rel_inv</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R_inv_inj" class="idref" href="#R_inv_inj"><span class="id" title="lemma">R_inv_inj</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#rel_injection"><span class="id" title="inductive">rel_injection</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#rel_inv"><span class="id" title="definition">rel_inv</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R"><span class="id" title="variable">R</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection"><span class="id" title="section">surjection2injection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="General_Relations.elagage" class="idref" href="#General_Relations.elagage"><span class="id" title="section">elagage</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="General_Relations.elagage.to_nat_elagage" class="idref" href="#General_Relations.elagage.to_nat_elagage"><span class="id" title="section">to_nat_elagage</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="General_Relations.elagage.to_nat_elagage.R" class="idref" href="#General_Relations.elagage.to_nat_elagage.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#GRelation"><span class="id" title="definition">GRelation</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="R_nat_elaguee" class="idref" href="#R_nat_elaguee"><span class="id" title="definition">R_nat_elaguee</span></a> (<a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.A"><span class="id" title="variable">A</span></a>) (<a id="n:20" class="idref" href="#n:20"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="p:21" class="idref" href="#p:21"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#p:21"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#n:20"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#p:21"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R_nat_elaguee_fun" class="idref" href="#R_nat_elaguee_fun"><span class="id" title="lemma">R_nat_elaguee_fun</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_functional"><span class="id" title="definition">rel_functional</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#R_nat_elaguee"><span class="id" title="definition">R_nat_elaguee</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R_nat_elaguee_domain" class="idref" href="#R_nat_elaguee_domain"><span class="id" title="lemma">R_nat_elaguee_domain</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:22" class="idref" href="#y:22"><span class="id" title="binder">y</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#y:22"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#n:23"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p:24" class="idref" href="#p:24"><span class="id" title="binder">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#R_nat_elaguee"><span class="id" title="definition">R_nat_elaguee</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#y:22"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.GRelations.html#p:24"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage"><span class="id" title="section">to_nat_elagage</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations.elagage"><span class="id" title="section">elagage</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.GRelations.html#General_Relations"><span class="id" title="section">General_Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.GRelations.html#rel_injection"><span class="id" title="inductive">rel_injection</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>}.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.GRelations.html#rel_surjection"><span class="id" title="inductive">rel_surjection</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>}.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.Lub.html b/theories/html/hydras.Schutte.Lub.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Schutte.Lub.html
@@ -0,0 +1,63 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Lub</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Lub</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="upper_bound" class="idref" href="#upper_bound"><span class="id" title="definition">upper_bound</span></a> (<a id="M:1" class="idref" href="#M:1"><span class="id" title="binder">M</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="D:2" class="idref" href="#D:2"><span class="id" title="binder">D</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:1"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:3" class="idref" href="#lt:3"><span class="id" title="binder">lt</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:1"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="X:4" class="idref" href="#X:4"><span class="id" title="binder">X</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:1"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Lub.html#M:1"><span class="id" title="variable">M</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:6" class="idref" href="#x:6"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Schutte.Lub.html#D:2"><span class="id" title="variable">D</span></a> <a class="idref" href="hydras.Schutte.Lub.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Schutte.Lub.html#X:4"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Lub.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Lub.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Lub.html#lt:3"><span class="id" title="variable">lt</span></a>  <a class="idref" href="hydras.Schutte.Lub.html#x:6"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:5"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="is_lub" class="idref" href="#is_lub"><span class="id" title="definition">is_lub</span></a> (<a id="M:7" class="idref" href="#M:7"><span class="id" title="binder">M</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="D:8" class="idref" href="#D:8"><span class="id" title="binder">D</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:7"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:9" class="idref" href="#lt:9"><span class="id" title="binder">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:7"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="X:10" class="idref" href="#X:10"><span class="id" title="binder">X</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Lub.html#M:7"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Lub.html#M:7"><span class="id" title="variable">M</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Schutte.Lub.html#D:8"><span class="id" title="variable">D</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:11"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Lub.html#upper_bound"><span class="id" title="definition">upper_bound</span></a>  <a class="idref" href="hydras.Schutte.Lub.html#D:8"><span class="id" title="variable">D</span></a> <a class="idref" href="hydras.Schutte.Lub.html#lt:9"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Schutte.Lub.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:11"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.Schutte.Lub.html#D:8"><span class="id" title="variable">D</span></a> <a class="idref" href="hydras.Schutte.Lub.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Lub.html#upper_bound"><span class="id" title="definition">upper_bound</span></a>  <a class="idref" href="hydras.Schutte.Lub.html#D:8"><span class="id" title="variable">D</span></a> <a class="idref" href="hydras.Schutte.Lub.html#lt:9"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Schutte.Lub.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Lub.html#y:12"><span class="id" title="variable">y</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Lub.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Lub.html#lt:9"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.Schutte.Lub.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Lub.html#y:12"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.MoreEpsilonIota.html b/theories/html/hydras.Schutte.MoreEpsilonIota.html
new file mode 100644
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.MoreEpsilonIota</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.MoreEpsilonIota</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Complements to Coq.Logic.Epsilon
+
+<div class="paragraph"> </div>
+
+  Pierre Casteran,
+  LaBRI, University of Bordeaux
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="library">Logic.Epsilon</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon_statement"><span class="id" title="axiom">epsilon_statement</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#constructive_indefinite_description"><span class="id" title="lemma">constructive_indefinite_description</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota_statement"><span class="id" title="lemma">iota_statement</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Print</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota_spec"><span class="id" title="definition">iota_spec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> {<span class="id" title="var">U</span>} <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="epsilon_ind" class="idref" href="#epsilon_ind"><span class="id" title="lemma">epsilon_ind</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} (<a id="inh:2" class="idref" href="#inh:2"><span class="id" title="binder">inh</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="P:3" class="idref" href="#P:3"><span class="id" title="binder">P</span></a> <a id="Q:4" class="idref" href="#Q:4"><span class="id" title="binder">Q</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ex"><span class="id" title="inductive">ex</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:3"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:5" class="idref" href="#a:5"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:3"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:5"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#Q:4"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:5"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#Q:4"><span class="id" title="variable">Q</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#inh:2"><span class="id" title="variable">inh</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:3"><span class="id" title="variable">P</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="epsilon_equiv" class="idref" href="#epsilon_equiv"><span class="id" title="lemma">epsilon_equiv</span></a>  {<a id="A:6" class="idref" href="#A:6"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>}(<a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:6"><span class="id" title="variable">A</span></a>)(<a id="P:8" class="idref" href="#P:8"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:6"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ex"><span class="id" title="inductive">ex</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:8"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)&lt;-&gt;</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:8"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:8"><span class="id" title="variable">P</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">epsilon_elim_aux</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> [ |- (?<span class="id" title="var">P</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> (<span class="id" title="var">A</span>:=?<span class="id" title="var">X</span>) ?<span class="id" title="var">a</span> ?<span class="id" title="var">Q</span>))] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#epsilon_ind"><span class="id" title="lemma">epsilon_ind</span></a>; <span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">epsilon_elim</span> := <span class="id" title="var">epsilon_elim_aux</span> ||<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;[ |- (?<span class="id" title="var">P</span> (?<span class="id" title="var">k</span> ?<span class="id" title="var">d</span>))] ⇒<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">let</span> <span class="id" title="var">v</span> := <span class="id" title="tactic">eval</span> <span class="id" title="tactic">cbv</span> <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">k</span>]  <span class="id" title="tactic">in</span> (<span class="id" title="var">k</span> <span class="id" title="var">d</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> ?<span class="id" title="var">w</span> ?<span class="id" title="var">d</span>) ⇒ <span class="id" title="tactic">change</span> (<span class="id" title="var">P</span> <span class="id" title="var">v</span>); <span class="id" title="var">epsilon_elim_aux</span> <span class="id" title="keyword">end</span>))<br/>
+&nbsp;&nbsp;| [ |- (?<span class="id" title="var">P</span> (?<span class="id" title="var">k</span> ?<span class="id" title="var">arg</span> ?<span class="id" title="var">arg1</span>))] ⇒<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">let</span> <span class="id" title="var">v</span> := <span class="id" title="tactic">eval</span> <span class="id" title="tactic">cbv</span> <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">k</span>]  <span class="id" title="tactic">in</span> (<span class="id" title="var">k</span> <span class="id" title="var">arg</span> <span class="id" title="var">arg1</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> ?<span class="id" title="var">w</span> ?<span class="id" title="var">d</span>) ⇒ <span class="id" title="tactic">change</span> (<span class="id" title="var">P</span> <span class="id" title="var">v</span>); <span class="id" title="var">epsilon_elim_aux</span> <span class="id" title="keyword">end</span>))<br/>
+&nbsp;&nbsp;| [ |- (?<span class="id" title="var">P</span> ?<span class="id" title="var">k</span>)] ⇒<br/>
+&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">let</span> <span class="id" title="var">v</span> := <span class="id" title="tactic">eval</span> <span class="id" title="tactic">cbv</span> <span class="id" title="keyword">zeta</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">k</span>]  <span class="id" title="tactic">in</span> <span class="id" title="var">k</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> <span class="id" title="var">v</span> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> ?<span class="id" title="var">w</span> ?<span class="id" title="var">d</span>) ⇒ <span class="id" title="tactic">change</span> (<span class="id" title="var">P</span> <span class="id" title="var">v</span>); <span class="id" title="var">epsilon_elim_aux</span> <span class="id" title="keyword">end</span>))<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="On_Iota" class="idref" href="#On_Iota"><span class="id" title="section">On_Iota</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="On_Iota.A" class="idref" href="#On_Iota.A"><span class="id" title="variable">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="On_Iota.P" class="idref" href="#On_Iota.P"><span class="id" title="variable">P</span></a>: <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:9"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span>  <a id="On_Iota.inhA" class="idref" href="#On_Iota.inhA"><span class="id" title="variable">inhA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_Iota.unique_P" class="idref" href="#On_Iota.unique_P"><span class="id" title="variable">unique_P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#x:12"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_spec_1" class="idref" href="#iota_spec_1"><span class="id" title="lemma">iota_spec_1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#unique"><span class="id" title="definition">unique</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.inhA"><span class="id" title="variable">inhA</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_eq" class="idref" href="#iota_eq"><span class="id" title="lemma">iota_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.inhA"><span class="id" title="variable">inhA</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_ind" class="idref" href="#iota_ind"><span class="id" title="lemma">iota_ind</span></a> (<a id="Q:15" class="idref" href="#Q:15"><span class="id" title="binder">Q</span></a>:<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#unique"><span class="id" title="definition">unique</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#Q:15"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#a:16"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#Q:15"><span class="id" title="variable">Q</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.inhA"><span class="id" title="variable">inhA</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P"><span class="id" title="variable">P</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#On_Iota"><span class="id" title="section">On_Iota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Ltac</span> <span class="id" title="var">iota_elim</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span> |- <span class="id" title="keyword">context</span> <span class="id" title="var">c</span> [(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> ?<span class="id" title="var">b</span> ?<span class="id" title="var">P</span>)] ⇒<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#iota_ind"><span class="id" title="lemma">iota_ind</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="InH" class="idref" href="#InH"><span class="id" title="record">InH</span></a> (<a id="A:17" class="idref" href="#A:17"><span class="id" title="binder">A</span></a>: <span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;<a id="InHWit" class="idref" href="#InHWit"><span class="id" title="projection">InHWit</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:17"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="some" class="idref" href="#some"><span class="id" title="definition">some</span></a> {<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="H:21" class="idref" href="#H:21"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:20"><span class="id" title="variable">A</span></a>} (<a id="P:22" class="idref" href="#P:22"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#epsilon"><span class="id" title="definition">epsilon</span></a> (@<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InHWit"><span class="id" title="definition">InHWit</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#H:21"><span class="id" title="variable">H</span></a>) <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:22"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="the" class="idref" href="#the"><span class="id" title="definition">the</span></a> {<a id="A:23" class="idref" href="#A:23"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>} {<a id="H:24" class="idref" href="#H:24"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:23"><span class="id" title="variable">A</span></a>} (<a id="P:25" class="idref" href="#P:25"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:23"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> (@<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InHWit"><span class="id" title="definition">InHWit</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#A:23"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#H:24"><span class="id" title="variable">H</span></a>) <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#P:25"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="inhNat" class="idref" href="#inhNat"><span class="id" title="instance">inhNat</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A small example 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="some_pos" class="idref" href="#some_pos"><span class="id" title="definition">some_pos</span></a> := <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some"><span class="id" title="definition">some</span></a> (<span class="id" title="keyword">fun</span> <a id="i:26" class="idref" href="#i:26"><span class="id" title="binder">i</span></a> ⇒ 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#i:26"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some_pos"><span class="id" title="definition">some_pos</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some_pos"><span class="id" title="definition">some_pos</span></a>, <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some"><span class="id" title="definition">some</span></a>; <span class="id" title="var">epsilon_elim</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> 42;   <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">arith</span>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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diff --git a/theories/html/hydras.Schutte.Ordering_Functions.html b/theories/html/hydras.Schutte.Ordering_Functions.html
new file mode 100644
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@@ -0,0 +1,519 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Ordering_Functions</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Ordering_Functions</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Ordering functions (after Schutte) 
+<div class="paragraph"> </div>
+
+   Pierre Casteran, LaBRI, University of Bordeaux
+
+<div class="paragraph"> </div>
+
+Every subset <span class="inlinecode"><span class="id" title="var">A</span></span> of <span class="inlinecode"><span class="id" title="var">Ord</span></span> can be enumerated in an unique way
+ by a segment of <span class="inlinecode"><span class="id" title="var">Ord</span></span>.
+
+<div class="paragraph"> </div>
+
+Thus it makes sense to consider the <span class="inlinecode"><span class="id" title="var">alpha</span></span>-th element of <span class="inlinecode"><span class="id" title="var">A</span></span>
+
+<div class="paragraph"> </div>
+
+This module shows the construction of the <i>ordering function</i>  of <span class="inlinecode"><span class="id" title="var">A</span></span>, following
+Schutte's definitions.
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">CountableTypes</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="module">Ensembles</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#"><span class="id" title="module">Well_Orders</span></a>  <a class="idref" href="hydras.Schutte.Countable.html#"><span class="id" title="module">Countable</span></a>  <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="module">PartialFun</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="module">Classical</span></a>  <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="module">MoreEpsilonIota</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#"><span class="id" title="module">Epsilon</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab203"></a><h2 class="section">Main definitions</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="segment" class="idref" href="#segment"><span class="id" title="definition">segment</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> <a id="beta:3" class="idref" href="#beta:3"><span class="id" title="binder">beta</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:3"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:1"><span class="id" title="variable">A</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:3"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="proper_segment" class="idref" href="#proper_segment"><span class="id" title="definition">proper_segment</span></a> (<a id="A:4" class="idref" href="#A:4"><span class="id" title="binder">A</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:4"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Same_set"><span class="id" title="definition">Same_set</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:4"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Class</span> <a id="ordering_function" class="idref" href="#ordering_function"><span class="id" title="record">ordering_function</span></a> (<a id="f:5" class="idref" href="#f:5"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:6" class="idref" href="#A:6"><span class="id" title="binder">A</span></a> <a id="B:7" class="idref" href="#B:7"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a id="Build_OF" class="idref" href="#Build_OF"><span class="id" title="constructor">Build_OF</span></a> {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="OF_segment" class="idref" href="#OF_segment"><span class="id" title="projection">OF_segment</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:6"><span class="id" title="variable">A</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="OF_total" class="idref" href="#OF_total"><span class="id" title="projection">OF_total</span></a> : <span class="id" title="keyword">∀</span> <a id="a:10" class="idref" href="#a:10"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:6"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:10"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:7"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:10"><span class="id" title="variable">a</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="OF_onto" class="idref" href="#OF_onto"><span class="id" title="projection">OF_onto</span></a> : <span class="id" title="keyword">∀</span> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:7"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:12"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:6"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:12"><span class="id" title="variable">b</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="OF_mono" class="idref" href="#OF_mono"><span class="id" title="projection">OF_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a> <a id="b:16" class="idref" href="#b:16"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:6"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:6"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:16"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:16"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:15"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:5"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:16"><span class="id" title="variable">b</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ordering_segment" class="idref" href="#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> <a id="B:19" class="idref" href="#B:19"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:20" class="idref" href="#f:20"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:20"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:18"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:19"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="the_ordering_segment" class="idref" href="#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> (<a id="B:21" class="idref" href="#B:21"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#the"><span class="id" title="definition">the</span></a>  (<span class="id" title="keyword">fun</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:21"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ord" class="idref" href="#ord"><span class="id" title="definition">ord</span></a>   (<a id="B:23" class="idref" href="#B:23"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some"><span class="id" title="definition">some</span></a> (<span class="id" title="keyword">fun</span> <a id="f:24" class="idref" href="#f:24"><span class="id" title="binder">f</span></a> ⇒ <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:24"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:23"><span class="id" title="variable">B</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:23"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="proper_segment_of" class="idref" href="#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> (<a id="B:25" class="idref" href="#B:25"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="beta:26" class="idref" href="#beta:26"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>  :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:25"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:26"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:25"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:26"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="normal" class="idref" href="#normal"><span class="id" title="definition">normal</span></a> (<a id="f:28" class="idref" href="#f:28"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="B:29" class="idref" href="#B:29"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:28"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:29"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#continuous"><span class="id" title="definition">continuous</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:28"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:29"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fun_equiv" class="idref" href="#fun_equiv"><span class="id" title="definition">fun_equiv</span></a> (<a id="f:30" class="idref" href="#f:30"><span class="id" title="binder">f</span></a> <a id="g:31" class="idref" href="#g:31"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a> <a id="B:33" class="idref" href="#B:33"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Same_set"><span class="id" title="definition">Same_set</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:33"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:30"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g:31"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:34"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab204"></a><h2 class="section">Properties of  segments</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordinal_segment" class="idref" href="#ordinal_segment"><span class="id" title="lemma">ordinal_segment</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="members_proper" class="idref" href="#members_proper"><span class="id" title="lemma">members_proper</span></a> (<a id="alpha:35" class="idref" href="#alpha:35"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment"><span class="id" title="definition">proper_segment</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:35"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="proper_members" class="idref" href="#proper_members"><span class="id" title="lemma">proper_members</span></a> (<a id="A:36" class="idref" href="#A:36"><span class="id" title="binder">A</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="H:37" class="idref" href="#H:37"><span class="id" title="binder">H</span></a> :  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment"><span class="id" title="definition">proper_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:36"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:38" class="idref" href="#a:38"><span class="id" title="binder">a</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Same_set"><span class="id" title="definition">Same_set</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:36"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:38"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="countable_segment_proper" class="idref" href="#countable_segment_proper"><span class="id" title="lemma">countable_segment_proper</span></a> : <span class="id" title="keyword">∀</span> <a id="A:43" class="idref" href="#A:43"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:43"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:43"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment"><span class="id" title="definition">proper_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:43"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_In" class="idref" href="#ordering_function_In"><span class="id" title="lemma">ordering_function_In</span></a> <a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> <a id="A:45" class="idref" href="#A:45"><span class="id" title="binder">A</span></a> <a id="B:46" class="idref" href="#B:46"><span class="id" title="binder">B</span></a> <a id="a:47" class="idref" href="#a:47"><span class="id" title="binder">a</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:45"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:46"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:45"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:47"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:46"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:44"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:47"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_mono" class="idref" href="#ordering_function_mono"><span class="id" title="lemma">ordering_function_mono</span></a> (<a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="A:49" class="idref" href="#A:49"><span class="id" title="binder">A</span></a> <a id="B:50" class="idref" href="#B:50"><span class="id" title="binder">B</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:49"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:50"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:51" class="idref" href="#alpha:51"><span class="id" title="binder">alpha</span></a> <a id="beta:52" class="idref" href="#beta:52"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:49"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:49"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:52"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:51"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:52"><span class="id" title="variable">beta</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">ordering_function_mono</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="ordering_function_mono_weak" class="idref" href="#ordering_function_mono_weak"><span class="id" title="lemma">ordering_function_mono_weak</span></a> (<a id="f:53" class="idref" href="#f:53"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="A:54" class="idref" href="#A:54"><span class="id" title="binder">A</span></a> <a id="B:55" class="idref" href="#B:55"><span class="id" title="binder">B</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:53"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:54"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:55"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a> <a id="b:57" class="idref" href="#b:57"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:54"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:54"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:57"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:53"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:53"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:57"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">ordering_function_mono_weak</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_monoR" class="idref" href="#ordering_function_monoR"><span class="id" title="lemma">ordering_function_monoR</span></a> : <span class="id" title="keyword">∀</span> <a id="f:58" class="idref" href="#f:58"><span class="id" title="binder">f</span></a> <a id="A:59" class="idref" href="#A:59"><span class="id" title="binder">A</span></a> <a id="B:60" class="idref" href="#B:60"><span class="id" title="binder">B</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:58"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:59"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:60"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:61" class="idref" href="#a:61"><span class="id" title="binder">a</span></a> <a id="b:62" class="idref" href="#b:62"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:59"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:59"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:62"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:58"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:58"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:62"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:61"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:62"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">ordering_function_monoR</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Ordering_bijection" class="idref" href="#Ordering_bijection"><span class="id" title="lemma">Ordering_bijection</span></a> : <span class="id" title="keyword">∀</span> <a id="f:63" class="idref" href="#f:63"><span class="id" title="binder">f</span></a> <a id="A:64" class="idref" href="#A:64"><span class="id" title="binder">A</span></a> <a id="B:65" class="idref" href="#B:65"><span class="id" title="binder">B</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:63"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:64"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:65"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:64"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:65"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:63"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="ordering_function_mono_weakR" class="idref" href="#ordering_function_mono_weakR"><span class="id" title="lemma">ordering_function_mono_weakR</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:66" class="idref" href="#f:66"><span class="id" title="binder">f</span></a> <a id="A:67" class="idref" href="#A:67"><span class="id" title="binder">A</span></a> <a id="B:68" class="idref" href="#B:68"><span class="id" title="binder">B</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:66"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:68"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a> <a id="b:70" class="idref" href="#b:70"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:66"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:66"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:70"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">ordering_function_mono_weakR</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_seg" class="idref" href="#ordering_function_seg"><span class="id" title="lemma">ordering_function_seg</span></a> : <span class="id" title="keyword">∀</span> <a id="A:71" class="idref" href="#A:71"><span class="id" title="binder">A</span></a> <a id="B:72" class="idref" href="#B:72"><span class="id" title="binder">B</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:71"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:72"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:71"><span class="id" title="variable">A</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="empty_ordering" class="idref" href="#empty_ordering"><span class="id" title="lemma">empty_ordering</span></a> : <span class="id" title="keyword">∀</span> <a id="B:73" class="idref" href="#B:73"><span class="id" title="binder">B</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:74" class="idref" href="#b:74"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:73"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:74"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> (<span class="id" title="keyword">fun</span> <a id="o:75" class="idref" href="#o:75"><span class="id" title="binder">o</span></a> ⇒ <a class="idref" href="hydras.Schutte.Ordering_Functions.html#o:75"><span class="id" title="variable">o</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:73"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="segment_lt" class="idref" href="#segment_lt"><span class="id" title="lemma">segment_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="A:76" class="idref" href="#A:76"><span class="id" title="binder">A</span></a> <a id="a:77" class="idref" href="#a:77"><span class="id" title="binder">a</span></a> <a id="b:78" class="idref" href="#b:78"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:76"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:76"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:78"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:77"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:76"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:78"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="segment_unbounded" class="idref" href="#segment_unbounded"><span class="id" title="lemma">segment_unbounded</span></a> : <span class="id" title="keyword">∀</span> <a id="A:79" class="idref" href="#A:79"><span class="id" title="binder">A</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:79"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:79"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:79"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="ordering_le" class="idref" href="#ordering_le"><span class="id" title="lemma">ordering_le</span></a> : <span class="id" title="keyword">∀</span> <a id="f:80" class="idref" href="#f:80"><span class="id" title="binder">f</span></a> <a id="A:81" class="idref" href="#A:81"><span class="id" title="binder">A</span></a> <a id="B:82" class="idref" href="#B:82"><span class="id" title="binder">B</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:80"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:81"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:82"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:83" class="idref" href="#alpha:83"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:81"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:83"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:83"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:80"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:83"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ordering_function_unicity" class="idref" href="#ordering_function_unicity"><span class="id" title="section">ordering_function_unicity</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> <a id="ordering_function_unicity.B" class="idref" href="#ordering_function_unicity.B"><span class="id" title="variable">B</span></a> <a id="ordering_function_unicity.A1" class="idref" href="#ordering_function_unicity.A1"><span class="id" title="variable">A1</span></a> <a id="ordering_function_unicity.A2" class="idref" href="#ordering_function_unicity.A2"><span class="id" title="variable">A2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> <a id="ordering_function_unicity.f1" class="idref" href="#ordering_function_unicity.f1"><span class="id" title="variable">f1</span></a> <a id="ordering_function_unicity.f2" class="idref" href="#ordering_function_unicity.f2"><span class="id" title="variable">f2</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="ordering_function_unicity.O1" class="idref" href="#ordering_function_unicity.O1"><span class="id" title="variable">O1</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.B"><span class="id" title="variable">B</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="ordering_function_unicity.O2" class="idref" href="#ordering_function_unicity.O2"><span class="id" title="variable">O2</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="ordering_function_unicity" class="idref" href="#ordering_function_unicity"><span class="id" title="lemma">ordering_function_unicity</span></a>  : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#fun_equiv"><span class="id" title="definition">fun_equiv</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A2"><span class="id" title="variable">A2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity"><span class="id" title="section">ordering_function_unicity</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_seg_unicity" class="idref" href="#ordering_function_seg_unicity"><span class="id" title="lemma">ordering_function_seg_unicity</span></a> : <span class="id" title="keyword">∀</span> <a id="A1:114" class="idref" href="#A1:114"><span class="id" title="binder">A1</span></a> <a id="A2:115" class="idref" href="#A2:115"><span class="id" title="binder">A2</span></a> <a id="B:116" class="idref" href="#B:116"><span class="id" title="binder">B</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A1:114"><span class="id" title="variable">A1</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:116"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A2:115"><span class="id" title="variable">A2</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:116"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A1:114"><span class="id" title="variable">A1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A2:115"><span class="id" title="variable">A2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us build now an ordering function, and the associated ordering segment
+ of any subset B composed of ordinals 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="proper_of_proper" class="idref" href="#proper_of_proper"><span class="id" title="lemma">proper_of_proper</span></a> : <span class="id" title="keyword">∀</span> <a id="B:117" class="idref" href="#B:117"><span class="id" title="binder">B</span></a> <a id="beta:118" class="idref" href="#beta:118"><span class="id" title="binder">beta</span></a> <a id="beta':119" class="idref" href="#beta':119"><span class="id" title="binder">beta'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:118"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:117"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:118"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:117"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:118"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta':119"><span class="id" title="variable">beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:117"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta':119"><span class="id" title="variable">beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:117"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:118"><span class="id" title="variable">beta</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta':119"><span class="id" title="variable">beta'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="building_ordering_function_1" class="idref" href="#building_ordering_function_1"><span class="id" title="section">building_ordering_function_1</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="building_ordering_function_1.B" class="idref" href="#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="building_ordering_function_1.H_B" class="idref" href="#building_ordering_function_1.H_B"><span class="id" title="variable">H_B</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:121" class="idref" href="#beta:121"><span class="id" title="binder">beta</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:121"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="A:122" class="idref" href="#A:122"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:122"><span class="id" title="variable">A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:121"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Section</span> <a id="building_ordering_function_1.beta_fixed" class="idref" href="#building_ordering_function_1.beta_fixed"><span class="id" title="section">beta_fixed</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="building_ordering_function_1.beta_fixed.beta" class="idref" href="#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="building_ordering_function_1.beta_fixed.beta_B" class="idref" href="#building_ordering_function_1.beta_fixed.beta_B"><span class="id" title="variable">beta_B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us build an ordering function for (proper_segment_of B beta) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="_A" class="idref" href="#_A"><span class="id" title="definition">_A</span></a> := <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#the"><span class="id" title="definition">the</span></a>  (<span class="id" title="keyword">fun</span> <a id="E:126" class="idref" href="#E:126"><span class="id" title="binder">E</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#E:126"><span class="id" title="variable">E</span></a>  (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a>)).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="_f" class="idref" href="#_f"><span class="id" title="definition">_f</span></a> := <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#some"><span class="id" title="definition">some</span></a>  (<span class="id" title="keyword">fun</span> <a id="f:127" class="idref" href="#f:127"><span class="id" title="binder">f</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:127"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a>)).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="of_beta'" class="idref" href="#of_beta'"><span class="id" title="lemma">of_beta'</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_f"><span class="id" title="definition">_f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Remark</span> <a id="Bbeta_denum" class="idref" href="#Bbeta_denum"><span class="id" title="lemma">Bbeta_denum</span></a> : <span class="id" title="definition">Countable</span> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">of_beta'</span>: <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="A_denum" class="idref" href="#A_denum"><span class="id" title="lemma">A_denum</span></a> : <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Proper_A" class="idref" href="#Proper_A"><span class="id" title="lemma">Proper_A</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment"><span class="id" title="definition">proper_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_def1" class="idref" href="#g_def1"><span class="id" title="lemma">g_def1</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="g_beta:128" class="idref" href="#g_beta:128"><span class="id" title="binder">g_beta</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta:128"><span class="id" title="variable">g_beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta:128"><span class="id" title="variable">g_beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_unic" class="idref" href="#g_unic"><span class="id" title="lemma">g_unic</span></a> : <span class="id" title="keyword">∀</span> <a id="g_beta:129" class="idref" href="#g_beta:129"><span class="id" title="binder">g_beta</span></a> <a id="g_beta':130" class="idref" href="#g_beta':130"><span class="id" title="binder">g_beta'</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta:129"><span class="id" title="variable">g_beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta':130"><span class="id" title="variable">g_beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta:129"><span class="id" title="variable">g_beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta':130"><span class="id" title="variable">g_beta'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta:129"><span class="id" title="variable">g_beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g_beta':130"><span class="id" title="variable">g_beta'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="g" class="idref" href="#g"><span class="id" title="definition">g</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#inh_Ord"><span class="id" title="axiom">inh_Ord</span></a> (<span class="id" title="keyword">fun</span> <a id="o:131" class="idref" href="#o:131"><span class="id" title="binder">o</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#o:131"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#o:131"><span class="id" title="variable">o</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed"><span class="id" title="section">beta_fixed</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_def" class="idref" href="#g_def"><span class="id" title="lemma">g_def</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:132" class="idref" href="#beta:132"><span class="id" title="binder">beta</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:132"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#_A"><span class="id" title="definition">_A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:132"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:132"><span class="id" title="variable">beta</span></a> ).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_lemma" class="idref" href="#g_lemma"><span class="id" title="lemma">g_lemma</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="beta:133" class="idref" href="#beta:133"><span class="id" title="binder">beta</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:133"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#_f"><span class="id" title="definition">_f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:133"><span class="id" title="variable">beta</span></a>) (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:133"><span class="id" title="variable">beta</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:133"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_mono" class="idref" href="#g_mono"><span class="id" title="lemma">g_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="beta1:134" class="idref" href="#beta1:134"><span class="id" title="binder">beta1</span></a> <a id="beta2:135" class="idref" href="#beta2:135"><span class="id" title="binder">beta2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta1:134"><span class="id" title="variable">beta1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta2:135"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta1:134"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta2:135"><span class="id" title="variable">beta2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta1:134"><span class="id" title="variable">beta1</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta2:135"><span class="id" title="variable">beta2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L3a" class="idref" href="#L3a"><span class="id" title="lemma">L3a</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="g_bij" class="idref" href="#g_bij"><span class="id" title="lemma">g_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>.<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">g_bij</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a id="building_ordering_function_1.g_1" class="idref" href="#building_ordering_function_1.g_1"><span class="id" title="variable">g_1</span></a> := <a class="idref" href="hydras.Schutte.PartialFun.html#inv_fun"><span class="id" title="definition">inv_fun</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#inh_Ord"><span class="id" title="axiom">inh_Ord</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_1_bij" class="idref" href="#g_1_bij"><span class="id" title="lemma">g_1_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.g_1"><span class="id" title="variable">g_1</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">g_1_bij</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="g_1_of" class="idref" href="#g_1_of"><span class="id" title="lemma">g_1_of</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.g_1"><span class="id" title="variable">g_1</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="image_B_g_seg" class="idref" href="#image_B_g_seg"><span class="id" title="lemma">image_B_g_seg</span></a>  : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#g"><span class="id" title="definition">g</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Corresponds to Lemma 3 of Schutte's chapter :
+    It is used twice in the building of ordering function for any subset B of ordinal 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="L3_u" class="idref" href="#L3_u"><span class="id" title="lemma">L3_u</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="S:140" class="idref" href="#S:140"><span class="id" title="binder">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#S:140"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1"><span class="id" title="section">building_ordering_function_1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For any set <span class="inlinecode"><span class="id" title="var">B</span></span>, we build by transfinite induction the ordering segment of
+  <span class="inlinecode"><span class="id" title="var">B</span></span> and the (unique upto extensionnality) ordering function of B
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="building_ordering_function_by_induction" class="idref" href="#building_ordering_function_by_induction"><span class="id" title="section">building_ordering_function_by_induction</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="building_ordering_function_by_induction.B" class="idref" href="#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ordering_segments_of_B" class="idref" href="#ordering_segments_of_B"><span class="id" title="lemma">ordering_segments_of_B</span></a> (<a id="beta:142" class="idref" href="#beta:142"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:142"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="A:143" class="idref" href="#A:143"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:143"><span class="id" title="variable">A</span></a>  (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#proper_segment_of"><span class="id" title="definition">proper_segment_of</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#beta:142"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Theorem</span> <a id="ordering_segment_ex_unique" class="idref" href="#ordering_segment_ex_unique"><span class="id" title="lemma">ordering_segment_ex_unique</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="S:144" class="idref" href="#S:144"><span class="id" title="binder">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_segment"><span class="id" title="definition">ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#S:144"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="ordering_function_ex" class="idref" href="#ordering_function_ex"><span class="id" title="lemma">ordering_function_ex</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="S:145" class="idref" href="#S:145"><span class="id" title="binder">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:146" class="idref" href="#f:146"><span class="id" title="binder">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:146"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#S:145"><span class="id" title="variable">S</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ord_ok" class="idref" href="#ord_ok"><span class="id" title="lemma">ord_ok</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a>  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="segment_the_ordering_segment" class="idref" href="#segment_the_ordering_segment"><span class="id" title="lemma">segment_the_ordering_segment</span></a>  :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ord_eq" class="idref" href="#ord_eq"><span class="id" title="lemma">ord_eq</span></a> (<a id="A:147" class="idref" href="#A:147"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="f:148" class="idref" href="#f:148"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:148"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:147"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#fun_equiv"><span class="id" title="definition">fun_equiv</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:148"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:147"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction"><span class="id" title="section">building_ordering_function_by_induction</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_ex"><span class="id" title="lemma">ordering_function_ex</span></a>.<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity"><span class="id" title="lemma">ordering_function_unicity</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="of_image" class="idref" href="#of_image"><span class="id" title="lemma">of_image</span></a> : <span class="id" title="keyword">∀</span> <a id="f:149" class="idref" href="#f:149"><span class="id" title="binder">f</span></a> <a id="A:150" class="idref" href="#A:150"><span class="id" title="binder">A</span></a> <a id="B:151" class="idref" href="#B:151"><span class="id" title="binder">B</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:149"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:150"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:151"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:149"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:150"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:150"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:149"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Th13_5" class="idref" href="#Th13_5"><span class="id" title="section">Th13_5</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Variables</span> (<a id="Th13_5.A" class="idref" href="#Th13_5.A"><span class="id" title="variable">A</span></a> <a id="Th13_5.B" class="idref" href="#Th13_5.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Variable</span> <a id="Th13_5.f" class="idref" href="#Th13_5.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Th13_5.f_ord" class="idref" href="#Th13_5.f_ord"><span class="id" title="variable">f_ord</span></a> : <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Section</span> <a id="Th13_5.recto" class="idref" href="#Th13_5.recto"><span class="id" title="section">recto</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Th13_5.recto.f_cont" class="idref" href="#Th13_5.recto.f_cont"><span class="id" title="variable">f_cont</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#continuous"><span class="id" title="definition">continuous</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Th_13_5_1" class="idref" href="#Th_13_5_1"><span class="id" title="lemma">Th_13_5_1</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto"><span class="id" title="section">recto</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Th13_5.verso" class="idref" href="#Th13_5.verso"><span class="id" title="section">verso</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Th13_5.verso.B_closed" class="idref" href="#Th13_5.verso.B_closed"><span class="id" title="variable">B_closed</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Th_13_5_2" class="idref" href="#Th_13_5_2"><span class="id" title="lemma">Th_13_5_2</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#continuous"><span class="id" title="definition">continuous</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.A"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso"><span class="id" title="section">verso</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th13_5"><span class="id" title="section">Th13_5</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="TH_13_6" class="idref" href="#TH_13_6"><span class="id" title="lemma">TH_13_6</span></a> (<a id="B:174" class="idref" href="#B:174"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="f:175" class="idref" href="#f:175"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:175"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:174"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:174"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:174"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_unbounded_unbounded" class="idref" href="#ordering_unbounded_unbounded"><span class="id" title="lemma">ordering_unbounded_unbounded</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:176" class="idref" href="#A:176"><span class="id" title="binder">A</span></a> <a id="B:177" class="idref" href="#B:177"><span class="id" title="binder">B</span></a> <a id="f:178" class="idref" href="#f:178"><span class="id" title="binder">f</span></a>,  <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:178"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:176"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:177"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:177"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:176"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="TH_13_6R" class="idref" href="#TH_13_6R"><span class="id" title="lemma">TH_13_6R</span></a> (<a id="A:179" class="idref" href="#A:179"><span class="id" title="binder">A</span></a> <a id="B:180" class="idref" href="#B:180"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="f:181" class="idref" href="#f:181"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:181"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:179"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:180"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:180"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:180"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <a class="idref" href="hydras.Schutte.Ordering_Functions.html#normal"><span class="id" title="definition">normal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:181"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:180"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If <span class="inlinecode"><span class="id" title="var">f</span></span> is the ordering function of <span class="inlinecode"><span class="id" title="var">B</span></span>, then <span class="inlinecode"><span class="id" title="var">f</span></span> <span class="inlinecode">0</span> is the least element of
+   <span class="inlinecode"><span class="id" title="var">B</span></span> 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ordering_function_least_least" class="idref" href="#ordering_function_least_least"><span class="id" title="lemma">ordering_function_least_least</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="B:182" class="idref" href="#B:182"><span class="id" title="binder">B</span></a> <a id="f:183" class="idref" href="#f:183"><span class="id" title="binder">f</span></a>  , <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ordering_function"><span class="id" title="class">ordering_function</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:183"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:182"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#B:182"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#f:183"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="segment_lt_closed" class="idref" href="#segment_lt_closed"><span class="id" title="lemma">segment_lt_closed</span></a> : <span class="id" title="keyword">∀</span> <a id="A:184" class="idref" href="#A:184"><span class="id" title="binder">A</span></a> <a id="a:185" class="idref" href="#a:185"><span class="id" title="binder">a</span></a> <a id="b:186" class="idref" href="#b:186"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.Ordering_Functions.html#segment"><span class="id" title="definition">segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:186"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:185"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#b:186"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:184"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#a:185"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="th_In" class="idref" href="#th_In"><span class="id" title="lemma">th_In</span></a> <a id="A:187" class="idref" href="#A:187"><span class="id" title="binder">A</span></a>  <a id="alpha:188" class="idref" href="#alpha:188"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:187"><span class="id" title="variable">A</span></a>) <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:188"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:187"><span class="id" title="variable">A</span></a> (<a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#A:187"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#alpha:188"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">About</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#Th_13_5_2"><span class="id" title="lemma">Th_13_5_2</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#ord"><span class="id" title="definition">ord</span></a>  : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment"><span class="id" title="definition">the_ordering_segment</span></a> : <span class="id" title="tactic">clear</span> <span class="id" title="var">implicits</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.PartialFun.html b/theories/html/hydras.Schutte.PartialFun.html
new file mode 100644
index 00000000..5d302635
--- /dev/null
+++ b/theories/html/hydras.Schutte.PartialFun.html
@@ -0,0 +1,266 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.PartialFun</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.PartialFun</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab215"></a><h1 class="section">Partial functions</h1>
+
+<div class="paragraph"> </div>
+
+ Pierre Casteran, Univ. Bordeaux and LaBRI  
+<div class="paragraph"> </div>
+
+  We study the relationship between  two representations of partial
+   functions : relational presentation, and with the  <span class="inlinecode"><span class="id" title="keyword">iota</span></span> description operator 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.ClassicalDescription.html#"><span class="id" title="library">ClassicalDescription</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#"><span class="id" title="library">Image</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.ProofIrrelevance.html#"><span class="id" title="library">ProofIrrelevance</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.ProofIrrelevanceFacts.html#"><span class="id" title="module">ProofIrrelevanceFacts</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">EnsemblesImplicit</span> <span class="id" title="library">FunctionProperties</span> <span class="id" title="library">ImageImplicit</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="library">MoreEpsilonIota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="AB_given" class="idref" href="#AB_given"><span class="id" title="section">AB_given</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="AB_given.A" class="idref" href="#AB_given.A"><span class="id" title="variable">A</span></a> <a id="AB_given.B" class="idref" href="#AB_given.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="AB_given.Ha" class="idref" href="#AB_given.Ha"><span class="id" title="variable">Ha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:2"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="AB_given.Hb" class="idref" href="#AB_given.Hb"><span class="id" title="variable">Hb</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:1"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="AB_given.DA" class="idref" href="#AB_given.DA"><span class="id" title="variable">DA</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:2"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="AB_given.DB" class="idref" href="#AB_given.DB"><span class="id" title="variable">DB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:1"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab216"></a><h2 class="section">Transformation of a functional relation into a function</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="iota_fun" class="idref" href="#iota_fun"><span class="id" title="definition">iota_fun</span></a> (<a id="R:7" class="idref" href="#R:7"><span class="id" title="binder">R</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) : <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.Hb"><span class="id" title="variable">Hb</span></a> (<span class="id" title="keyword">fun</span> <a id="b':9" class="idref" href="#b':9"><span class="id" title="binder">b'</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:8"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#R:7"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:8"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b':9"><span class="id" title="variable">b'</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b':9"><span class="id" title="variable">b'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_fun_e" class="idref" href="#iota_fun_e"><span class="id" title="lemma">iota_fun_e</span></a> (<a id="R:10" class="idref" href="#R:10"><span class="id" title="binder">R</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="b:12" class="idref" href="#b:12"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:12"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:12"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#unique"><span class="id" title="definition">unique</span></a> (<span class="id" title="keyword">fun</span> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> ⇒ (<a class="idref" href="hydras.Schutte.PartialFun.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:13"><span class="id" title="variable">b</span></a>)) (<a class="idref" href="hydras.Schutte.PartialFun.html#iota_fun"><span class="id" title="definition">iota_fun</span></a>   <a class="idref" href="hydras.Schutte.PartialFun.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:11"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_fun_ind" class="idref" href="#iota_fun_ind"><span class="id" title="lemma">iota_fun_ind</span></a> (<a id="P:14" class="idref" href="#P:14"><span class="id" title="binder">P</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Q:15" class="idref" href="#Q:15"><span class="id" title="binder">Q</span></a> <a id="R:16" class="idref" href="#R:16"><span class="id" title="binder">R</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>):<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a> <a id="x:18" class="idref" href="#x:18"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:18"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="b:19" class="idref" href="#b:19"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#R:16"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:19"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:19"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#P:14"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:20" class="idref" href="#b:20"><span class="id" title="binder">b</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#unique"><span class="id" title="definition">unique</span></a> (<span class="id" title="keyword">fun</span> <a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="hydras.Schutte.PartialFun.html#R:16"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:21"><span class="id" title="variable">b</span></a>) <a class="idref" href="hydras.Schutte.PartialFun.html#b:20"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#Q:15"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:20"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#Q:15"><span class="id" title="variable">Q</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:18"><span class="id" title="variable">x</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#iota_fun"><span class="id" title="definition">iota_fun</span></a>   <a class="idref" href="hydras.Schutte.PartialFun.html#R:16"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:17"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="AB_given.f_given" class="idref" href="#AB_given.f_given"><span class="id" title="section">f_given</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="AB_given.f_given.f" class="idref" href="#AB_given.f_given.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+relational representation 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="AB_given.f_given.Rf" class="idref" href="#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+abstract properties of a function (relational representation ) 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_domain" class="idref" href="#rel_domain"><span class="id" title="definition">rel_domain</span></a> := <span class="id" title="keyword">∀</span> <a id="a:24" class="idref" href="#a:24"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:24"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:25"><span class="id" title="variable">b</span></a> .<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_codomain" class="idref" href="#rel_codomain"><span class="id" title="definition">rel_codomain</span></a> := <span class="id" title="keyword">∀</span> <a id="a:26" class="idref" href="#a:26"><span class="id" title="binder">a</span></a> <a id="b:27" class="idref" href="#b:27"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:26"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:27"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:27"><span class="id" title="variable">b</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_functional" class="idref" href="#rel_functional"><span class="id" title="definition">rel_functional</span></a> := <span class="id" title="keyword">∀</span> <a id="a:28" class="idref" href="#a:28"><span class="id" title="binder">a</span></a> <a id="b:29" class="idref" href="#b:29"><span class="id" title="binder">b</span></a> <a id="b':30" class="idref" href="#b':30"><span class="id" title="binder">b'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:28"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b':30"><span class="id" title="variable">b'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:29"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b':30"><span class="id" title="variable">b'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_onto" class="idref" href="#rel_onto"><span class="id" title="definition">rel_onto</span></a> := <span class="id" title="keyword">∀</span> <a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:31"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:32" class="idref" href="#a:32"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:32"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:31"><span class="id" title="variable">b</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_inj" class="idref" href="#rel_inj"><span class="id" title="definition">rel_inj</span></a> := <span class="id" title="keyword">∀</span> <a id="a:33" class="idref" href="#a:33"><span class="id" title="binder">a</span></a> <a id="a':34" class="idref" href="#a':34"><span class="id" title="binder">a'</span></a> <a id="b:35" class="idref" href="#b:35"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':34"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':34"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#a:33"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':34"><span class="id" title="variable">a'</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span>  <a id="rel_injection" class="idref" href="#rel_injection"><span class="id" title="inductive">rel_injection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;<a id="rel_inj_i" class="idref" href="#rel_inj_i"><span class="id" title="constructor">rel_inj_i</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#rel_domain"><span class="id" title="definition">rel_domain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.PartialFun.html#rel_codomain"><span class="id" title="definition">rel_codomain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_functional"><span class="id" title="definition">rel_functional</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_inj"><span class="id" title="definition">rel_inj</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_injection:36"><span class="id" title="inductive">rel_injection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span>  <a id="rel_bijection" class="idref" href="#rel_bijection"><span class="id" title="inductive">rel_bijection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;<a id="rel_bij_i" class="idref" href="#rel_bij_i"><span class="id" title="constructor">rel_bij_i</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#rel_domain"><span class="id" title="definition">rel_domain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.PartialFun.html#rel_codomain"><span class="id" title="definition">rel_codomain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_functional"><span class="id" title="definition">rel_functional</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_onto"><span class="id" title="definition">rel_onto</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_inj"><span class="id" title="definition">rel_inj</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#rel_bijection:38"><span class="id" title="inductive">rel_bijection</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Abstract properties of f: A-&gt;B (wrt. domain DA and codomain DB 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="fun_codomain" class="idref" href="#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> := <span class="id" title="keyword">∀</span> <a id="a:40" class="idref" href="#a:40"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:40"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:40"><span class="id" title="variable">a</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="fun_onto" class="idref" href="#fun_onto"><span class="id" title="definition">fun_onto</span></a> := <span class="id" title="keyword">∀</span> <a id="b:41" class="idref" href="#b:41"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:41"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:42" class="idref" href="#a:42"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:42"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:41"><span class="id" title="variable">b</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="fun_inj" class="idref" href="#fun_inj"><span class="id" title="definition">fun_inj</span></a> := <span class="id" title="keyword">∀</span> <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="a':44" class="idref" href="#a':44"><span class="id" title="binder">a'</span></a> , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':44"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':44"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a':44"><span class="id" title="variable">a'</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="image" class="idref" href="#image"><span class="id" title="definition">image</span></a> := <span class="id" title="keyword">fun</span> <a id="b:45" class="idref" href="#b:45"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:46" class="idref" href="#a:46"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:46"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:45"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span>  <a id="fun_injection" class="idref" href="#fun_injection"><span class="id" title="inductive">fun_injection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;<a id="fun_inj_i" class="idref" href="#fun_inj_i"><span class="id" title="constructor">fun_inj_i</span></a> :  <a class="idref" href="hydras.Schutte.PartialFun.html#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#fun_inj"><span class="id" title="definition">fun_inj</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#fun_injection:47"><span class="id" title="inductive">fun_injection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span>  <a id="fun_bijection" class="idref" href="#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;<a id="fun_bij_i" class="idref" href="#fun_bij_i"><span class="id" title="constructor">fun_bij_i</span></a> :  <a class="idref" href="hydras.Schutte.PartialFun.html#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#fun_onto"><span class="id" title="definition">fun_onto</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#fun_inj"><span class="id" title="definition">fun_inj</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection:49"><span class="id" title="inductive">fun_bijection</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="rel_inv" class="idref" href="#rel_inv"><span class="id" title="definition">rel_inv</span></a> := <span class="id" title="keyword">fun</span> <a id="b:51" class="idref" href="#b:51"><span class="id" title="binder">b</span></a> <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:51"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf"><span class="id" title="variable">Rf</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#a:52"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:51"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.f_given"><span class="id" title="section">f_given</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Conversion from a relational definition : A-&gt;B-&gt;Prop into a partial
+    function of type A-&gt;B 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;<span class="id" title="keyword">Section</span> <a id="AB_given.rel_to_fun" class="idref" href="#AB_given.rel_to_fun"><span class="id" title="section">rel_to_fun</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Variables</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="AB_given.rel_to_fun.Rf" class="idref" href="#AB_given.rel_to_fun.Rf"><span class="id" title="variable">Rf</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="r2i" class="idref" href="#r2i"><span class="id" title="definition">r2i</span></a> := <a class="idref" href="hydras.Schutte.PartialFun.html#iota_fun"><span class="id" title="definition">iota_fun</span></a>   <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun.Rf"><span class="id" title="variable">Rf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun"><span class="id" title="section">rel_to_fun</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.PartialFun.html#AB_given"><span class="id" title="section">AB_given</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="inversion_of_bijection" class="idref" href="#inversion_of_bijection"><span class="id" title="section">inversion_of_bijection</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="inversion_of_bijection.A" class="idref" href="#inversion_of_bijection.A"><span class="id" title="variable">A</span></a> <a id="inversion_of_bijection.B" class="idref" href="#inversion_of_bijection.B"><span class="id" title="variable">B</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="inversion_of_bijection.inhA" class="idref" href="#inversion_of_bijection.inhA"><span class="id" title="variable">inhA</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:55"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="inversion_of_bijection.DA" class="idref" href="#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:55"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="inversion_of_bijection.DB" class="idref" href="#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:54"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="inversion_of_bijection.f" class="idref" href="#inversion_of_bijection.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#A:55"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:54"><span class="id" title="variable">B</span></a>).<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="inversion_of_bijection.inv_spec" class="idref" href="#inversion_of_bijection.inv_spec"><span class="id" title="variable">inv_spec</span></a> := <span class="id" title="keyword">fun</span> <a id="y:60" class="idref" href="#y:60"><span class="id" title="binder">y</span></a> <a id="x:61" class="idref" href="#x:61"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#y:60"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:61"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#y:60"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="inv_fun" class="idref" href="#inv_fun"><span class="id" title="definition">inv_fun</span></a> := <a class="idref" href="hydras.Schutte.PartialFun.html#r2i"><span class="id" title="definition">r2i</span></a>   <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inhA"><span class="id" title="variable">inhA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inv_spec"><span class="id" title="variable">inv_spec</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="inversion_of_bijection.f_b" class="idref" href="#inversion_of_bijection.f_b"><span class="id" title="variable">f_b</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inv_compose" class="idref" href="#inv_compose"><span class="id" title="lemma">inv_compose</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:63" class="idref" href="#x:63"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:63"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inv_fun"><span class="id" title="definition">inv_fun</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:63"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:63"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inv_composeR" class="idref" href="#inv_composeR"><span class="id" title="lemma">inv_composeR</span></a> :  <span class="id" title="keyword">∀</span> <a id="b:64" class="idref" href="#b:64"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:64"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#inv_fun"><span class="id" title="definition">inv_fun</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:64"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#b:64"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inv_fun_bij" class="idref" href="#inv_fun_bij"><span class="id" title="lemma">inv_fun_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB"><span class="id" title="variable">DB</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA"><span class="id" title="variable">DA</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#inv_fun"><span class="id" title="definition">inv_fun</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.PartialFun.html#inversion_of_bijection"><span class="id" title="section">inversion_of_bijection</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="image_as_Im" class="idref" href="#image_as_Im"><span class="id" title="lemma">image_as_Im</span></a> {<a id="A:65" class="idref" href="#A:65"><span class="id" title="binder">A</span></a> <a id="B:66" class="idref" href="#B:66"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="U:67" class="idref" href="#U:67"><span class="id" title="binder">U</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:65"><span class="id" title="variable">A</span></a>) (<a id="f:68" class="idref" href="#f:68"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#A:65"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:66"><span class="id" title="variable">B</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:67"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#f:68"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#Im"><span class="id" title="inductive">Im</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:67"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#f:68"><span class="id" title="variable">f</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+convert a <span class="inlinecode"><span class="id" title="var">fun_bijection</span></span> to a <span class="inlinecode"><span class="id" title="var">bijection</span></span> in the sense of the
+   <span class="inlinecode"><span class="id" title="var">ZornsLemma</span></span> library. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="fun_restr" class="idref" href="#fun_restr"><span class="id" title="definition">fun_restr</span></a> {<a id="U:69" class="idref" href="#U:69"><span class="id" title="binder">U</span></a> <a id="V:70" class="idref" href="#V:70"><span class="id" title="binder">V</span></a> : <span class="id" title="keyword">Type</span>} (<a id="f:71" class="idref" href="#f:71"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#U:69"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:70"><span class="id" title="variable">V</span></a>)<br/>
+&nbsp;&nbsp;{<a id="A:72" class="idref" href="#A:72"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:69"><span class="id" title="variable">U</span></a>} {<a id="B:73" class="idref" href="#B:73"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:70"><span class="id" title="variable">V</span></a>}<br/>
+&nbsp;&nbsp;(<a id="Hf:74" class="idref" href="#Hf:74"><span class="id" title="binder">Hf</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:73"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#f:71"><span class="id" title="variable">f</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="x:75" class="idref" href="#x:75"><span class="id" title="binder">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:69"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:75"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="y:76" class="idref" href="#y:76"><span class="id" title="binder">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:70"><span class="id" title="variable">V</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:73"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#y:76"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="p:78" class="idref" href="#p:78"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="x:77" class="idref" href="#x:77"><span class="id" title="binder">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:69"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#x:77"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> (<span class="id" title="keyword">fun</span> <a id="y:79" class="idref" href="#y:79"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#V:70"><span class="id" title="variable">V</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:73"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#y:79"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.PartialFun.html#f:71"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#p:78"><span class="id" title="variable">p</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.PartialFun.html#Hf:74"><span class="id" title="variable">Hf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj1_sig"><span class="id" title="definition">proj1_sig</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#p:78"><span class="id" title="variable">p</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#proj2_sig"><span class="id" title="definition">proj2_sig</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#p:78"><span class="id" title="variable">p</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fun_bijection_codomain" class="idref" href="#fun_bijection_codomain"><span class="id" title="lemma">fun_bijection_codomain</span></a><br/>
+&nbsp;&nbsp;{<a id="U:80" class="idref" href="#U:80"><span class="id" title="binder">U</span></a> <a id="V:81" class="idref" href="#V:81"><span class="id" title="binder">V</span></a> : <span class="id" title="keyword">Type</span>} {<a id="A:82" class="idref" href="#A:82"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:80"><span class="id" title="variable">U</span></a>} {<a id="B:83" class="idref" href="#B:83"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:81"><span class="id" title="variable">V</span></a>}<br/>
+&nbsp;&nbsp;(<a id="g:84" class="idref" href="#g:84"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#U:80"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:81"><span class="id" title="variable">V</span></a>) (<a id="g_bij:85" class="idref" href="#g_bij:85"><span class="id" title="binder">g_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:82"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:83"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#g:84"><span class="id" title="variable">g</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:82"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:83"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#g:84"><span class="id" title="variable">g</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="fun_bijection_is_ZL_bijection" class="idref" href="#fun_bijection_is_ZL_bijection"><span class="id" title="lemma">fun_bijection_is_ZL_bijection</span></a><br/>
+&nbsp;&nbsp;{<a id="U:86" class="idref" href="#U:86"><span class="id" title="binder">U</span></a> <a id="V:87" class="idref" href="#V:87"><span class="id" title="binder">V</span></a> : <span class="id" title="keyword">Type</span>} {<a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#U:86"><span class="id" title="variable">U</span></a>} {<a id="B:89" class="idref" href="#B:89"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:87"><span class="id" title="variable">V</span></a>}<br/>
+&nbsp;&nbsp;(<a id="g:90" class="idref" href="#g:90"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#U:86"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#V:87"><span class="id" title="variable">V</span></a>) (<a id="g_bij:91" class="idref" href="#g_bij:91"><span class="id" title="binder">g_bij</span></a> : <a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection"><span class="id" title="inductive">fun_bijection</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#A:88"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#B:89"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#g:90"><span class="id" title="variable">g</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">FunctionProperties.bijective</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Schutte.PartialFun.html#fun_restr"><span class="id" title="definition">fun_restr</span></a> (<a class="idref" href="hydras.Schutte.PartialFun.html#fun_bijection_codomain"><span class="id" title="lemma">fun_bijection_codomain</span></a> <a class="idref" href="hydras.Schutte.PartialFun.html#g_bij:91"><span class="id" title="variable">g_bij</span></a>)).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.Schutte.html b/theories/html/hydras.Schutte.Schutte.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Schutte.Schutte.html
@@ -0,0 +1,162 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Schutte</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Schutte</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+
+<div class="paragraph"> </div>
+
+  Axiomatic definition of countable ordinal numbers (after Kurt Schutte's
+  "Proof Theory"
+
+<div class="paragraph"> </div>
+
+ Pierre Casteran (LaBRI, University of Bordeaux)
+ with contributions by Florian Hatat (formerly student at ENS Lyon)
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#"><span class="id" title="library">Schutte_basics</span></a> <a class="idref" href="hydras.Schutte.Ordering_Functions.html#"><span class="id" title="library">Ordering_Functions</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Addition.html#"><span class="id" title="library">Addition</span></a> <a class="idref" href="hydras.Schutte.AP.html#"><span class="id" title="library">AP</span></a> <a class="idref" href="hydras.Schutte.CNF.html#"><span class="id" title="library">CNF</span></a> <a class="idref" href="hydras.Schutte.Critical.html#"><span class="id" title="library">Critical</span></a> <a class="idref" href="hydras.Schutte.Correctness_E0.html#"><span class="id" title="library">Correctness_E0</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab205"></a><h1 class="section">Warning</h1>
+
+
+<div class="paragraph"> </div>
+
+This directory contains an adaptation to Coq of the mathematical
+presentation of the set of countable ordinal numbers by K. Schutte.
+
+<div class="paragraph"> </div>
+
+In order to respect as most as possible the style of that presentation,
+we chosed to work in classical logic augmented by Hilbert's <span class="inlinecode">[<span class="id" title="var">epsilon</span>]</span>
+operator.
+
+<div class="paragraph"> </div>
+
+So, all the construction herein is powered by six axioms, three of them are
+Schutte's axioms, and the other ones allow us to work in that "classical" framework, still using the Coq proof assistant and its libraries.
+
+<div class="paragraph"> </div>
+
+<a id="lab206"></a><h2 class="section">Schutte's Axioms</h2>
+
+
+<div class="paragraph"> </div>
+
+ We consider a type <span class="inlinecode"><span class="id" title="var">ON</span></span> (Ordinal numbers), well-ordered by some relation
+ <span class="inlinecode"><span class="id" title="var">lt</span></span>, and such that every subset <span class="inlinecode"><span class="id" title="var">X</span></span> of <span class="inlinecode"><span class="id" title="var">Ord</span></span> is countable iff <span class="inlinecode"><span class="id" title="var">X</span></span> is bounded.
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><br/>
+<span class="id" title="keyword">Axiom</span> <span class="id" title="var">AX1</span> : <span class="id" title="var">WO</span> <span class="id" title="var">lt</span>.<br/>
+<br/>
+<span class="id" title="keyword">Axiom</span> <span class="id" title="var">AX2</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span>: <span class="id" title="var">Ensemble</span> <span class="id" title="var">Ord</span>, (<span class="id" title="tactic">∃</span> <span class="id" title="var">a</span>,  (<span class="id" title="keyword">∀</span> <span class="id" title="var">y</span>, <span class="id" title="var">X</span> <span class="id" title="var">y</span> → <span class="id" title="var">y</span> &lt; <span class="id" title="var">a</span>)) →<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">countable</span> <span class="id" title="var">X</span>.<br/>
+<br/>
+<br/>
+<span class="id" title="keyword">Axiom</span> <span class="id" title="var">AX3</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span> : <span class="id" title="var">Ensemble</span> <span class="id" title="var">Ord</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">countable</span> <span class="id" title="var">X</span> →<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">a</span>,  <span class="id" title="keyword">∀</span> <span class="id" title="var">y</span>, <span class="id" title="var">X</span> <span class="id" title="var">y</span> → <span class="id" title="var">y</span> &lt; <span class="id" title="var">a</span>.<br/>
+
+<div class="paragraph"> </div>
+
+</span>
+<div class="paragraph"> </div>
+
+<a id="lab207"></a><h2 class="section">Classical logic and Hilbert style</h2>
+
+
+<div class="paragraph"> </div>
+
+<a id="lab208"></a><h3 class="section">From <span class="inlinecode"><span class="id" title="var">Coq.Logic.Classical</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><span class="id" title="keyword">Axiom</span> <span class="id" title="var">classic</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span>:<span class="id" title="keyword">Prop</span>, <span class="id" title="var">P</span> ∨ ¬ <span class="id" title="var">P</span>.
+<div class="paragraph"> </div>
+
+</span>
+<div class="paragraph"> </div>
+
+<a id="lab209"></a><h3 class="section">From <span class="inlinecode"><span class="id" title="var">Coq.Logic.Epsilon</span></span></h3>
+
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><br/>
+<span class="id" title="keyword">Axiom</span> <span class="id" title="var">epsilon_statement</span> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">A</span>→<span class="id" title="keyword">Prop</span>), <span class="id" title="var">inhabited</span> <span class="id" title="var">A</span> →<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{ <span class="id" title="var">x</span> : <span class="id" title="var">A</span> | (<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>, <span class="id" title="var">P</span> <span class="id" title="var">x</span>) → <span class="id" title="var">P</span> <span class="id" title="var">x</span> }.<br/>
+
+<div class="paragraph"> </div>
+
+</span>
+<div class="paragraph"> </div>
+
+<a id="lab210"></a><h3 class="section">Needed for <span class="inlinecode"><span class="id" title="var">epsilon</span></span> to work properly.</h3>
+
+
+<div class="paragraph"> </div>
+
+<br/>
+<span class="inlinecode"><span class="id" title="keyword">Axiom</span> <span class="id" title="var">inh_Ord</span> : <span class="id" title="var">inhabited</span> <span class="id" title="var">Ord</span>.<br/>
+
+<div class="paragraph"> </div>
+
+</span>
+</div>
+<div class="code">
+<span class="id" title="keyword">Example</span> <a id="Ex42" class="idref" href="#Ex42"><span class="id" title="definition">Ex42</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> 42 <a class="idref" href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Schutte.AP.html#6b5cc3fabae1095df9f5452df6b2c0df"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.Schutte.AP.html#6b5cc3fabae1095df9f5452df6b2c0df"><span class="id" title="notation">^2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.AP.html#6b5cc3fabae1095df9f5452df6b2c0df"><span class="id" title="notation">omega</span></a><a class="idref" href="hydras.Schutte.AP.html#6b5cc3fabae1095df9f5452df6b2c0df"><span class="id" title="notation">^2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Schutte.AP.html#AP_plus_closed"><span class="id" title="lemma">AP_plus_closed</span></a>. 
+<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.Schutte_basics.html b/theories/html/hydras.Schutte.Schutte_basics.html
new file mode 100644
index 00000000..bf27873b
--- /dev/null
+++ b/theories/html/hydras.Schutte.Schutte_basics.html
@@ -0,0 +1,824 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Schutte_basics</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Schutte_basics</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+ Ordinals of the first and second class,
+   After Kurt Schuttes book : Proof theory
+
+<div class="paragraph"> </div>
+
+ 
+<div class="paragraph"> </div>
+
+   Pierre Casteran, Univ. Bordeaux and  LaBRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="library">Classical</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Classical_sets.html#"><span class="id" title="library">Classical_sets</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">ZornsLemma</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">CountableTypes</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Schutte.Well_Orders.html#"><span class="id" title="library">Well_Orders</span></a> <a class="idref" href="hydras.Schutte.Lub.html#"><span class="id" title="library">Lub</span></a> <a class="idref" href="hydras.Schutte.Countable.html#"><span class="id" title="library">Countable</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="module">Compare_dec</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#"><span class="id" title="module">Coq.Sets.Image</span></a>  <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="module">PartialFun</span></a> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="module">MoreEpsilonIota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Scope</span> <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">schutte_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">sch</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab188"></a><h1 class="section">Definitions</h1>
+
+<div class="paragraph"> </div>
+
+   The type of countable ordinal numbers  
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Parameter</span> <a id="Ord" class="idref" href="#Ord"><span class="id" title="axiom">Ord</span></a> : <span class="id" title="keyword">Type</span>.<br/>
+<span class="id" title="keyword">Parameter</span> <a id="lt" class="idref" href="#lt"><span class="id" title="axiom">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+<span class="id" title="keyword">Infix</span> <a id="::schutte_scope:x_'&lt;'_x" class="idref" href="#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&quot;</span></a>&lt;" := <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="ordinal" class="idref" href="#ordinal"><span class="id" title="abbreviation">ordinal</span></a> := (@<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Full_set"><span class="id" title="inductive">Full_set</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a id="big0" class="idref" href="#big0"><span class="id" title="definition">big0</span></a> <a id="alpha:3" class="idref" href="#alpha:3"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <span class="id" title="keyword">fun</span> <a id="beta:4" class="idref" href="#beta:4"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:4"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:3"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">ordinal_ok</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab189"></a><h1 class="section">The three axioms by Schutte</h1>
+
+<div class="paragraph"> </div>
+
+ First Schutte's axiom  : Ord is well-ordered wrt lt 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="AX1" class="idref" href="#AX1"><span class="id" title="axiom">AX1</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">AX1</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span> <a id="WO_ord" class="idref" href="#WO_ord"><span class="id" title="instance">WO_ord</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> := <a class="idref" href="hydras.Schutte.Schutte_basics.html#AX1"><span class="id" title="axiom">AX1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Stuff for using Coq.Logic.Epsilon 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="inh_Ord" class="idref" href="#inh_Ord"><span class="id" title="axiom">inh_Ord</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#inhabited"><span class="id" title="inductive">inhabited</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="InH_Ord" class="idref" href="#InH_Ord"><span class="id" title="instance">InH_Ord</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>.  
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Inh_OSets" class="idref" href="#Inh_OSets"><span class="id" title="instance">Inh_OSets</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>).<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="Inh_Ord_Ord" class="idref" href="#Inh_Ord_Ord"><span class="id" title="instance">Inh_Ord_Ord</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span>  <span class="id" title="var">AX1</span> <span class="id" title="var">Inh_Ord_Ord</span> <span class="id" title="var">Inh_OSets</span> <span class="id" title="var">inh_Ord</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="le" class="idref" href="#le"><span class="id" title="definition">le</span></a> := <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Infix</span> <a id="13c3c4e605a8e463a22f054036895bb7" class="idref" href="#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">&quot;</span></a>≤" := <a class="idref" href="hydras.Schutte.Schutte_basics.html#le"><span class="id" title="definition">le</span></a> : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Second and third axioms from Schutte
+
+<div class="paragraph"> </div>
+
+A subset X of Ord is bounded iff X is countable 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="AX2" class="idref" href="#AX2"><span class="id" title="axiom">AX2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="X:10" class="idref" href="#X:10"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:12" class="idref" href="#y:12"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:10"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:12"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:11"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:10"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="AX3" class="idref" href="#AX3"><span class="id" title="axiom">AX3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="X:14" class="idref" href="#X:14"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:14"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:15" class="idref" href="#a:15"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <span class="id" title="keyword">∀</span> <a id="y:16" class="idref" href="#y:16"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:14"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:16"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:16"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:15"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab190"></a><h1 class="section">First definitions</h1>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="ge" class="idref" href="#ge"><span class="id" title="definition">ge</span></a> <a id="alpha:18" class="idref" href="#alpha:18"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <span class="id" title="keyword">fun</span> <a id="beta:19" class="idref" href="#beta:19"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:18"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:19"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Unbounded" class="idref" href="#Unbounded"><span class="id" title="definition">Unbounded</span></a> (<a id="X:20" class="idref" href="#X:20"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:22" class="idref" href="#y:22"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:20"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:22"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:21"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:22"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="zero" class="idref" href="#zero"><span class="id" title="definition">zero</span></a> := <a class="idref" href="hydras.Schutte.Well_Orders.html#the_least"><span class="id" title="definition">the_least</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succ" class="idref" href="#succ"><span class="id" title="definition">succ</span></a> (<a id="alpha:23" class="idref" href="#alpha:23"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.Schutte.Well_Orders.html#the_least"><span class="id" title="definition">the_least</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:23"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:24"><span class="id" title="variable">beta</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Finite ordinals 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="finite" class="idref" href="#finite"><span class="id" title="definition">finite</span></a> (<a id="i:25" class="idref" href="#i:25"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:25"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span> 0 ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">i</span> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#finite:26"><span class="id" title="definition">finite</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:25"><span class="id" title="variable">i</span></a>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="F" class="idref" href="#F"><span class="id" title="abbreviation">F</span></a> <span class="id" title="var">i</span> := (<a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">finite</span></a> <span class="id" title="var">i</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">finite</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">:</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">nat</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">Ord</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="is_finite" class="idref" href="#is_finite"><span class="id" title="definition">is_finite</span></a>  := <a class="idref" href="hydras.Schutte.Countable.html#seq_range"><span class="id" title="definition">seq_range</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">finite</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab191"></a><h2 class="section">Limits</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sup_spec" class="idref" href="#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a id="X:28" class="idref" href="#X:28"><span class="id" title="binder">X</span></a> <a id="lambda:29" class="idref" href="#lambda:29"><span class="id" title="binder">lambda</span></a> := <a class="idref" href="hydras.Schutte.Lub.html#is_lub"><span class="id" title="definition">is_lub</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:28"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lambda:29"><span class="id" title="variable">lambda</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="sup" class="idref" href="#sup"><span class="id" title="definition">sup</span></a> (<a id="X:30" class="idref" href="#X:30"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#the"><span class="id" title="definition">the</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:30"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="abe2b27667df40e2c3f74853287f2063" class="idref" href="#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">&quot;</span></a>'|_|' X" := (<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> <span class="id" title="var">X</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 29) : <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Limit of omega-sequences 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="omega_limit" class="idref" href="#omega_limit"><span class="id" title="definition">omega_limit</span></a> (<a id="s:31" class="idref" href="#s:31"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Countable.html#seq_range"><span class="id" title="definition">seq_range</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:31"><span class="id" title="variable">s</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="_omega" class="idref" href="#_omega"><span class="id" title="definition">_omega</span></a> := <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#finite"><span class="id" title="definition">finite</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a id="omega" class="idref" href="#omega"><span class="id" title="abbreviation">omega</span></a> := (<a class="idref" href="hydras.Schutte.Schutte_basics.html#_omega"><span class="id" title="definition">_omega</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Successor and limit ordinals 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="is_succ" class="idref" href="#is_succ"><span class="id" title="definition">is_succ</span></a> (<a id="alpha:32" class="idref" href="#alpha:32"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:33" class="idref" href="#beta:33"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:32"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:33"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="is_limit" class="idref" href="#is_limit"><span class="id" title="definition">is_limit</span></a> (<a id="alpha:34" class="idref" href="#alpha:34"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:34"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_succ"><span class="id" title="definition">is_succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:34"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Ordinals considered as sets 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="members" class="idref" href="#members"><span class="id" title="definition">members</span></a> (<a id="a:35" class="idref" href="#a:35"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) := (<span class="id" title="keyword">fun</span> <a id="b:36" class="idref" href="#b:36"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:36"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:35"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="set_eq" class="idref" href="#set_eq"><span class="id" title="definition">set_eq</span></a> (<a id="X:37" class="idref" href="#X:37"><span class="id" title="binder">X</span></a> <a id="Y:38" class="idref" href="#Y:38"><span class="id" title="binder">Y</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) := <span class="id" title="keyword">∀</span> <a id="a:39" class="idref" href="#a:39"><span class="id" title="binder">a</span></a>,  (<a class="idref" href="hydras.Schutte.Schutte_basics.html#X:37"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:39"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:38"><span class="id" title="variable">Y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:39"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Induction (after Schutte) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="progressive" class="idref" href="#progressive"><span class="id" title="definition">progressive</span></a> (<a id="P:40" class="idref" href="#P:40"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:41" class="idref" href="#a:41"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:42" class="idref" href="#b:42"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:42"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:41"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#P:40"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:42"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#P:40"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:41"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Closed" class="idref" href="#Closed"><span class="id" title="definition">Closed</span></a> (<a id="B:43" class="idref" href="#B:43"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="M:44" class="idref" href="#M:44"><span class="id" title="binder">M</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#M:44"><span class="id" title="variable">M</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#B:43"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#M:44"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#M:44"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#B:43"><span class="id" title="variable">B</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#M:44"><span class="id" title="variable">M</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span>  <a id="continuous" class="idref" href="#continuous"><span class="id" title="definition">continuous</span></a> (<a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a> <a id="B:47" class="idref" href="#B:47"><span class="id" title="binder">B</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.PartialFun.html#fun_codomain"><span class="id" title="definition">fun_codomain</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#A:46"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#B:47"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#Closed"><span class="id" title="definition">Closed</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#A:46"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="U:48" class="idref" href="#U:48"><span class="id" title="binder">U</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#U:48"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#A:46"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#U:48"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#U:48"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.PartialFun.html#image"><span class="id" title="definition">image</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#U:48"><span class="id" title="variable">U</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#f:45"><span class="id" title="variable">f</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#f:45"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#U:48"><span class="id" title="variable">U</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab192"></a><h1 class="section">Basic properties</h1>
+
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Unbounded_not_countable" class="idref" href="#Unbounded_not_countable"><span class="id" title="lemma">Unbounded_not_countable</span></a> (<a id="X:49" class="idref" href="#X:49"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:49"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#not"><span class="id" title="definition">not</span></a> (<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:49"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="countable_not_Unbounded" class="idref" href="#countable_not_Unbounded"><span class="id" title="lemma">countable_not_Unbounded</span></a> : <span class="id" title="keyword">∀</span> <a id="X:50" class="idref" href="#X:50"><span class="id" title="binder">X</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:50"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#not"><span class="id" title="definition">not</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:50"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Progressive_Acc" class="idref" href="#Progressive_Acc"><span class="id" title="lemma">Progressive_Acc</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#progressive"><span class="id" title="definition">progressive</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="transfinite_induction" class="idref" href="#transfinite_induction"><span class="id" title="lemma">transfinite_induction</span></a> (<a id="P:51" class="idref" href="#P:51"><span class="id" title="binder">P</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><span class="id" title="keyword">Prop</span>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#progressive"><span class="id" title="definition">progressive</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#P:51"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:52" class="idref" href="#a:52"><span class="id" title="binder">a</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#P:51"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:52"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab193"></a><h2 class="section">Properties of <span class="inlinecode"><span class="id" title="var">le</span></span> and <span class="inlinecode"><span class="id" title="var">lt</span></span></h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_refl" class="idref" href="#le_refl"><span class="id" title="lemma">le_refl</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:61" class="idref" href="#alpha:61"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:61"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:61"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">le_refl</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="eq_le" class="idref" href="#eq_le"><span class="id" title="lemma">eq_le</span></a> : <span class="id" title="keyword">∀</span> <a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a> <a id="b:63" class="idref" href="#b:63"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:62"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:63"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:62"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:63"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_le" class="idref" href="#lt_le"><span class="id" title="lemma">lt_le</span></a> : <span class="id" title="keyword">∀</span> <a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a> <a id="b:65" class="idref" href="#b:65"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:65"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:65"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_trans" class="idref" href="#lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:66" class="idref" href="#a:66"><span class="id" title="binder">a</span></a> <a id="b:67" class="idref" href="#b:67"><span class="id" title="binder">b</span></a> <a id="c:68" class="idref" href="#c:68"><span class="id" title="binder">c</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:67"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:67"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:68"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:66"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:68"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_lt_trans" class="idref" href="#le_lt_trans"><span class="id" title="lemma">le_lt_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a> <a id="b:70" class="idref" href="#b:70"><span class="id" title="binder">b</span></a> <a id="c:71" class="idref" href="#c:71"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:71"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:71"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_le_trans" class="idref" href="#lt_le_trans"><span class="id" title="lemma">lt_le_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:72" class="idref" href="#a:72"><span class="id" title="binder">a</span></a> <a id="b:73" class="idref" href="#b:73"><span class="id" title="binder">b</span></a> <a id="c:74" class="idref" href="#c:74"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:73"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:73"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:74"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:72"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:74"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_trans" class="idref" href="#le_trans"><span class="id" title="lemma">le_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="a:75" class="idref" href="#a:75"><span class="id" title="binder">a</span></a> <a id="b:76" class="idref" href="#b:76"><span class="id" title="binder">b</span></a> <a id="c:77" class="idref" href="#c:77"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:76"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:77"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:75"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#c:77"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_irrefl" class="idref" href="#lt_irrefl"><span class="id" title="lemma">lt_irrefl</span></a> : <span class="id" title="keyword">∀</span> <a id="a:78" class="idref" href="#a:78"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#a:78"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:78"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_antisym" class="idref" href="#le_antisym"><span class="id" title="lemma">le_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="a:79" class="idref" href="#a:79"><span class="id" title="binder">a</span></a> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:80"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:80"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:80"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_eq_or_lt" class="idref" href="#le_eq_or_lt"><span class="id" title="lemma">le_eq_or_lt</span></a> : <span class="id" title="keyword">∀</span> <a id="a:81" class="idref" href="#a:81"><span class="id" title="binder">a</span></a> <a id="b:82" class="idref" href="#b:82"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:81"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:82"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:81"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:82"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:81"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:82"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_not_gt" class="idref" href="#le_not_gt"><span class="id" title="lemma">le_not_gt</span></a> : <span class="id" title="keyword">∀</span> <a id="a:83" class="idref" href="#a:83"><span class="id" title="binder">a</span></a> <a id="b:84" class="idref" href="#b:84"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:83"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:84"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:84"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:83"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">eq_le</span> <span class="id" title="var">lt_le</span> <span class="id" title="var">lt_trans</span> <span class="id" title="var">le_trans</span> <span class="id" title="var">le_lt_trans</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lt_le_trans</span> <span class="id" title="var">lt_irrefl</span> <span class="id" title="var">le_not_gt</span>:<br/>
+&nbsp;&nbsp;<span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_disj" class="idref" href="#le_disj"><span class="id" title="lemma">le_disj</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:85" class="idref" href="#alpha:85"><span class="id" title="binder">alpha</span></a> <a id="beta:86" class="idref" href="#beta:86"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:85"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:86"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:85"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:86"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:85"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:86"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="all_ord_acc" class="idref" href="#all_ord_acc"><span class="id" title="lemma">all_ord_acc</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:87" class="idref" href="#alpha:87"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:87"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="trichotomy" class="idref" href="#trichotomy"><span class="id" title="lemma">trichotomy</span></a> : <span class="id" title="keyword">∀</span> <a id="a:88" class="idref" href="#a:88"><span class="id" title="binder">a</span></a> <a id="b:89" class="idref" href="#b:89"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:88"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:89"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:88"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_or_ge" class="idref" href="#lt_or_ge"><span class="id" title="lemma">lt_or_ge</span></a> : <span class="id" title="keyword">∀</span> <a id="a:90" class="idref" href="#a:90"><span class="id" title="binder">a</span></a> <a id="b:91" class="idref" href="#b:91"><span class="id" title="binder">b</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:90"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:91"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:90"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_gt_le" class="idref" href="#not_gt_le"><span class="id" title="lemma">not_gt_le</span></a> : <span class="id" title="keyword">∀</span> <a id="a:92" class="idref" href="#a:92"><span class="id" title="binder">a</span></a> <a id="b:93" class="idref" href="#b:93"><span class="id" title="binder">b</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:93"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:92"><span class="id" title="variable">a</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#b:93"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab194"></a><h2 class="section">Global properties</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="Non_denum" class="idref" href="#Non_denum"><span class="id" title="lemma">Non_denum</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Inh_ord" class="idref" href="#Inh_ord"><span class="id" title="lemma">Inh_ord</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="unbounded" class="idref" href="#unbounded"><span class="id" title="lemma">unbounded</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:94" class="idref" href="#alpha:94"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:95" class="idref" href="#beta:95"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:94"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:95"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="the_least_ok" class="idref" href="#the_least_ok"><span class="id" title="lemma">the_least_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="X:98" class="idref" href="#X:98"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:98"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="a:99" class="idref" href="#a:99"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:98"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:99"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_least"><span class="id" title="definition">the_least</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:98"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#a:99"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab195"></a><h2 class="section">About zero</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_le" class="idref" href="#zero_le"><span class="id" title="lemma">zero_le</span></a> (<a id="alpha:100" class="idref" href="#alpha:100"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:100"><span class="id" title="variable">alpha</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_lt_zero" class="idref" href="#not_lt_zero"><span class="id" title="lemma">not_lt_zero</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:101" class="idref" href="#alpha:101"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:101"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_or_greater" class="idref" href="#zero_or_greater"><span class="id" title="lemma">zero_or_greater</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:102" class="idref" href="#alpha:102"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:102"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:103" class="idref" href="#beta:103"><span class="id" title="binder">beta</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:103"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:102"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_or_positive" class="idref" href="#zero_or_positive"><span class="id" title="lemma">zero_or_positive</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:104" class="idref" href="#alpha:104"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:104"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:104"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab196"></a><h2 class="section">Properties of successor</h2>
+
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="succ_spec" class="idref" href="#succ_spec"><span class="id" title="definition">succ_spec</span></a> (<a id="alpha:105" class="idref" href="#alpha:105"><span class="id" title="binder">alpha</span></a>:<a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> (<span class="id" title="keyword">fun</span> <a id="z:106" class="idref" href="#z:106"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:105"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#z:106"><span class="id" title="variable">z</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_ok" class="idref" href="#succ_ok"><span class="id" title="lemma">succ_ok</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:107" class="idref" href="#alpha:107"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ_spec"><span class="id" title="definition">succ_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:107"><span class="id" title="variable">alpha</span></a>  (<a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:107"><span class="id" title="variable">alpha</span></a>).  
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ" class="idref" href="#lt_succ"><span class="id" title="lemma">lt_succ</span></a> (<a id="alpha:108" class="idref" href="#alpha:108"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:108"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:108"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">lt_succ</span> : <span class="id" title="var">schutte</span>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le" class="idref" href="#lt_succ_le"><span class="id" title="lemma">lt_succ_le</span></a> (<a id="alpha:109" class="idref" href="#alpha:109"><span class="id" title="binder">alpha</span></a> <a id="beta:110" class="idref" href="#beta:110"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:110"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:109"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:110"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_le_2" class="idref" href="#lt_succ_le_2"><span class="id" title="lemma">lt_succ_le_2</span></a> (<a id="alpha:111" class="idref" href="#alpha:111"><span class="id" title="binder">alpha</span></a> <a id="beta:112" class="idref" href="#beta:112"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:112"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:111"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:112"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_mono" class="idref" href="#succ_mono"><span class="id" title="lemma">succ_mono</span></a> (<a id="alpha:113" class="idref" href="#alpha:113"><span class="id" title="binder">alpha</span></a> <a id="beta:114" class="idref" href="#beta:114"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:114"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:113"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:114"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ_mono"><span class="id" title="lemma">succ_mono</span></a> [ <span class="id" title="var">alpha</span> <span class="id" title="keyword">beta</span>].<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_monoR" class="idref" href="#succ_monoR"><span class="id" title="lemma">succ_monoR</span></a> (<a id="alpha:115" class="idref" href="#alpha:115"><span class="id" title="binder">alpha</span></a> <a id="beta:116" class="idref" href="#beta:116"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:115"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:116"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:115"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:116"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_injection" class="idref" href="#succ_injection"><span class="id" title="lemma">succ_injection</span></a> (<a id="alpha:117" class="idref" href="#alpha:117"><span class="id" title="binder">alpha</span></a> <a id="beta:118" class="idref" href="#beta:118"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:117"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:118"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:117"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:118"><span class="id" title="variable">beta</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="succ_zero_diff" class="idref" href="#succ_zero_diff"><span class="id" title="lemma">succ_zero_diff</span></a> (<a id="alpha:119" class="idref" href="#alpha:119"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:119"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_lt_succ" class="idref" href="#zero_lt_succ"><span class="id" title="lemma">zero_lt_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="alpha:120" class="idref" href="#alpha:120"><span class="id" title="binder">alpha</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:120"><span class="id" title="variable">alpha</span></a>. 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_succ_lt" class="idref" href="#lt_succ_lt"><span class="id" title="lemma">lt_succ_lt</span></a> (<a id="alpha:121" class="idref" href="#alpha:121"><span class="id" title="binder">alpha</span></a> <a id="beta:122" class="idref" href="#beta:122"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:122"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:121"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:122"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:121"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:122"><span class="id" title="variable">beta</span></a>. 
+<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">zero_lt_succ</span> <span class="id" title="var">zero_le</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Less than finite is finite ... 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_lt_inv" class="idref" href="#finite_lt_inv"><span class="id" title="lemma">finite_lt_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="i:123" class="idref" href="#i:123"><span class="id" title="binder">i</span></a> <a id="o:124" class="idref" href="#o:124"><span class="id" title="binder">o</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#o:124"><span class="id" title="variable">o</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:123"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="j:125" class="idref" href="#j:125"><span class="id" title="binder">j</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#j:125"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#i:123"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#o:124"><span class="id" title="variable">o</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:125"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_mono" class="idref" href="#finite_mono"><span class="id" title="lemma">finite_mono</span></a> : <span class="id" title="keyword">∀</span> <a id="i:126" class="idref" href="#i:126"><span class="id" title="binder">i</span></a> <a id="j:127" class="idref" href="#j:127"><span class="id" title="binder">j</span></a>, (<a class="idref" href="hydras.Schutte.Schutte_basics.html#i:126"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#j:127"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:126"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:127"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">finite_mono</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_inj" class="idref" href="#finite_inj"><span class="id" title="lemma">finite_inj</span></a> : <span class="id" title="keyword">∀</span> <a id="i:128" class="idref" href="#i:128"><span class="id" title="binder">i</span></a> <a id="j:129" class="idref" href="#j:129"><span class="id" title="binder">j</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:128"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:129"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:128"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:129"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab197"></a><h2 class="section">Building limits</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_exists" class="idref" href="#sup_exists"><span class="id" title="lemma">sup_exists</span></a> : <span class="id" title="keyword">∀</span> <a id="X:130" class="idref" href="#X:130"><span class="id" title="binder">X</span></a>, <span class="id" title="definition">Countable</span>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:130"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ex"><span class="id" title="inductive">ex</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:130"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_unicity" class="idref" href="#sup_unicity"><span class="id" title="lemma">sup_unicity</span></a> : <span class="id" title="keyword">∀</span> <a id="X:137" class="idref" href="#X:137"><span class="id" title="binder">X</span></a> <a id="l:138" class="idref" href="#l:138"><span class="id" title="binder">l</span></a> <a id="l':139" class="idref" href="#l':139"><span class="id" title="binder">l'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="definition">Countable</span>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:137"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="x:140" class="idref" href="#x:140"><span class="id" title="binder">x</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:137"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:140"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:140"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:137"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#l:138"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:137"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#l':139"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#l:138"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#l':139"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_spec_unicity" class="idref" href="#sup_spec_unicity"><span class="id" title="lemma">sup_spec_unicity</span></a> (<a id="X:141" class="idref" href="#X:141"><span class="id" title="binder">X</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="HX:142" class="idref" href="#HX:142"><span class="id" title="binder">HX</span></a> : <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:141"><span class="id" title="variable">X</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="u:143" class="idref" href="#u:143"><span class="id" title="binder">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:141"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#u:143"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_ok1" class="idref" href="#sup_ok1"><span class="id" title="lemma">sup_ok1</span></a> (<a id="X:144" class="idref" href="#X:144"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)(<a id="HX:145" class="idref" href="#HX:145"><span class="id" title="binder">HX</span></a> : <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:144"><span class="id" title="variable">X</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup_spec"><span class="id" title="definition">sup_spec</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:144"><span class="id" title="variable">X</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:144"><span class="id" title="variable">X</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_upper_bound" class="idref" href="#sup_upper_bound"><span class="id" title="lemma">sup_upper_bound</span></a> (<a id="X:146" class="idref" href="#X:146"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="alpha:147" class="idref" href="#alpha:147"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:146"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:146"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:147"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:147"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:146"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_least_upper_bound" class="idref" href="#sup_least_upper_bound"><span class="id" title="lemma">sup_least_upper_bound</span></a> (<a id="X:148" class="idref" href="#X:148"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="alpha:149" class="idref" href="#alpha:149"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:148"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:150" class="idref" href="#y:150"><span class="id" title="binder">y</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:148"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:150"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:150"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:149"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:148"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:149"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_of_leq" class="idref" href="#sup_of_leq"><span class="id" title="lemma">sup_of_leq</span></a> (<a id="alpha:151" class="idref" href="#alpha:151"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:151"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <a id="x:152" class="idref" href="#x:152"><span class="id" title="binder">x</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> ⇒  <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:152"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:151"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_mono" class="idref" href="#sup_mono"><span class="id" title="lemma">sup_mono</span></a> (<a id="X:155" class="idref" href="#X:155"><span class="id" title="binder">X</span></a> <a id="Y:156" class="idref" href="#Y:156"><span class="id" title="binder">Y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:155"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:156"><span class="id" title="variable">Y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:157" class="idref" href="#x:157"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:155"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:158" class="idref" href="#y:158"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:156"><span class="id" title="variable">Y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:158"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:157"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:158"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:155"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:156"><span class="id" title="variable">Y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_eq_intro" class="idref" href="#sup_eq_intro"><span class="id" title="lemma">sup_eq_intro</span></a> (<a id="X:159" class="idref" href="#X:159"><span class="id" title="binder">X</span></a> <a id="Y:160" class="idref" href="#Y:160"><span class="id" title="binder">Y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:159"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:160"><span class="id" title="variable">Y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:159"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:160"><span class="id" title="variable">Y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:160"><span class="id" title="variable">Y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:159"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:159"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Y:160"><span class="id" title="variable">Y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_sup_exists_leq" class="idref" href="#lt_sup_exists_leq"><span class="id" title="lemma">lt_sup_exists_leq</span></a> (<a id="X:161" class="idref" href="#X:161"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:161"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:162" class="idref" href="#y:162"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:162"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:161"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:163" class="idref" href="#x:163"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:161"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:163"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:162"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:163"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_sup_exists_lt" class="idref" href="#lt_sup_exists_lt"><span class="id" title="lemma">lt_sup_exists_lt</span></a> (<a id="X:168" class="idref" href="#X:168"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:168"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:169" class="idref" href="#y:169"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:169"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:168"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:170" class="idref" href="#x:170"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:168"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:170"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:169"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:170"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="members_eq" class="idref" href="#members_eq"><span class="id" title="lemma">members_eq</span></a> (<a id="alpha:175" class="idref" href="#alpha:175"><span class="id" title="binder">alpha</span></a> <a id="beta:176" class="idref" href="#beta:176"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:175"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:176"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:175"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:176"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_members_succ" class="idref" href="#sup_members_succ"><span class="id" title="lemma">sup_members_succ</span></a> (<a id="alpha:177" class="idref" href="#alpha:177"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:177"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:177"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:177"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:177"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_members_not_succ" class="idref" href="#sup_members_not_succ"><span class="id" title="lemma">sup_members_not_succ</span></a> (<a id="alpha:178" class="idref" href="#alpha:178"><span class="id" title="binder">alpha</span></a> <a id="beta:179" class="idref" href="#beta:179"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:178"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:178"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:178"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:179"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab198"></a><h2 class="section">Sequences of ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="seq_mono" class="idref" href="#seq_mono"><span class="id" title="definition">seq_mono</span></a> (<a id="s:180" class="idref" href="#s:180"><span class="id" title="binder">s</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:181" class="idref" href="#i:181"><span class="id" title="binder">i</span></a> <a id="j:182" class="idref" href="#j:182"><span class="id" title="binder">j</span></a>, (<a class="idref" href="hydras.Schutte.Schutte_basics.html#i:181"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:182"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">nat</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:180"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:181"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:180"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:182"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="seq_mono_intro" class="idref" href="#seq_mono_intro"><span class="id" title="lemma">seq_mono_intro</span></a> (<a id="s:183" class="idref" href="#s:183"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:184" class="idref" href="#i:184"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:183"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:184"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:183"><span class="id" title="variable">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:184"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#seq_mono"><span class="id" title="definition">seq_mono</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:183"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="seq_mono_inj" class="idref" href="#seq_mono_inj"><span class="id" title="lemma">seq_mono_inj</span></a> (<a id="s:185" class="idref" href="#s:185"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#seq_mono"><span class="id" title="definition">seq_mono</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:185"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:185"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">Countable.seq_range_countable</span> <span class="id" title="var">seq_mono_intro</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Full_set"><span class="id" title="inductive">Full_set</span></a>: <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_limit" class="idref" href="#lt_omega_limit"><span class="id" title="lemma">lt_omega_limit</span></a> (<a id="s:186" class="idref" href="#s:186"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#seq_mono"><span class="id" title="definition">seq_mono</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:186"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="i:187" class="idref" href="#i:187"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:186"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:187"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:186"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_limit_least" class="idref" href="#omega_limit_least"><span class="id" title="lemma">omega_limit_least</span></a> (<a id="s:188" class="idref" href="#s:188"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#seq_mono"><span class="id" title="definition">seq_mono</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:188"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:189" class="idref" href="#y:189"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:190" class="idref" href="#i:190"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:188"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:190"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:189"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:188"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#y:189"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_limit_lt_exists_lt" class="idref" href="#lt_omega_limit_lt_exists_lt"><span class="id" title="lemma">lt_omega_limit_lt_exists_lt</span></a> (<a id="alpha:191" class="idref" href="#alpha:191"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="s:192" class="idref" href="#s:192"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:193" class="idref" href="#i:193"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:193"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:192"><span class="id" title="variable">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:193"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:191"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="j:194" class="idref" href="#j:194"><span class="id" title="binder">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:191"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:192"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:194"><span class="id" title="variable">j</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="omega_limit_least_gt" class="idref" href="#omega_limit_least_gt"><span class="id" title="lemma">omega_limit_least_gt</span></a> (<a id="alpha:197" class="idref" href="#alpha:197"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) (<a id="s:198" class="idref" href="#s:198"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="Hmono:200" class="idref" href="#Hmono:200"><span class="id" title="binder">Hmono</span></a> : <span class="id" title="keyword">∀</span> <a id="i:199" class="idref" href="#i:199"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:198"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:199"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:198"><span class="id" title="variable">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:199"><span class="id" title="variable">i</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="H:201" class="idref" href="#H:201"><span class="id" title="binder">H</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:197"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega_limit"><span class="id" title="definition">omega_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:198"><span class="id" title="variable">s</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:202" class="idref" href="#i:202"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a> (<span class="id" title="keyword">fun</span> <a id="j:203" class="idref" href="#j:203"><span class="id" title="binder">j</span></a> ⇒  <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:197"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#s:198"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#j:203"><span class="id" title="variable">j</span></a>) <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:202"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab199"></a><h2 class="section">Properties of omega</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_lt_omega" class="idref" href="#finite_lt_omega"><span class="id" title="lemma">finite_lt_omega</span></a> (<a id="i:206" class="idref" href="#i:206"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:206"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="zero_lt_omega" class="idref" href="#zero_lt_omega"><span class="id" title="lemma">zero_lt_omega</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_omega_finite" class="idref" href="#lt_omega_finite"><span class="id" title="lemma">lt_omega_finite</span></a> (<a id="alpha:207" class="idref" href="#alpha:207"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:207"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:208" class="idref" href="#i:208"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:207"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:208"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_omega" class="idref" href="#is_limit_omega"><span class="id" title="lemma">is_limit_omega</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab200"></a><h2 class="section">About zero, is_succ and is_limit</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_is_succ_zero" class="idref" href="#not_is_succ_zero"><span class="id" title="lemma">not_is_succ_zero</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_succ"><span class="id" title="definition">is_succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_is_limit_zero" class="idref" href="#not_is_limit_zero"><span class="id" title="lemma">not_is_limit_zero</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_is_limit_succ" class="idref" href="#not_is_limit_succ"><span class="id" title="lemma">not_is_limit_succ</span></a> (<a id="alpha:209" class="idref" href="#alpha:209"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :  <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_succ"><span class="id" title="definition">is_succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:209"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:209"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_is_succ_limit" class="idref" href="#not_is_succ_limit"><span class="id" title="lemma">not_is_succ_limit</span></a> (<a id="alpha:210" class="idref" href="#alpha:210"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) : <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:210"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_succ"><span class="id" title="definition">is_succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:210"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab201"></a><h2 class="section">About members</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="countable_members" class="idref" href="#countable_members"><span class="id" title="lemma">countable_members</span></a> (<a id="alpha:211" class="idref" href="#alpha:211"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>): <span class="id" title="definition">Countable</span> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:211"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">countable_members</span> : <span class="id" title="var">schutte</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_sup_members" class="idref" href="#le_sup_members"><span class="id" title="lemma">le_sup_members</span></a> (<a id="alpha:212" class="idref" href="#alpha:212"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :  <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:212"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:212"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="is_limit_sup_members" class="idref" href="#is_limit_sup_members"><span class="id" title="lemma">is_limit_sup_members</span></a> (<a id="alpha:213" class="idref" href="#alpha:213"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:213"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:213"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">_</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">|</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:213"><span class="id" title="variable">alpha</span></a><a class="idref" href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="sup_members_disj" class="idref" href="#sup_members_disj"><span class="id" title="lemma">sup_members_disj</span></a> (<a id="alpha:214" class="idref" href="#alpha:214"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:214"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#sup"><span class="id" title="definition">sup</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:214"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:214"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:214"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="classification" class="idref" href="#classification"><span class="id" title="lemma">classification</span></a> (<a id="alpha:215" class="idref" href="#alpha:215"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:215"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_succ"><span class="id" title="definition">is_succ</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:215"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a>  <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:215"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab202"></a><h2 class="section">Extensional equalities on sets of ordinals</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="members_omega" class="idref" href="#members_omega"><span class="id" title="lemma">members_omega</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Same_set"><span class="id" title="definition">Same_set</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#omega"><span class="id" title="abbreviation">omega</span></a>) <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_finite"><span class="id" title="definition">is_finite</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Not_Unbounded_bounded" class="idref" href="#Not_Unbounded_bounded"><span class="id" title="lemma">Not_Unbounded_bounded</span></a> (<a id="X:216" class="idref" href="#X:216"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:216"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="beta:217" class="idref" href="#beta:217"><span class="id" title="binder">beta</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <a id="x:218" class="idref" href="#x:218"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:216"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:218"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#x:218"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#beta:217"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Not_Unbounded_countable" class="idref" href="#Not_Unbounded_countable"><span class="id" title="lemma">Not_Unbounded_countable</span></a> (<a id="X:227" class="idref" href="#X:227"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:227"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:227"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="not_countable_unbounded" class="idref" href="#not_countable_unbounded"><span class="id" title="lemma">not_countable_unbounded</span></a> (<a id="X:228" class="idref" href="#X:228"><span class="id" title="binder">X</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>):<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <span class="id" title="definition">Countable</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:228"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#Unbounded"><span class="id" title="definition">Unbounded</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#X:228"><span class="id" title="variable">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="le_alpha_zero" class="idref" href="#le_alpha_zero"><span class="id" title="lemma">le_alpha_zero</span></a> (<a id="alpha:229" class="idref" href="#alpha:229"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:229"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:229"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#zero"><span class="id" title="definition">zero</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="finite_not_limit" class="idref" href="#finite_not_limit"><span class="id" title="lemma">finite_not_limit</span></a> (<a id="i:230" class="idref" href="#i:230"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#is_limit"><span class="id" title="definition">is_limit</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#i:230"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="iota_demo.zero" class="idref" href="#iota_demo.zero"><span class="id" title="definition">zero</span></a>: <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Epsilon.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#inh_Ord"><span class="id" title="axiom">inh_Ord</span></a> (<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#ordinal"><span class="id" title="abbreviation">ordinal</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="iota_demo.zero_le" class="idref" href="#iota_demo.zero_le"><span class="id" title="lemma">zero_le</span></a> (<a id="alpha:232" class="idref" href="#alpha:232"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#Ord"><span class="id" title="axiom">Ord</span></a>) :  <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#alpha:232"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="iota_demo.Bad" class="idref" href="#iota_demo.Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="iota_demo.Bad.bottom" class="idref" href="#iota_demo.Bad.bottom"><span class="id" title="definition">bottom</span></a> := <a class="idref" href="hydras.Schutte.Well_Orders.html#the_least"><span class="id" title="definition">the_least</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Empty_set"><span class="id" title="inductive">Empty_set</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_demo.Bad.le_zero_bottom" class="idref" href="#iota_demo.Bad.le_zero_bottom"><span class="id" title="lemma">le_zero_bottom</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.zero"><span class="id" title="definition">zero</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom"><span class="id" title="definition">bottom</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_demo.Bad.bottom_eq" class="idref" href="#iota_demo.Bad.bottom_eq"><span class="id" title="lemma">bottom_eq</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom"><span class="id" title="definition">bottom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom"><span class="id" title="definition">bottom</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="iota_demo.Bad.le_bottom_zero" class="idref" href="#iota_demo.Bad.le_bottom_zero"><span class="id" title="lemma">le_bottom_zero</span></a> : <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom"><span class="id" title="definition">bottom</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.zero"><span class="id" title="definition">zero</span></a>.  
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad"><span class="id" title="module">Bad</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Schutte_basics.html#iota_demo"><span class="id" title="module">iota_demo</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.Schutte.Well_Orders.html b/theories/html/hydras.Schutte.Well_Orders.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.Schutte.Well_Orders.html
@@ -0,0 +1,173 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.Schutte.Well_Orders</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.Schutte.Well_Orders</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ Well ordered sets (after Schutte) 
+<div class="paragraph"> </div>
+
+  Pierre Casteran LaBRI, Universite de Bordeaux  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Logic.Classical.html#"><span class="id" title="library">Classical</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Classical_sets.html#"><span class="id" title="library">Classical_sets</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#"><span class="id" title="library">RelationClasses</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.PartialFun.html#"><span class="id" title="library">PartialFun</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#"><span class="id" title="module">MoreEpsilonIota</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> [<span class="id" title="var">U</span>].<br/>
+<span class="id" title="keyword">Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> [<span class="id" title="var">U</span>].<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="the_context" class="idref" href="#the_context"><span class="id" title="section">the_context</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="the_context.M" class="idref" href="#the_context.M"><span class="id" title="variable">M</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="the_context.Lt" class="idref" href="#the_context.Lt"><span class="id" title="variable">Lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#M:1"><span class="id" title="variable">M</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Le" class="idref" href="#Le"><span class="id" title="definition">Le</span></a> (<a id="a:3" class="idref" href="#a:3"><span class="id" title="binder">a</span></a> <a id="b:4" class="idref" href="#b:4"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) :=  <a class="idref" href="hydras.Schutte.Well_Orders.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:4"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:3"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:4"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="least_member" class="idref" href="#least_member"><span class="id" title="definition">least_member</span></a>  (<a id="X:5" class="idref" href="#X:5"><span class="id" title="binder">X</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) (<a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:5"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:5"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:6"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:7"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="least_fixpoint" class="idref" href="#least_fixpoint"><span class="id" title="definition">least_fixpoint</span></a> (<a id="f:8" class="idref" href="#f:8"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) (<a id="x:9" class="idref" href="#x:9"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="y:10" class="idref" href="#y:10"><span class="id" title="binder">y</span></a>: <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>,  <a class="idref" href="hydras.Schutte.Well_Orders.html#f:8"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:10"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:9"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:10"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Well Ordering 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a id="WO" class="idref" href="#WO"><span class="id" title="record">WO</span></a> : <span class="id" title="keyword">Type</span>:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Lt_trans" class="idref" href="#Lt_trans"><span class="id" title="projection">Lt_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Lt_irreflexive" class="idref" href="#Lt_irreflexive"><span class="id" title="projection">Lt_irreflexive</span></a> : <span class="id" title="keyword">∀</span> <a id="a:13" class="idref" href="#a:13"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:13"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:13"><span class="id" title="variable">a</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="well_order" class="idref" href="#well_order"><span class="id" title="projection">well_order</span></a> : <span class="id" title="keyword">∀</span> (<a id="X:15" class="idref" href="#X:15"><span class="id" title="binder">X</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>)(<a id="a:16" class="idref" href="#a:16"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:15"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:16"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a0:17" class="idref" href="#a0:17"><span class="id" title="binder">a0</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:15"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a0:17"><span class="id" title="variable">a0</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="the_context.About_WO" class="idref" href="#the_context.About_WO"><span class="id" title="section">About_WO</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Context</span> (<a id="Wo:19" class="idref" href="#Wo:19"><span class="id" title="binder">Wo</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span>  <a id="Lt_connect" class="idref" href="#Lt_connect"><span class="id" title="lemma">Lt_connect</span></a> : <span class="id" title="keyword">∀</span> <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> <a id="b:21" class="idref" href="#b:21"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:20"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:21"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:20"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Le_refl" class="idref" href="#Le_refl"><span class="id" title="lemma">Le_refl</span></a> : <span class="id" title="keyword">∀</span> <a id="x:22" class="idref" href="#x:22"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>, <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:22"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:22"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Le_antisym" class="idref" href="#Le_antisym"><span class="id" title="lemma">Le_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a>,  <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:24"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Instance</span>  <a id="Le_trans" class="idref" href="#Le_trans"><span class="id" title="instance">Le_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Le_Lt_trans" class="idref" href="#Le_Lt_trans"><span class="id" title="lemma">Le_Lt_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="x:25" class="idref" href="#x:25"><span class="id" title="binder">x</span></a> <a id="y:26" class="idref" href="#y:26"><span class="id" title="binder">y</span></a> <a id="z:27" class="idref" href="#z:27"><span class="id" title="binder">z</span></a>, <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:26"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:26"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#z:27"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#z:27"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lt_Le_trans" class="idref" href="#Lt_Le_trans"><span class="id" title="lemma">Lt_Le_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="x:28" class="idref" href="#x:28"><span class="id" title="binder">x</span></a> <a id="y:29" class="idref" href="#y:29"><span class="id" title="binder">y</span></a> <a id="z:30" class="idref" href="#z:30"><span class="id" title="binder">z</span></a>, <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:29"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:29"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#z:30"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:28"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#z:30"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Lt_not_Gt" class="idref" href="#Lt_not_Gt"><span class="id" title="lemma">Lt_not_Gt</span></a> : <span class="id" title="keyword">∀</span> <a id="x:31" class="idref" href="#x:31"><span class="id" title="binder">x</span></a> <a id="y:32" class="idref" href="#y:32"><span class="id" title="binder">y</span></a>,  <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:32"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.Lt"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#y:32"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:31"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="least_member_lower_bound" class="idref" href="#least_member_lower_bound"><span class="id" title="lemma">least_member_lower_bound</span></a> : <span class="id" title="keyword">∀</span> <a id="X:33" class="idref" href="#X:33"><span class="id" title="binder">X</span></a> <a id="a:34" class="idref" href="#a:34"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:33"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="b:35" class="idref" href="#b:35"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:33"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:35"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:34"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:35"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="least_member_glb" class="idref" href="#least_member_glb"><span class="id" title="lemma">least_member_glb</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="X:36" class="idref" href="#X:36"><span class="id" title="binder">X</span></a> <a id="a:37" class="idref" href="#a:37"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:36"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:37"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="b:38" class="idref" href="#b:38"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="c:39" class="idref" href="#c:39"><span class="id" title="binder">c</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:36"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#c:39"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:38"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#c:39"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:38"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:37"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="least_member_unicity" class="idref" href="#least_member_unicity"><span class="id" title="lemma">least_member_unicity</span></a> : <span class="id" title="keyword">∀</span>  <a id="X:40" class="idref" href="#X:40"><span class="id" title="binder">X</span></a> <a id="a:41" class="idref" href="#a:41"><span class="id" title="binder">a</span></a> <a id="b:42" class="idref" href="#b:42"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:40"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:41"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:40"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:42"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:41"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:42"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="least_member_ex_unique" class="idref" href="#least_member_ex_unique"><span class="id" title="lemma">least_member_ex_unique</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span>   <a id="X:43" class="idref" href="#X:43"><span class="id" title="binder">X</span></a>  <a id="x:44" class="idref" href="#x:44"><span class="id" title="binder">x</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="inhX:45" class="idref" href="#inhX:45"><span class="id" title="binder">inhX</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:43"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#x:44"><span class="id" title="variable">x</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="a:46" class="idref" href="#a:46"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:43"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:46"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="least_member_of_eq" class="idref" href="#least_member_of_eq"><span class="id" title="lemma">least_member_of_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="X:47" class="idref" href="#X:47"><span class="id" title="binder">X</span></a> <a id="Y:48" class="idref" href="#Y:48"><span class="id" title="binder">Y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.M"><span class="id" title="variable">M</span></a>) <a id="a:49" class="idref" href="#a:49"><span class="id" title="binder">a</span></a> <a id="b:50" class="idref" href="#b:50"><span class="id" title="binder">b</span></a> ,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:47"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Y:48"><span class="id" title="variable">Y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Included"><span class="id" title="definition">Included</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Y:48"><span class="id" title="variable">Y</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:47"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#X:47"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>  <a class="idref" href="hydras.Schutte.Well_Orders.html#Y:48"><span class="id" title="variable">Y</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:50"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.Well_Orders.html#a:49"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#b:50"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context.About_WO"><span class="id" title="section">About_WO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.Schutte.Well_Orders.html#the_context"><span class="id" title="section">the_context</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="the_least" class="idref" href="#the_least"><span class="id" title="definition">the_least</span></a> {<a id="M:51" class="idref" href="#M:51"><span class="id" title="binder">M</span></a>: <span class="id" title="keyword">Type</span>} {<a id="Lt:52" class="idref" href="#Lt:52"><span class="id" title="binder">Lt</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="inh:53" class="idref" href="#inh:53"><span class="id" title="binder">inh</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#M:51"><span class="id" title="variable">M</span></a>} {<a id="WO:54" class="idref" href="#WO:54"><span class="id" title="binder">WO</span></a>: <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Lt:52"><span class="id" title="variable">Lt</span></a>} (<a id="X:55" class="idref" href="#X:55"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#M:51"><span class="id" title="variable">M</span></a>)  : <a class="idref" href="hydras.Schutte.Well_Orders.html#M:51"><span class="id" title="variable">M</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#the"><span class="id" title="definition">the</span></a> (<a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Lt:52"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:55"><span class="id" title="variable">X</span></a> ).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="the_least_unicity" class="idref" href="#the_least_unicity"><span class="id" title="lemma">the_least_unicity</span></a> {<a id="M:56" class="idref" href="#M:56"><span class="id" title="binder">M</span></a>: <span class="id" title="keyword">Type</span>} {<a id="Lt:57" class="idref" href="#Lt:57"><span class="id" title="binder">Lt</span></a>}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<a id="inh:58" class="idref" href="#inh:58"><span class="id" title="binder">inh</span></a> : <a class="idref" href="hydras.Schutte.MoreEpsilonIota.html#InH"><span class="id" title="class">InH</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#M:56"><span class="id" title="variable">M</span></a>} {<a id="WO:59" class="idref" href="#WO:59"><span class="id" title="binder">WO</span></a>: <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#Lt:57"><span class="id" title="variable">Lt</span></a>} (<a id="X:60" class="idref" href="#X:60"><span class="id" title="binder">X</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#M:56"><span class="id" title="variable">M</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="HX:61" class="idref" href="#HX:61"><span class="id" title="binder">HX</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Inhabited"><span class="id" title="inductive">Inhabited</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:60"><span class="id" title="variable">X</span></a>)<br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">∃</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">!</span></a> <a id="l:62" class="idref" href="#l:62"><span class="id" title="binder">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a3726bd940c3c0e96cb8e13291e9ded4"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#least_member"><span class="id" title="definition">least_member</span></a>   <a class="idref" href="hydras.Schutte.Well_Orders.html#Lt:57"><span class="id" title="variable">Lt</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#X:60"><span class="id" title="variable">X</span></a> <a class="idref" href="hydras.Schutte.Well_Orders.html#l:62"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+#[ <span class="id" title="var">global</span> ] <span class="id" title="keyword">Instance</span> <a id="WO_nat" class="idref" href="#WO_nat"><span class="id" title="instance">WO_nat</span></a> : <a class="idref" href="hydras.Schutte.Well_Orders.html#WO"><span class="id" title="class">WO</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">Peano.lt</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.closure.html b/theories/html/hydras.rpo.closure.html
new file mode 100644
index 00000000..396d8d09
--- /dev/null
+++ b/theories/html/hydras.rpo.closure.html
@@ -0,0 +1,73 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.closure</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.closure</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ by Evelyne Contejean, LRI 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span>  <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="trans_clos" class="idref" href="#trans_clos"><span class="id" title="inductive">trans_clos</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="R:2" class="idref" href="#R:2"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.closure.html#A:1"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <span class="id" title="var">A</span>:=<br/>
+&nbsp;&nbsp;| <a id="t_step" class="idref" href="#t_step"><span class="id" title="constructor">t_step</span></a> : <span class="id" title="keyword">∀</span> <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a>, <a class="idref" href="hydras.rpo.closure.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.rpo.closure.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos:3"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.rpo.closure.html#y:6"><span class="id" title="variable">y</span></a><br/>
+&nbsp;&nbsp;| <a id="t_trans" class="idref" href="#t_trans"><span class="id" title="constructor">t_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> <a id="y:8" class="idref" href="#y:8"><span class="id" title="binder">y</span></a> <a id="z:9" class="idref" href="#z:9"><span class="id" title="binder">z</span></a>, <a class="idref" href="hydras.rpo.closure.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.rpo.closure.html#y:8"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos:3"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#y:8"><span class="id" title="variable">y</span></a> <a class="idref" href="hydras.rpo.closure.html#z:9"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos:3"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:2"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="hydras.rpo.closure.html#z:9"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="trans_clos_is_trans" class="idref" href="#trans_clos_is_trans"><span class="id" title="lemma">trans_clos_is_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:10" class="idref" href="#A:10"><span class="id" title="binder">A</span></a> :<span class="id" title="keyword">Set</span>) (<a id="R:11" class="idref" href="#R:11"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.closure.html#A:10"><span class="id" title="variable">A</span></a>) <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> <a id="c:14" class="idref" href="#c:14"><span class="id" title="binder">c</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:11"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.closure.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:11"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.closure.html#c:14"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:11"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.closure.html#c:14"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="acc_trans" class="idref" href="#acc_trans"><span class="id" title="lemma">acc_trans</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:15" class="idref" href="#A:15"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="R:16" class="idref" href="#R:16"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.closure.html#A:15"><span class="id" title="variable">A</span></a>) <a id="a:17" class="idref" href="#a:17"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.closure.html#R:16"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#a:17"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:16"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.closure.html#a:17"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wf_trans" class="idref" href="#wf_trans"><span class="id" title="lemma">wf_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:18" class="idref" href="#A:18"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="R:19" class="idref" href="#R:19"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.closure.html#A:18"><span class="id" title="variable">A</span></a>) , <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.closure.html#R:19"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:19"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="inv_trans" class="idref" href="#inv_trans"><span class="id" title="lemma">inv_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="R:21" class="idref" href="#R:21"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.closure.html#A:20"><span class="id" title="variable">A</span></a>) (<a id="P:22" class="idref" href="#P:22"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.closure.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="b:24" class="idref" href="#b:24"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.rpo.closure.html#P:22"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.closure.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#R:21"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.closure.html#b:24"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#P:22"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.closure.html#b:24"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:25" class="idref" href="#a:25"><span class="id" title="binder">a</span></a> <a id="b:26" class="idref" href="#b:26"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.rpo.closure.html#P:22"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.closure.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> <a class="idref" href="hydras.rpo.closure.html#R:21"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.closure.html#a:25"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.closure.html#b:26"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.closure.html#P:22"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.closure.html#b:26"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.decidable_set.html b/theories/html/hydras.rpo.decidable_set.html
new file mode 100644
index 00000000..fadd9501
--- /dev/null
+++ b/theories/html/hydras.rpo.decidable_set.html
@@ -0,0 +1,48 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.decidable_set</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.decidable_set</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ by Evelyne Contejean 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="S" class="idref" href="#S"><span class="id" title="module">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Parameter</span> <a id="S.A" class="idref" href="#S.A"><span class="id" title="axiom">A</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Axiom</span> <a id="S.eq_A_dec" class="idref" href="#S.eq_A_dec"><span class="id" title="axiom">eq_A_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:2" class="idref" href="#a1:2"><span class="id" title="binder">a1</span></a> <a id="a2:3" class="idref" href="#a2:3"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.decidable_set.html#S.A"><span class="id" title="axiom">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.decidable_set.html#a1:2"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.decidable_set.html#a2:3"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.decidable_set.html#a1:2"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.decidable_set.html#a2:3"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">S</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.dickson.html b/theories/html/hydras.rpo.dickson.html
new file mode 100644
index 00000000..149dd566
--- /dev/null
+++ b/theories/html/hydras.rpo.dickson.html
@@ -0,0 +1,190 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.dickson</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.dickson</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+ by Evelyne Contejean 
+<div class="paragraph"> </div>
+
+<a id="lab120"></a><h1 class="section">Dickson Lemma: the multiset extension of a well-founded ordering is well-founded.</h1>
+
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#"><span class="id" title="library">Multiset</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.closure.html#"><span class="id" title="library">closure</span></a> <a class="idref" href="hydras.rpo.more_list.html#"><span class="id" title="library">more_list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#"><span class="id" title="library">list_permut</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">DS1</span> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.DS" class="idref" href="#Make.DS"><span class="id" title="module">DS</span></a> := <a class="idref" href="hydras.rpo.dickson.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.LP" class="idref" href="#Make.LP"><span class="id" title="module">LP</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Make"><span class="id" title="module">list_permut.Make</span></a> (<a class="idref" href="hydras.rpo.dickson.html#DS1"><span class="id" title="module">DS1</span></a>).<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.dickson.html#Make.LP"><span class="id" title="module">LP</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab121"></a><h2 class="section">Definition of the multiset extension of a relation.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Inductive</span> <a id="Make.multiset_extension_step" class="idref" href="#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> (<a id="R:1" class="idref" href="#R:1"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.rmv_case" class="idref" href="#Make.rmv_case"><span class="id" title="constructor">rmv_case</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:4" class="idref" href="#l1:4"><span class="id" title="binder">l1</span></a> <a id="l2:5" class="idref" href="#l2:5"><span class="id" title="binder">l2</span></a> <a id="l:6" class="idref" href="#l:6"><span class="id" title="binder">l</span></a> <a id="la:7" class="idref" href="#la:7"><span class="id" title="binder">la</span></a> <a id="a:8" class="idref" href="#a:8"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:9" class="idref" href="#b:9"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:9"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#la:7"><span class="id" title="variable">la</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:1"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:9"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:8"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:4"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.rpo.dickson.html#la:7"><span class="id" title="variable">la</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:6"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:5"><span class="id" title="variable">l2</span></a> (<a class="idref" href="hydras.rpo.dickson.html#a:8"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:6"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#multiset_extension_step:2"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:1"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:4"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:5"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+If <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">&lt;&lt;</span> <span class="inlinecode">{<span class="id" title="var">a</span>}</span> <span class="inlinecode"><span class="id" title="var">U</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> , then 
+      either, there exists n' such that n = {a} U n' and <span class="inlinecode"><span class="id" title="var">n'</span></span> <span class="inlinecode">&lt;&lt;</span> <span class="inlinecode"><span class="id" title="var">m</span></span>,
+      or, there exists k, such that n = k U m, and <span class="inlinecode"><span class="id" title="var">k</span></span> <span class="inlinecode">&lt;&lt;</span> <span class="inlinecode">{<span class="id" title="var">a</span>}</span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.two_cases" class="idref" href="#Make.two_cases"><span class="id" title="lemma">two_cases</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="R:10" class="idref" href="#R:10"><span class="id" title="binder">R</span></a> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a> <a id="m:12" class="idref" href="#m:12"><span class="id" title="binder">m</span></a> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#n:13"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.rpo.dickson.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:12"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n':14" class="idref" href="#n':14"><span class="id" title="binder">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#n:13"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.rpo.dickson.html#a:11"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#n':14"><span class="id" title="variable">n'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#n':14"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:12"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="k:15" class="idref" href="#k:15"><span class="id" title="binder">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:16" class="idref" href="#b:16"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:16"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#k:15"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:10"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:16"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:11"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#n:13"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.rpo.dickson.html#k:15"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:12"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">multiset_extension_step</span></span> is compatible with permutation. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_multiset_extension_step_1" class="idref" href="#Make.list_permut_multiset_extension_step_1"><span class="id" title="lemma">list_permut_multiset_extension_step_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:17" class="idref" href="#R:17"><span class="id" title="binder">R</span></a> <a id="l1:18" class="idref" href="#l1:18"><span class="id" title="binder">l1</span></a> <a id="l2:19" class="idref" href="#l2:19"><span class="id" title="binder">l2</span></a> <a id="l:20" class="idref" href="#l:20"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:18"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:19"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:17"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:18"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:20"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:17"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:19"><span class="id" title="variable">l2</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:20"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_multiset_extension_step_2" class="idref" href="#Make.list_permut_multiset_extension_step_2"><span class="id" title="lemma">list_permut_multiset_extension_step_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:21" class="idref" href="#R:21"><span class="id" title="binder">R</span></a> <a id="l1:22" class="idref" href="#l1:22"><span class="id" title="binder">l1</span></a> <a id="l2:23" class="idref" href="#l2:23"><span class="id" title="binder">l2</span></a> <a id="l:24" class="idref" href="#l:24"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:22"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:23"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:21"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:24"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:22"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:21"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:24"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:23"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_multiset_extension_step_app1" class="idref" href="#Make.context_multiset_extension_step_app1"><span class="id" title="lemma">context_multiset_extension_step_app1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:25" class="idref" href="#R:25"><span class="id" title="binder">R</span></a> <a id="l1:26" class="idref" href="#l1:26"><span class="id" title="binder">l1</span></a> <a id="l2:27" class="idref" href="#l2:27"><span class="id" title="binder">l2</span></a> <a id="l:28" class="idref" href="#l:28"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:25"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:26"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:27"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:25"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.rpo.dickson.html#l:28"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:26"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#l:28"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:27"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_trans_clos_multiset_extension_step_app1" class="idref" href="#Make.context_trans_clos_multiset_extension_step_app1"><span class="id" title="lemma">context_trans_clos_multiset_extension_step_app1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:29" class="idref" href="#R:29"><span class="id" title="binder">R</span></a> <a id="l1:30" class="idref" href="#l1:30"><span class="id" title="binder">l1</span></a> <a id="l2:31" class="idref" href="#l2:31"><span class="id" title="binder">l2</span></a> <a id="l:32" class="idref" href="#l:32"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:29"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#l1:30"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:31"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">trans_clos</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:29"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#l:32"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:30"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#l:32"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:31"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_multiset_extension_step_app2" class="idref" href="#Make.context_multiset_extension_step_app2"><span class="id" title="lemma">context_multiset_extension_step_app2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:33" class="idref" href="#R:33"><span class="id" title="binder">R</span></a> <a id="l1:34" class="idref" href="#l1:34"><span class="id" title="binder">l1</span></a> <a id="l2:35" class="idref" href="#l2:35"><span class="id" title="binder">l2</span></a> <a id="l:36" class="idref" href="#l:36"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:33"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:34"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:35"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:33"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.rpo.dickson.html#l1:34"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:36"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#l2:35"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#l:36"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Parametric</span> <span class="id" title="var">Morphism</span> <a id="R:37" class="idref" href="#R:37"><span class="id" title="binder">R</span></a> : (@<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:37"><span class="id" title="variable">R</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">mult_morph</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab122"></a><h3 class="section">Accessibility lemmata.</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_acc" class="idref" href="#Make.list_permut_acc"><span class="id" title="lemma">list_permut_acc</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:38" class="idref" href="#R:38"><span class="id" title="binder">R</span></a> <a id="l1:39" class="idref" href="#l1:39"><span class="id" title="binder">l1</span></a> <a id="l2:40" class="idref" href="#l2:40"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#l2:40"><span class="id" title="variable">l2</span></a> <a class="idref" href="hydras.rpo.dickson.html#l1:39"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:38"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#l1:39"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:38"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#l2:40"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Parametric</span> <span class="id" title="var">Morphism</span> <a id="R:41" class="idref" href="#R:41"><span class="id" title="binder">R</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:41"><span class="id" title="variable">R</span></a>)) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">acc_morph</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.dickson_aux1" class="idref" href="#Make.dickson_aux1"><span class="id" title="lemma">dickson_aux1</span></a> :<br/>
+<span class="id" title="keyword">∀</span> (<a id="R:42" class="idref" href="#R:42"><span class="id" title="binder">R</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.dickson.html#Make.elt"><span class="id" title="definition">elt</span></a>) <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m:45" class="idref" href="#m:45"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m:45"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:45"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="m:46" class="idref" href="#m:46"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m:46"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m':47" class="idref" href="#m':47"><span class="id" title="binder">m'</span></a>, (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m':47"><span class="id" title="variable">m'</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:46"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m':47"><span class="id" title="variable">m'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:42"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:46"><span class="id" title="variable">m</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.dickson_aux2" class="idref" href="#Make.dickson_aux2"><span class="id" title="lemma">dickson_aux2</span></a> :<br/>
+<span class="id" title="keyword">∀</span> <a id="R:48" class="idref" href="#R:48"><span class="id" title="binder">R</span></a> <a id="m:49" class="idref" href="#m:49"><span class="id" title="binder">m</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:48"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m:49"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:50" class="idref" href="#a:50"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:51" class="idref" href="#b:51"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.rpo.dickson.html#R:48"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:51"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="m:52" class="idref" href="#m:52"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:48"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m:52"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:48"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#b:51"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:52"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:48"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#a:50"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:49"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.dickson_aux3" class="idref" href="#Make.dickson_aux3"><span class="id" title="lemma">dickson_aux3</span></a> :<br/>
+<span class="id" title="keyword">∀</span> <a id="R:55" class="idref" href="#R:55"><span class="id" title="binder">R</span></a> <a id="a:56" class="idref" href="#a:56"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:55"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="m:57" class="idref" href="#m:57"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:55"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#m:57"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:55"><span class="id" title="variable">R</span></a>) (<a class="idref" href="hydras.rpo.dickson.html#a:56"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.dickson.html#m:57"><span class="id" title="variable">m</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Main lemma. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.dickson" class="idref" href="#Make.dickson"><span class="id" title="lemma">dickson</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:62" class="idref" href="#R:62"><span class="id" title="binder">R</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:62"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:62"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.multiset_closure" class="idref" href="#Make.multiset_closure"><span class="id" title="lemma">multiset_closure</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="R:63" class="idref" href="#R:63"><span class="id" title="binder">R</span></a> <a id="p:64" class="idref" href="#p:64"><span class="id" title="binder">p</span></a> <a id="q:65" class="idref" href="#q:65"><span class="id" title="binder">q</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.dickson.html#R:63"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.closure.html#trans_clos"><span class="id" title="inductive">closure.trans_clos</span></a> (<a class="idref" href="hydras.rpo.dickson.html#Make.multiset_extension_step"><span class="id" title="inductive">multiset_extension_step</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:63"><span class="id" title="variable">R</span></a>) <a class="idref" href="hydras.rpo.dickson.html#p:64"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.rpo.dickson.html#q:65"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="p':66" class="idref" href="#p':66"><span class="id" title="binder">p'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="q':67" class="idref" href="#q':67"><span class="id" title="binder">q'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="pq:68" class="idref" href="#pq:68"><span class="id" title="binder">pq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#p:64"><span class="id" title="variable">p</span></a> (<a class="idref" href="hydras.rpo.dickson.html#p':66"><span class="id" title="variable">p'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#pq:68"><span class="id" title="variable">pq</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#q:65"><span class="id" title="variable">q</span></a> (<a class="idref" href="hydras.rpo.dickson.html#q':67"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.dickson.html#pq:68"><span class="id" title="variable">pq</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.dickson.html#q':67"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.dickson.html#p':66"><span class="id" title="variable">p'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:70" class="idref" href="#b:70"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:70"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.dickson.html#q':67"><span class="id" title="variable">q'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.dickson.html#R:63"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.rpo.dickson.html#a:69"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.dickson.html#b:70"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.dickson.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.list_permut.html b/theories/html/hydras.rpo.list_permut.html
new file mode 100644
index 00000000..2152e2f9
--- /dev/null
+++ b/theories/html/hydras.rpo.list_permut.html
@@ -0,0 +1,356 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.list_permut</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.list_permut</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+by Evelyne Contejean, LRI 
+<div class="paragraph"> </div>
+
+<a id="lab125"></a><h1 class="section">Permutation over lists, and finite multisets.</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#"><span class="id" title="library">Multiset</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.decidable_set.html#"><span class="id" title="library">decidable_set</span></a> <a class="idref" href="hydras.rpo.more_list.html#"><span class="id" title="library">more_list</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="Permut" class="idref" href="#Permut"><span class="id" title="module">Permut</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Module</span> <a id="Permut.DS" class="idref" href="#Permut.DS"><span class="id" title="module">DS</span></a> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Permut.elt" class="idref" href="#Permut.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Permut.A"><span class="id" title="axiom">DS.A</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Permut.eq_elt_dec" class="idref" href="#Permut.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Permut.eq_A_dec"><span class="id" title="axiom">DS.eq_A_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Permut.list_to_multiset" class="idref" href="#Permut.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> (<a id="l:1" class="idref" href="#l:1"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Permut.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiset"><span class="id" title="inductive">multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Permut.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_permut.html#l:1"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#EmptyBag"><span class="id" title="definition">EmptyBag</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Permut.elt"><span class="id" title="definition">elt</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#munion"><span class="id" title="definition">munion</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#SingletonBag"><span class="id" title="definition">SingletonBag</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.list_permut.html#Permut.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">h</span>) (<a class="idref" href="hydras.rpo.list_permut.html#list_to_multiset:2"><span class="id" title="definition">list_to_multiset</span></a> <span class="id" title="var">tl</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Permut.list_permut" class="idref" href="#Permut.list_permut"><span class="id" title="definition">list_permut</span></a> (<a id="l1:4" class="idref" href="#l1:4"><span class="id" title="binder">l1</span></a> <a id="l2:5" class="idref" href="#l2:5"><span class="id" title="binder">l2</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Permut.elt"><span class="id" title="definition">elt</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#meq"><span class="id" title="definition">meq</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Permut.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:4"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#Permut.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:5"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.list_permut.html#Permut"><span class="id" title="module">Permut</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab126"></a><h2 class="section">Definition of permutation over lists.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">DS1</span> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>) &lt;: <a class="idref" href="hydras.rpo.list_permut.html#Permut"><span class="id" title="module">Permut</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">DS</span>:= <a class="idref" href="hydras.rpo.list_permut.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.DS" class="idref" href="#Make.DS"><span class="id" title="module">DS</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.list_permut.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.elt" class="idref" href="#Make.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Make.A"><span class="id" title="axiom">DS.A</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.eq_elt_dec" class="idref" href="#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="t1:6" class="idref" href="#t1:6"><span class="id" title="binder">t1</span></a> <a id="t2:7" class="idref" href="#t2:7"><span class="id" title="binder">t2</span></a> : <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_permut.html#t1:6"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_permut.html#t2:7"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_permut.html#t1:6"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.list_permut.html#t2:7"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Make.eq_A_dec"><span class="id" title="axiom">DS.eq_A_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.list_to_multiset" class="idref" href="#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> (<a id="l:8" class="idref" href="#l:8"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiset"><span class="id" title="inductive">multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_permut.html#l:8"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#EmptyBag"><span class="id" title="definition">EmptyBag</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#munion"><span class="id" title="definition">munion</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#SingletonBag"><span class="id" title="definition">SingletonBag</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.list_permut.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">h</span>) (<a class="idref" href="hydras.rpo.list_permut.html#list_to_multiset:9"><span class="id" title="definition">list_to_multiset</span></a> <span class="id" title="var">tl</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.list_permut" class="idref" href="#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a id="l1:11" class="idref" href="#l1:11"><span class="id" title="binder">l1</span></a> <a id="l2:12" class="idref" href="#l2:12"><span class="id" title="binder">l2</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#meq"><span class="id" title="definition">meq</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:11"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:12"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Properties over the multiplicity. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.multiplicity_app" class="idref" href="#Make.multiplicity_app"><span class="id" title="lemma">multiplicity_app</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="l1:13" class="idref" href="#l1:13"><span class="id" title="binder">l1</span></a> <a id="l2:14" class="idref" href="#l2:14"><span class="id" title="binder">l2</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="t:15" class="idref" href="#t:15"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiplicity"><span class="id" title="definition">multiplicity</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:13"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:14"><span class="id" title="variable">l2</span></a>)) <a class="idref" href="hydras.rpo.list_permut.html#t:15"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiplicity"><span class="id" title="definition">multiplicity</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:13"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#t:15"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiplicity"><span class="id" title="definition">multiplicity</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:14"><span class="id" title="variable">l2</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#t:15"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.out_mult_O" class="idref" href="#Make.out_mult_O"><span class="id" title="lemma">out_mult_O</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:16" class="idref" href="#t:16"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="l:17" class="idref" href="#l:17"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_permut.html#t:16"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:17"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiplicity"><span class="id" title="definition">multiplicity</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:17"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#t:16"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.in_mult_S" class="idref" href="#Make.in_mult_S"><span class="id" title="lemma">in_mult_S</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="t:18" class="idref" href="#t:18"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_permut.html#t:18"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:19"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Multiset.html#multiplicity"><span class="id" title="definition">multiplicity</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_to_multiset"><span class="id" title="definition">list_to_multiset</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:19"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#t:18"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> 1.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab127"></a><h2 class="section">Permutation is a equivalence relation.</h2>
+
+Reflexivity. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Theorem</span> <a id="Make.list_permut_refl" class="idref" href="#Make.list_permut_refl"><span class="id" title="lemma">list_permut_refl</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:20" class="idref" href="#l:20"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:20"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:20"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Symetry. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Theorem</span> <a id="Make.list_permut_sym" class="idref" href="#Make.list_permut_sym"><span class="id" title="lemma">list_permut_sym</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:21" class="idref" href="#l1:21"><span class="id" title="binder">l1</span></a> <a id="l2:22" class="idref" href="#l2:22"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:21"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:22"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:22"><span class="id" title="variable">l2</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:21"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Immediate</span> <span class="id" title="var">list_permut_refl</span> : <span class="id" title="var">core</span>.<br/>
+#[<span class="id" title="var">global</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">list_permut_sym</span> : <span class="id" title="var">core</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Transitivity. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Theorem</span> <a id="Make.list_permut_trans" class="idref" href="#Make.list_permut_trans"><span class="id" title="lemma">list_permut_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:23" class="idref" href="#l1:23"><span class="id" title="binder">l1</span></a> <a id="l2:24" class="idref" href="#l2:24"><span class="id" title="binder">l2</span></a> <a id="l3:25" class="idref" href="#l3:25"><span class="id" title="binder">l3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:23"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:24"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:24"><span class="id" title="variable">l2</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l3:25"><span class="id" title="variable">l3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:23"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l3:25"><span class="id" title="variable">l3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Relation</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <br/>
+<span class="id" title="tactic">reflexivity</span> <span class="id" title="var">proved</span> <span class="id" title="tactic">by</span> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut_refl"><span class="id" title="lemma">list_permut_refl</span></a><br/>
+<span class="id" title="tactic">symmetry</span> <span class="id" title="var">proved</span> <span class="id" title="tactic">by</span> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut_sym"><span class="id" title="lemma">list_permut_sym</span></a><br/>
+<span class="id" title="tactic">transitivity</span> <span class="id" title="var">proved</span> <span class="id" title="tactic">by</span> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut_trans"><span class="id" title="lemma">list_permut_trans</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">LP</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation of an empty list. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_nil" class="idref" href="#Make.list_permut_nil"><span class="id" title="lemma">list_permut_nil</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l:26" class="idref" href="#l:26"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:26"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:26"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab128"></a><h2 class="section">Compatibility Properties.</h2>
+
+ Permutation is compatible with In. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.in_permut_in" class="idref" href="#Make.in_permut_in"><span class="id" title="lemma">in_permut_in</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:27" class="idref" href="#l1:27"><span class="id" title="binder">l1</span></a> <a id="l2:28" class="idref" href="#l2:28"><span class="id" title="binder">l2</span></a> <a id="e:29" class="idref" href="#e:29"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_permut.html#e:29"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:27"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:27"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:28"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_permut.html#e:29"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:28"><span class="id" title="variable">l2</span></a>.<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Morphism</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> (<span class="id" title="var">A</span> :=<a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>)) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">in_morph</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cons_permut_in" class="idref" href="#Make.cons_permut_in"><span class="id" title="lemma">cons_permut_in</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:30" class="idref" href="#l1:30"><span class="id" title="binder">l1</span></a> <a id="l2:31" class="idref" href="#l2:31"><span class="id" title="binder">l2</span></a> <a id="e:32" class="idref" href="#e:32"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#e:32"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:30"><span class="id" title="variable">l1</span></a>) <a class="idref" href="hydras.rpo.list_permut.html#l2:31"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_permut.html#e:32"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:31"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with adding an element. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_list_permut_cons" class="idref" href="#Make.context_list_permut_cons"><span class="id" title="lemma">context_list_permut_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:33" class="idref" href="#e:33"><span class="id" title="binder">e</span></a> <a id="l1:34" class="idref" href="#l1:34"><span class="id" title="binder">l1</span></a> <a id="l2:35" class="idref" href="#l2:35"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:34"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:35"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#e:33"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:34"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#e:33"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:35"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Morphism</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#cons"><span class="id" title="abbreviation">List.cons</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>)) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a>) <br/>
+&nbsp;<span class="id" title="keyword">as</span> <span class="id" title="var">add_elt_morph</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_add_inside" class="idref" href="#Make.list_permut_add_inside"><span class="id" title="lemma">list_permut_add_inside</span></a> :<br/>
+<span class="id" title="keyword">∀</span> <a id="a:36" class="idref" href="#a:36"><span class="id" title="binder">a</span></a> <a id="l1:37" class="idref" href="#l1:37"><span class="id" title="binder">l1</span></a> <a id="l2:38" class="idref" href="#l2:38"><span class="id" title="binder">l2</span></a> <a id="l3:39" class="idref" href="#l3:39"><span class="id" title="binder">l3</span></a> <a id="l4:40" class="idref" href="#l4:40"><span class="id" title="binder">l4</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:37"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:38"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l3:39"><span class="id" title="variable">l3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l4:40"><span class="id" title="variable">l4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:37"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:38"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l3:39"><span class="id" title="variable">l3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a:36"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l4:40"><span class="id" title="variable">l4</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_add_cons_inside" class="idref" href="#Make.list_permut_add_cons_inside"><span class="id" title="lemma">list_permut_add_cons_inside</span></a> :<br/>
+<span class="id" title="keyword">∀</span> <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="l:44" class="idref" href="#l:44"><span class="id" title="binder">l</span></a> <a id="l1:45" class="idref" href="#l1:45"><span class="id" title="binder">l1</span></a> <a id="l2:46" class="idref" href="#l2:46"><span class="id" title="binder">l2</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:44"><span class="id" title="variable">l</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:45"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:46"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:44"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l1:45"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:46"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with append. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_list_permut_app1" class="idref" href="#Make.context_list_permut_app1"><span class="id" title="lemma">context_list_permut_app1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:47" class="idref" href="#l:47"><span class="id" title="binder">l</span></a> <a id="l1:48" class="idref" href="#l1:48"><span class="id" title="binder">l1</span></a> <a id="l2:49" class="idref" href="#l2:49"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:48"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:49"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l:47"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:48"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l:47"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:49"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.context_list_permut_app2" class="idref" href="#Make.context_list_permut_app2"><span class="id" title="lemma">context_list_permut_app2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:50" class="idref" href="#l:50"><span class="id" title="binder">l</span></a> <a id="l1:51" class="idref" href="#l1:51"><span class="id" title="binder">l1</span></a> <a id="l2:52" class="idref" href="#l2:52"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:51"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:52"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:51"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:50"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l2:52"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:50"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Morphism</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#app"><span class="id" title="abbreviation">List.app</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>)) <br/>
+<span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">app_morph</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_app_app" class="idref" href="#Make.list_permut_app_app"><span class="id" title="lemma">list_permut_app_app</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:53" class="idref" href="#l1:53"><span class="id" title="binder">l1</span></a> <a id="l2:54" class="idref" href="#l2:54"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:53"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:54"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l2:54"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:53"><span class="id" title="variable">l1</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with removal of common elements 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.remove_context_list_permut_cons" class="idref" href="#Make.remove_context_list_permut_cons"><span class="id" title="lemma">remove_context_list_permut_cons</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:55" class="idref" href="#e:55"><span class="id" title="binder">e</span></a> <a id="l1:56" class="idref" href="#l1:56"><span class="id" title="binder">l1</span></a> <a id="l2:57" class="idref" href="#l2:57"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#e:55"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:56"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#e:55"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:57"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:56"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:57"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.remove_context_list_permut_app2" class="idref" href="#Make.remove_context_list_permut_app2"><span class="id" title="lemma">remove_context_list_permut_app2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:58" class="idref" href="#l:58"><span class="id" title="binder">l</span></a> <a id="l1:59" class="idref" href="#l1:59"><span class="id" title="binder">l1</span></a> <a id="l2:60" class="idref" href="#l2:60"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:59"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:58"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l2:60"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:58"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:59"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:60"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_remove_hd" class="idref" href="#Make.list_permut_remove_hd"><span class="id" title="lemma">list_permut_remove_hd</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:61" class="idref" href="#l:61"><span class="id" title="binder">l</span></a> <a id="l1:62" class="idref" href="#l1:62"><span class="id" title="binder">l1</span></a> <a id="l2:63" class="idref" href="#l2:63"><span class="id" title="binder">l2</span></a> <a id="a:64" class="idref" href="#a:64"><span class="id" title="binder">a</span></a>,   <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:61"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l1:62"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a:64"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:63"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l:61"><span class="id" title="variable">l</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:62"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:63"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with length. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_length" class="idref" href="#Make.list_permut_length"><span class="id" title="lemma">list_permut_length</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:67" class="idref" href="#l1:67"><span class="id" title="binder">l1</span></a> <a id="l2:68" class="idref" href="#l2:68"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:67"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:68"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:67"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:68"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Morphism</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<span class="id" title="var">A</span>:=<a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>)) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">length_morph</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with size. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_size" class="idref" href="#Make.list_permut_size"><span class="id" title="lemma">list_permut_size</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="size:69" class="idref" href="#size:69"><span class="id" title="binder">size</span></a> <a id="l1:70" class="idref" href="#l1:70"><span class="id" title="binder">l1</span></a> <a id="l2:71" class="idref" href="#l2:71"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:70"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:71"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.list_permut.html#size:69"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:70"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.list_permut.html#size:69"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:71"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Parametric</span> <span class="id" title="var">Morphism</span> <a id="size:74" class="idref" href="#size:74"><span class="id" title="binder">size</span></a> : (@<a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="hydras.rpo.list_permut.html#size:74"><span class="id" title="variable">size</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> (<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> @<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">list_size_morph</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Permutation is compatible with map. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_map" class="idref" href="#Make.list_permut_map"><span class="id" title="lemma">list_permut_map</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:75" class="idref" href="#f:75"><span class="id" title="binder">f</span></a> <a id="l1:76" class="idref" href="#l1:76"><span class="id" title="binder">l1</span></a> <a id="l2:77" class="idref" href="#l2:77"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:76"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:77"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.list_permut.html#f:75"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:76"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.list_permut.html#f:75"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:77"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Add</span> <span class="id" title="var">Parametric</span> <span class="id" title="var">Morphism</span> <a id="f:78" class="idref" href="#f:78"><span class="id" title="binder">f</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.list_permut.html#f:78"><span class="id" title="variable">f</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">signature</span> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">map_morph</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab129"></a><h2 class="section">Permutation for short lists.</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_length_1" class="idref" href="#Make.list_permut_length_1"><span class="id" title="lemma">list_permut_length_1</span></a>:<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:79" class="idref" href="#a:79"><span class="id" title="binder">a</span></a> <a id="b:80" class="idref" href="#b:80"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#b:80"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a:79"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_permut.html#b:80"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_length_2" class="idref" href="#Make.list_permut_length_2"><span class="id" title="lemma">list_permut_length_2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a1:81" class="idref" href="#a1:81"><span class="id" title="binder">a1</span></a> <a id="b1:82" class="idref" href="#b1:82"><span class="id" title="binder">b1</span></a> <a id="a2:83" class="idref" href="#a2:83"><span class="id" title="binder">a2</span></a> <a id="b2:84" class="idref" href="#b2:84"><span class="id" title="binder">b2</span></a>, <a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#a1:81"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#b1:82"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#a2:83"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_permut.html#b2:84"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.list_permut.html#a1:81"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.list_permut.html#a2:83"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.list_permut.html#b1:82"><span class="id" title="variable">b1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.list_permut.html#b2:84"><span class="id" title="variable">b2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.list_permut.html#a1:81"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.list_permut.html#b2:84"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.list_permut.html#a2:83"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.list_permut.html#b1:82"><span class="id" title="variable">b1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab130"></a><h2 class="section">Link with AC syntactic decomposition.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.ac_syntactic_aux" class="idref" href="#Make.ac_syntactic_aux"><span class="id" title="lemma">ac_syntactic_aux</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="l1:85" class="idref" href="#l1:85"><span class="id" title="binder">l1</span></a> <a id="l2:86" class="idref" href="#l2:86"><span class="id" title="binder">l2</span></a> <a id="l3:87" class="idref" href="#l3:87"><span class="id" title="binder">l3</span></a> <a id="l4:88" class="idref" href="#l4:88"><span class="id" title="binder">l4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l1:85"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:86"><span class="id" title="variable">l2</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l3:87"><span class="id" title="variable">l3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l4:88"><span class="id" title="variable">l4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u1:89" class="idref" href="#u1:89"><span class="id" title="binder">u1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u2:90" class="idref" href="#u2:90"><span class="id" title="binder">u2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u3:91" class="idref" href="#u3:91"><span class="id" title="binder">u3</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u4:92" class="idref" href="#u4:92"><span class="id" title="binder">u4</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:85"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u1:89"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u2:90"><span class="id" title="variable">u2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:86"><span class="id" title="variable">l2</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u3:91"><span class="id" title="variable">u3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u4:92"><span class="id" title="variable">u4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l3:87"><span class="id" title="variable">l3</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u1:89"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u3:91"><span class="id" title="variable">u3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l4:88"><span class="id" title="variable">l4</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u2:90"><span class="id" title="variable">u2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u4:92"><span class="id" title="variable">u4</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.ac_syntactic" class="idref" href="#Make.ac_syntactic"><span class="id" title="lemma">ac_syntactic</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="l1:93" class="idref" href="#l1:93"><span class="id" title="binder">l1</span></a> <a id="l2:94" class="idref" href="#l2:94"><span class="id" title="binder">l2</span></a> <a id="l3:95" class="idref" href="#l3:95"><span class="id" title="binder">l3</span></a> <a id="l4:96" class="idref" href="#l4:96"><span class="id" title="binder">l4</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#Make.elt"><span class="id" title="definition">elt</span></a>),<br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#l2:94"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:93"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.list_permut.html#l4:96"><span class="id" title="variable">l4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l3:95"><span class="id" title="variable">l3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u1:97" class="idref" href="#u1:97"><span class="id" title="binder">u1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u2:98" class="idref" href="#u2:98"><span class="id" title="binder">u2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u3:99" class="idref" href="#u3:99"><span class="id" title="binder">u3</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u4:100" class="idref" href="#u4:100"><span class="id" title="binder">u4</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:93"><span class="id" title="variable">l1</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u1:97"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u2:98"><span class="id" title="variable">u2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:94"><span class="id" title="variable">l2</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u3:99"><span class="id" title="variable">u3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u4:100"><span class="id" title="variable">u4</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l3:95"><span class="id" title="variable">l3</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u1:97"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u3:99"><span class="id" title="variable">u3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;<a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l4:96"><span class="id" title="variable">l4</span></a> (<a class="idref" href="hydras.rpo.list_permut.html#u2:98"><span class="id" title="variable">u2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_permut.html#u4:100"><span class="id" title="variable">u4</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_dec" class="idref" href="#Make.list_permut_dec"><span class="id" title="lemma">list_permut_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="l1:101" class="idref" href="#l1:101"><span class="id" title="binder">l1</span></a> <a id="l2:102" class="idref" href="#l2:102"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:101"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:102"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.rpo.list_permut.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l1:101"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_permut.html#l2:102"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.list_permut.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.list_set.html b/theories/html/hydras.rpo.list_set.html
new file mode 100644
index 00000000..5fee66c0
--- /dev/null
+++ b/theories/html/hydras.rpo.list_set.html
@@ -0,0 +1,475 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.list_set</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.list_set</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+To do : Is it a clone of ListSet ? 
+<div class="paragraph"> </div>
+
+ Evelyne Contejean, LRI 
+<div class="paragraph"> </div>
+
+<a id="lab123"></a><h1 class="section">Sets built with lists</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.more_list.html#"><span class="id" title="library">more_list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#"><span class="id" title="library">list_permut</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="S" class="idref" href="#S"><span class="id" title="module">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Module</span> <a id="S.DS" class="idref" href="#S.DS"><span class="id" title="module">DS</span></a> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="S.elt" class="idref" href="#S.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="hydras.rpo.list_set.html#S.A"><span class="id" title="axiom">DS.A</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="S.eq_elt_dec" class="idref" href="#S.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> := <a class="idref" href="hydras.rpo.list_set.html#S.eq_A_dec"><span class="id" title="axiom">DS.eq_A_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="S.without_red" class="idref" href="#S.without_red"><span class="id" title="definition">without_red</span></a> (<a id="l:1" class="idref" href="#l:1"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#S.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l:1"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> ⇒ <span class="id" title="keyword">if</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#S.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">e</span> <span class="id" title="var">le</span>) <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.list_set.html#without_red:2"><span class="id" title="definition">without_red</span></a> <span class="id" title="var">le</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="S.t" class="idref" href="#S.t"><span class="id" title="record">t</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="S.mk_set" class="idref" href="#S.mk_set"><span class="id" title="constructor">mk_set</span></a> <br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="S.support" class="idref" href="#S.support"><span class="id" title="projection">support</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#S.elt"><span class="id" title="definition">elt</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="S.is_a_set" class="idref" href="#S.is_a_set"><span class="id" title="projection">is_a_set</span></a> : <a class="idref" href="hydras.rpo.list_set.html#S.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#support:5"><span class="id" title="method">support</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="S.cardinal" class="idref" href="#S.cardinal"><span class="id" title="definition">cardinal</span></a> <a id="s:7" class="idref" href="#s:7"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:7"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#S.support"><span class="id" title="projection">support</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="S.subset" class="idref" href="#S.subset"><span class="id" title="definition">subset</span></a> <a id="s1:8" class="idref" href="#s1:8"><span class="id" title="binder">s1</span></a> <a id="s2:9" class="idref" href="#s2:9"><span class="id" title="binder">s2</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:10" class="idref" href="#e:10"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:10"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:8"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#S.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:10"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:9"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#S.support"><span class="id" title="projection">support</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="S.cardinal_subset" class="idref" href="#S.cardinal_subset"><span class="id" title="axiom">cardinal_subset</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:11" class="idref" href="#s1:11"><span class="id" title="binder">s1</span></a> <a id="s2:12" class="idref" href="#s2:12"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#S.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:11"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:12"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#S.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:11"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.list_set.html#S.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:12"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.list_set.html#S"><span class="id" title="module">S</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab124"></a><h2 class="section">Definition of sets using lists.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">DS1</span> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>) &lt;: <a class="idref" href="hydras.rpo.list_set.html#S"><span class="id" title="module">S</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">DS</span>:= <a class="idref" href="hydras.rpo.list_set.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.DS" class="idref" href="#Make.DS"><span class="id" title="module">DS</span></a> := <a class="idref" href="hydras.rpo.list_set.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.list_set.html#DS1"><span class="id" title="module">DS1</span></a>.<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.LP" class="idref" href="#Make.LP"><span class="id" title="module">LP</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Make"><span class="id" title="module">list_permut.Make</span></a> (<a class="idref" href="hydras.rpo.list_set.html#DS1"><span class="id" title="module">DS1</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.elt" class="idref" href="#Make.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="hydras.rpo.list_set.html#Make.A"><span class="id" title="axiom">DS.A</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.eq_elt_dec" class="idref" href="#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="t1:14" class="idref" href="#t1:14"><span class="id" title="binder">t1</span></a> <a id="t2:15" class="idref" href="#t2:15"><span class="id" title="binder">t2</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#t1:14"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_set.html#t2:15"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#t1:14"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.list_set.html#t2:15"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> := <a class="idref" href="hydras.rpo.list_set.html#Make.eq_A_dec"><span class="id" title="axiom">DS.eq_A_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.without_red" class="idref" href="#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a id="l:16" class="idref" href="#l:16"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l:16"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> ⇒ <span class="id" title="keyword">if</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">e</span> <span class="id" title="var">le</span>) <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.list_set.html#without_red:17"><span class="id" title="definition">without_red</span></a> <span class="id" title="var">le</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="Make.t" class="idref" href="#Make.t"><span class="id" title="record">t</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;<a id="Make.mk_set" class="idref" href="#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> <br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Make.support" class="idref" href="#Make.support"><span class="id" title="projection">support</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Make.is_a_set" class="idref" href="#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#support:20"><span class="id" title="method">support</span></a><br/>
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.mem" class="idref" href="#Make.mem"><span class="id" title="definition">mem</span></a> <a id="e:22" class="idref" href="#e:22"><span class="id" title="binder">e</span></a> <a id="s:23" class="idref" href="#s:23"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:22"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:23"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.mem_dec" class="idref" href="#Make.mem_dec"><span class="id" title="lemma">mem_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="e:24" class="idref" href="#e:24"><span class="id" title="binder">e</span></a> <a id="s:25" class="idref" href="#s:25"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:24"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:25"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:24"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:25"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.add_prf" class="idref" href="#Make.add_prf"><span class="id" title="lemma">add_prf</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:26" class="idref" href="#e:26"><span class="id" title="binder">e</span></a> <a id="l:27" class="idref" href="#l:27"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:27"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:26"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:27"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#e:26"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:27"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.add" class="idref" href="#Make.add"><span class="id" title="definition">add</span></a> <a id="e:28" class="idref" href="#e:28"><span class="id" title="binder">e</span></a> <a id="s:29" class="idref" href="#s:29"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:28"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:29"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#left"><span class="id" title="constructor">left</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.rpo.list_set.html#s:29"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#right"><span class="id" title="constructor">right</span></a> <span class="id" title="var">R</span> ⇒ <a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#e:28"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:29"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>)) (<a class="idref" href="hydras.rpo.list_set.html#Make.add_prf"><span class="id" title="lemma">add_prf</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:28"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:29"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s:29"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a>) <span class="id" title="var">R</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.add_1" class="idref" href="#Make.add_1"><span class="id" title="lemma">add_1</span></a> : <span class="id" title="keyword">∀</span> <a id="e:30" class="idref" href="#e:30"><span class="id" title="binder">e</span></a> <a id="s:31" class="idref" href="#s:31"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:30"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:30"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:31"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.add_2" class="idref" href="#Make.add_2"><span class="id" title="lemma">add_2</span></a> : <span class="id" title="keyword">∀</span> <a id="e:32" class="idref" href="#e:32"><span class="id" title="binder">e</span></a> <a id="e':33" class="idref" href="#e':33"><span class="id" title="binder">e'</span></a> <a id="s:34" class="idref" href="#s:34"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:32"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:34"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:32"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e':33"><span class="id" title="variable">e'</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:34"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.add_12" class="idref" href="#Make.add_12"><span class="id" title="lemma">add_12</span></a> : <span class="id" title="keyword">∀</span> <a id="e:35" class="idref" href="#e:35"><span class="id" title="binder">e</span></a> <a id="e':36" class="idref" href="#e':36"><span class="id" title="binder">e'</span></a> <a id="s:37" class="idref" href="#s:37"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:35"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e':36"><span class="id" title="variable">e'</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:37"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:35"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_set.html#e':36"><span class="id" title="variable">e'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:35"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:37"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.filter_aux" class="idref" href="#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> (<a id="P:38" class="idref" href="#P:38"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="P_dec:40" class="idref" href="#P_dec:40"><span class="id" title="binder">P_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="e:39" class="idref" href="#e:39"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:39"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:39"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;(<a id="l:41" class="idref" href="#l:41"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l:41"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="hydras.rpo.list_set.html#P_dec:40"><span class="id" title="variable">P_dec</span></a> <span class="id" title="var">e</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.list_set.html#filter_aux:42"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:40"><span class="id" title="variable">P_dec</span></a> <span class="id" title="var">le</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.list_set.html#filter_aux:42"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:38"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:40"><span class="id" title="variable">P_dec</span></a> <span class="id" title="var">le</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.included_filter_aux" class="idref" href="#Make.included_filter_aux"><span class="id" title="lemma">included_filter_aux</span></a> : <br/>
+<span class="id" title="keyword">∀</span> <a id="P:44" class="idref" href="#P:44"><span class="id" title="binder">P</span></a> <a id="P_dec:45" class="idref" href="#P_dec:45"><span class="id" title="binder">P_dec</span></a> <a id="e:46" class="idref" href="#e:46"><span class="id" title="binder">e</span></a> <a id="l:47" class="idref" href="#l:47"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:46"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:44"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:45"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:47"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:46"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:47"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_filter_aux" class="idref" href="#Make.without_red_filter_aux"><span class="id" title="lemma">without_red_filter_aux</span></a> :  <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:48" class="idref" href="#P:48"><span class="id" title="binder">P</span></a> <a id="P_dec:49" class="idref" href="#P_dec:49"><span class="id" title="binder">P_dec</span></a> <a id="l:50" class="idref" href="#l:50"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:50"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:48"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:49"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:50"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.filter" class="idref" href="#Make.filter"><span class="id" title="definition">filter</span></a> <a id="P:51" class="idref" href="#P:51"><span class="id" title="binder">P</span></a> <a id="P_dec:52" class="idref" href="#P_dec:52"><span class="id" title="binder">P_dec</span></a> <a id="s:53" class="idref" href="#s:53"><span class="id" title="binder">s</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:51"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:52"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:53"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.rpo.list_set.html#Make.without_red_filter_aux"><span class="id" title="lemma">without_red_filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:51"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:52"><span class="id" title="variable">P_dec</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.list_set.html#s:53"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.filter_1_list" class="idref" href="#Make.filter_1_list"><span class="id" title="lemma">filter_1_list</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:54" class="idref" href="#P:54"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="P_dec:55" class="idref" href="#P_dec:55"><span class="id" title="binder">P_dec</span></a> <a id="l:56" class="idref" href="#l:56"><span class="id" title="binder">l</span></a> <a id="e:57" class="idref" href="#e:57"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:57"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:56"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:54"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:57"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:57"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:54"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:55"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:56"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.filter_1" class="idref" href="#Make.filter_1"><span class="id" title="lemma">filter_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:58" class="idref" href="#P:58"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="P_dec:59" class="idref" href="#P_dec:59"><span class="id" title="binder">P_dec</span></a> <a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> <a id="e:61" class="idref" href="#e:61"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:61"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:60"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:58"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:61"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:61"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:58"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:59"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:60"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.filter_2_list" class="idref" href="#Make.filter_2_list"><span class="id" title="lemma">filter_2_list</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:62" class="idref" href="#P:62"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="P_dec:63" class="idref" href="#P_dec:63"><span class="id" title="binder">P_dec</span></a> <a id="l:64" class="idref" href="#l:64"><span class="id" title="binder">l</span></a> <a id="e:65" class="idref" href="#e:65"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:65"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter_aux"><span class="id" title="definition">filter_aux</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:62"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:63"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:64"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:65"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:64"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:62"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:65"><span class="id" title="variable">e</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.filter_2" class="idref" href="#Make.filter_2"><span class="id" title="lemma">filter_2</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:66" class="idref" href="#P:66"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="P_dec:67" class="idref" href="#P_dec:67"><span class="id" title="binder">P_dec</span></a> <a id="s:68" class="idref" href="#s:68"><span class="id" title="binder">s</span></a> <a id="e:69" class="idref" href="#e:69"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:69"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:66"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:67"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:68"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:69"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:68"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:66"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:69"><span class="id" title="variable">e</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.remove_red" class="idref" href="#Make.remove_red"><span class="id" title="definition">remove_red</span></a> (<a id="l:70" class="idref" href="#l:70"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l:70"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">e</span> <span class="id" title="var">le</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.list_set.html#remove_red:71"><span class="id" title="definition">remove_red</span></a> <span class="id" title="var">le</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.list_set.html#remove_red:71"><span class="id" title="definition">remove_red</span></a> <span class="id" title="var">le</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.included_remove_red" class="idref" href="#Make.included_remove_red"><span class="id" title="lemma">included_remove_red</span></a> : <br/>
+<span class="id" title="keyword">∀</span> <a id="e:73" class="idref" href="#e:73"><span class="id" title="binder">e</span></a> <a id="l:74" class="idref" href="#l:74"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:73"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_red"><span class="id" title="definition">remove_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:74"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:73"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:74"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.remove_red_included" class="idref" href="#Make.remove_red_included"><span class="id" title="lemma">remove_red_included</span></a> : <span class="id" title="keyword">∀</span> <a id="e:75" class="idref" href="#e:75"><span class="id" title="binder">e</span></a> <a id="l:76" class="idref" href="#l:76"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:75"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:76"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:75"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_red"><span class="id" title="definition">remove_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:76"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_remove_red" class="idref" href="#Make.without_red_remove_red"><span class="id" title="lemma">without_red_remove_red</span></a> :  <span class="id" title="keyword">∀</span> <a id="l:77" class="idref" href="#l:77"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_red"><span class="id" title="definition">remove_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:77"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_remove" class="idref" href="#Make.without_red_remove"><span class="id" title="lemma">without_red_remove</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:78" class="idref" href="#e:78"><span class="id" title="binder">e</span></a> <a id="l1:79" class="idref" href="#l1:79"><span class="id" title="binder">l1</span></a> <a id="l2:80" class="idref" href="#l2:80"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#l1:79"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:78"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:80"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#l1:79"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:80"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_add" class="idref" href="#Make.without_red_add"><span class="id" title="lemma">without_red_add</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:81" class="idref" href="#e:81"><span class="id" title="binder">e</span></a> <a id="l1:82" class="idref" href="#l1:82"><span class="id" title="binder">l1</span></a> <a id="l2:83" class="idref" href="#l2:83"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#l1:82"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:83"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:81"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#l1:82"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:83"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#l1:82"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:81"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:83"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_nil" class="idref" href="#Make.without_red_nil"><span class="id" title="lemma">without_red_nil</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.empty" class="idref" href="#Make.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red_nil"><span class="id" title="lemma">without_red_nil</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_singleton" class="idref" href="#Make.without_red_singleton"><span class="id" title="lemma">without_red_singleton</span></a> : <span class="id" title="keyword">∀</span> <a id="e:84" class="idref" href="#e:84"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#e:84"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.singleton" class="idref" href="#Make.singleton"><span class="id" title="definition">singleton</span></a> (<a id="e:85" class="idref" href="#e:85"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="hydras.rpo.list_set.html#Make.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#e:85"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.without_red_singleton"><span class="id" title="lemma">without_red_singleton</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:85"><span class="id" title="variable">e</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.make_set" class="idref" href="#Make.make_set"><span class="id" title="definition">make_set</span></a> (<a id="l:86" class="idref" href="#l:86"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="hydras.rpo.list_set.html#Make.t"><span class="id" title="record">t</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_red"><span class="id" title="definition">remove_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:86"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.without_red_remove_red"><span class="id" title="lemma">without_red_remove_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l:86"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.add_without_red" class="idref" href="#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> (<a id="acc:87" class="idref" href="#acc:87"><span class="id" title="binder">acc</span></a> <a id="l:88" class="idref" href="#l:88"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l:88"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.rpo.list_set.html#acc:87"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#le"><span class="id" title="inductive">le</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">e</span> <a class="idref" href="hydras.rpo.list_set.html#acc:87"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.list_set.html#add_without_red:89"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:87"><span class="id" title="variable">acc</span></a> <span class="id" title="var">le</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.list_set.html#add_without_red:89"><span class="id" title="definition">add_without_red</span></a> (<span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:87"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">le</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_add_without_red" class="idref" href="#Make.without_red_add_without_red"><span class="id" title="lemma">without_red_add_without_red</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:91" class="idref" href="#l1:91"><span class="id" title="binder">l1</span></a> <a id="l2:92" class="idref" href="#l2:92"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:91"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:91"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:92"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.union" class="idref" href="#Make.union"><span class="id" title="definition">union</span></a> <a id="s1:93" class="idref" href="#s1:93"><span class="id" title="binder">s1</span></a> <a id="s2:94" class="idref" href="#s2:94"><span class="id" title="binder">s2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:93"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:94"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.rpo.list_set.html#Make.without_red_add_without_red"><span class="id" title="lemma">without_red_add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:93"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:94"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s1:93"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_1_aux" class="idref" href="#Make.union_1_aux"><span class="id" title="lemma">union_1_aux</span></a> :<br/>
+<span class="id" title="keyword">∀</span> (<a id="l1:95" class="idref" href="#l1:95"><span class="id" title="binder">l1</span></a> <a id="l2:96" class="idref" href="#l2:96"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="e:97" class="idref" href="#e:97"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:97"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:95"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:97"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:95"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:96"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_1" class="idref" href="#Make.union_1"><span class="id" title="lemma">union_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:98" class="idref" href="#s1:98"><span class="id" title="binder">s1</span></a> <a id="s2:99" class="idref" href="#s2:99"><span class="id" title="binder">s2</span></a> <a id="e:100" class="idref" href="#e:100"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:100"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:98"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:100"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:98"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:99"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_2_aux" class="idref" href="#Make.union_2_aux"><span class="id" title="lemma">union_2_aux</span></a> :<br/>
+<span class="id" title="keyword">∀</span> (<a id="l1:101" class="idref" href="#l1:101"><span class="id" title="binder">l1</span></a> <a id="l2:102" class="idref" href="#l2:102"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="e:103" class="idref" href="#e:103"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:103"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:102"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:103"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:101"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:102"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_2" class="idref" href="#Make.union_2"><span class="id" title="lemma">union_2</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:104" class="idref" href="#s1:104"><span class="id" title="binder">s1</span></a> <a id="s2:105" class="idref" href="#s2:105"><span class="id" title="binder">s2</span></a> <a id="e:106" class="idref" href="#e:106"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:106"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:105"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:106"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:104"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:105"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_12_aux" class="idref" href="#Make.union_12_aux"><span class="id" title="lemma">union_12_aux</span></a> :<br/>
+<span class="id" title="keyword">∀</span> (<a id="l1:107" class="idref" href="#l1:107"><span class="id" title="binder">l1</span></a> <a id="l2:108" class="idref" href="#l2:108"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) (<a id="e:109" class="idref" href="#e:109"><span class="id" title="binder">e</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:109"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add_without_red"><span class="id" title="definition">add_without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:107"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:108"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:109"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:107"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:109"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:108"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_12" class="idref" href="#Make.union_12"><span class="id" title="lemma">union_12</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:110" class="idref" href="#s1:110"><span class="id" title="binder">s1</span></a> <a id="s2:111" class="idref" href="#s2:111"><span class="id" title="binder">s2</span></a> <a id="e:112" class="idref" href="#e:112"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:112"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:110"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:111"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:112"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:110"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:112"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:111"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="Make.remove_not_common" class="idref" href="#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> (<a id="acc:113" class="idref" href="#acc:113"><span class="id" title="binder">acc</span></a> <a id="l1:114" class="idref" href="#l1:114"><span class="id" title="binder">l1</span></a> <a id="l2:115" class="idref" href="#l2:115"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l2</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.list_set.html#l2:115"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.rpo.list_set.html#acc:113"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In_dec"><span class="id" title="abbreviation">In_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_elt_dec"><span class="id" title="definition">eq_elt_dec</span></a> <span class="id" title="var">e</span> <a class="idref" href="hydras.rpo.list_set.html#l1:114"><span class="id" title="variable">l1</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.list_set.html#remove_not_common:116"><span class="id" title="definition">remove_not_common</span></a> (<span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:113"><span class="id" title="variable">acc</span></a>) <a class="idref" href="hydras.rpo.list_set.html#l1:114"><span class="id" title="variable">l1</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.list_set.html#remove_not_common:116"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:113"><span class="id" title="variable">acc</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:114"><span class="id" title="variable">l1</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_remove_not_common_aux" class="idref" href="#Make.without_red_remove_not_common_aux"><span class="id" title="lemma">without_red_remove_not_common_aux</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="acc:118" class="idref" href="#acc:118"><span class="id" title="binder">acc</span></a> <a id="l1:119" class="idref" href="#l1:119"><span class="id" title="binder">l1</span></a> <a id="l2:120" class="idref" href="#l2:120"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:121" class="idref" href="#e:121"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:121"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:118"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:121"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:120"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:118"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:119"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:120"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:118"><span class="id" title="variable">acc</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:119"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:120"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_remove_not_common" class="idref" href="#Make.without_red_remove_not_common"><span class="id" title="lemma">without_red_remove_not_common</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:122" class="idref" href="#l1:122"><span class="id" title="binder">l1</span></a> <a id="l2:123" class="idref" href="#l2:123"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:122"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:123"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:122"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:123"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.inter" class="idref" href="#Make.inter"><span class="id" title="definition">inter</span></a> <a id="s1:124" class="idref" href="#s1:124"><span class="id" title="binder">s1</span></a> <a id="s2:125" class="idref" href="#s2:125"><span class="id" title="binder">s2</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.mk_set"><span class="id" title="constructor">mk_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:124"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:125"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.rpo.list_set.html#Make.without_red_remove_not_common"><span class="id" title="lemma">without_red_remove_not_common</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.list_set.html#s1:124"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:125"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.is_a_set"><span class="id" title="projection">is_a_set</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_1_aux" class="idref" href="#Make.inter_1_aux"><span class="id" title="lemma">inter_1_aux</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="acc:126" class="idref" href="#acc:126"><span class="id" title="binder">acc</span></a> <a id="l1:127" class="idref" href="#l1:127"><span class="id" title="binder">l1</span></a> <a id="l2:128" class="idref" href="#l2:128"><span class="id" title="binder">l2</span></a> <a id="e:129" class="idref" href="#e:129"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:129"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:126"><span class="id" title="variable">acc</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:127"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:128"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:129"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:126"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:129"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:127"><span class="id" title="variable">l1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_1" class="idref" href="#Make.inter_1"><span class="id" title="lemma">inter_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:130" class="idref" href="#s1:130"><span class="id" title="binder">s1</span></a> <a id="s2:131" class="idref" href="#s2:131"><span class="id" title="binder">s2</span></a> <a id="e:132" class="idref" href="#e:132"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:132"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:130"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:131"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:132"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:130"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_2_aux" class="idref" href="#Make.inter_2_aux"><span class="id" title="lemma">inter_2_aux</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="acc:133" class="idref" href="#acc:133"><span class="id" title="binder">acc</span></a> <a id="l1:134" class="idref" href="#l1:134"><span class="id" title="binder">l1</span></a> <a id="l2:135" class="idref" href="#l2:135"><span class="id" title="binder">l2</span></a> <a id="e:136" class="idref" href="#e:136"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:136"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:133"><span class="id" title="variable">acc</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:134"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:135"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:136"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:133"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:136"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:135"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_2" class="idref" href="#Make.inter_2"><span class="id" title="lemma">inter_2</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:137" class="idref" href="#s1:137"><span class="id" title="binder">s1</span></a> <a id="s2:138" class="idref" href="#s2:138"><span class="id" title="binder">s2</span></a> <a id="e:139" class="idref" href="#e:139"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:139"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:137"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:138"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:139"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:138"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_12_aux" class="idref" href="#Make.inter_12_aux"><span class="id" title="lemma">inter_12_aux</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="acc:140" class="idref" href="#acc:140"><span class="id" title="binder">acc</span></a> <a id="l1:141" class="idref" href="#l1:141"><span class="id" title="binder">l1</span></a> <a id="l2:142" class="idref" href="#l2:142"><span class="id" title="binder">l2</span></a> <a id="e:143" class="idref" href="#e:143"><span class="id" title="binder">e</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:143"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:141"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:143"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:142"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:143"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.remove_not_common"><span class="id" title="definition">remove_not_common</span></a> <a class="idref" href="hydras.rpo.list_set.html#acc:140"><span class="id" title="variable">acc</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:141"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:142"><span class="id" title="variable">l2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.inter_12" class="idref" href="#Make.inter_12"><span class="id" title="lemma">inter_12</span></a> : <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:152" class="idref" href="#s1:152"><span class="id" title="binder">s1</span></a> <a id="s2:153" class="idref" href="#s2:153"><span class="id" title="binder">s2</span></a> <a id="e:154" class="idref" href="#e:154"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:154"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:152"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:154"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:153"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:154"><span class="id" title="variable">e</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:152"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:153"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.subset" class="idref" href="#Make.subset"><span class="id" title="definition">subset</span></a> <a id="s1:155" class="idref" href="#s1:155"><span class="id" title="binder">s1</span></a> <a id="s2:156" class="idref" href="#s2:156"><span class="id" title="binder">s2</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:157" class="idref" href="#e:157"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:157"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:155"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:157"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:156"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_dec" class="idref" href="#Make.subset_dec"><span class="id" title="lemma">subset_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:158" class="idref" href="#s1:158"><span class="id" title="binder">s1</span></a> <a id="s2:159" class="idref" href="#s2:159"><span class="id" title="binder">s2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:158"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:159"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:158"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:159"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_union_1" class="idref" href="#Make.subset_union_1"><span class="id" title="lemma">subset_union_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:160" class="idref" href="#s1:160"><span class="id" title="binder">s1</span></a> <a id="s2:161" class="idref" href="#s2:161"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:160"><span class="id" title="variable">s1</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:160"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:161"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_union_2" class="idref" href="#Make.subset_union_2"><span class="id" title="lemma">subset_union_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:162" class="idref" href="#s1:162"><span class="id" title="binder">s1</span></a> <a id="s2:163" class="idref" href="#s2:163"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:163"><span class="id" title="variable">s2</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:162"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:163"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_inter_1" class="idref" href="#Make.subset_inter_1"><span class="id" title="lemma">subset_inter_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:164" class="idref" href="#s1:164"><span class="id" title="binder">s1</span></a> <a id="s2:165" class="idref" href="#s2:165"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:164"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:165"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s1:164"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_inter_2" class="idref" href="#Make.subset_inter_2"><span class="id" title="lemma">subset_inter_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:166" class="idref" href="#s1:166"><span class="id" title="binder">s1</span></a> <a id="s2:167" class="idref" href="#s2:167"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:166"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:167"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:167"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.eq_set" class="idref" href="#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a id="s1:168" class="idref" href="#s1:168"><span class="id" title="binder">s1</span></a> <a id="s2:169" class="idref" href="#s2:169"><span class="id" title="binder">s2</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e:170" class="idref" href="#e:170"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:170"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:168"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:170"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:169"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.eq_set_dec" class="idref" href="#Make.eq_set_dec"><span class="id" title="lemma">eq_set_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:171" class="idref" href="#s1:171"><span class="id" title="binder">s1</span></a> <a id="s2:172" class="idref" href="#s2:172"><span class="id" title="binder">s2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:171"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:172"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:171"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:172"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.eq_set_refl" class="idref" href="#Make.eq_set_refl"><span class="id" title="lemma">eq_set_refl</span></a> : <span class="id" title="keyword">∀</span> <a id="s:173" class="idref" href="#s:173"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:173"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:173"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.eq_set_sym" class="idref" href="#Make.eq_set_sym"><span class="id" title="lemma">eq_set_sym</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:174" class="idref" href="#s1:174"><span class="id" title="binder">s1</span></a> <a id="s2:175" class="idref" href="#s2:175"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:174"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:175"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:175"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:174"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.eq_set_trans" class="idref" href="#Make.eq_set_trans"><span class="id" title="lemma">eq_set_trans</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:176" class="idref" href="#s1:176"><span class="id" title="binder">s1</span></a> <a id="s2:177" class="idref" href="#s2:177"><span class="id" title="binder">s2</span></a> <a id="s3:178" class="idref" href="#s3:178"><span class="id" title="binder">s3</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:176"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:177"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:177"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s3:178"><span class="id" title="variable">s3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:176"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s3:178"><span class="id" title="variable">s3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.add_comm" class="idref" href="#Make.add_comm"><span class="id" title="lemma">add_comm</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="e1:179" class="idref" href="#e1:179"><span class="id" title="binder">e1</span></a> <a id="e2:180" class="idref" href="#e2:180"><span class="id" title="binder">e2</span></a> <a id="s:181" class="idref" href="#s:181"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e1:179"><span class="id" title="variable">e1</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e2:180"><span class="id" title="variable">e2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:181"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e2:180"><span class="id" title="variable">e2</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="hydras.rpo.list_set.html#e1:179"><span class="id" title="variable">e1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:181"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_empty_1" class="idref" href="#Make.union_empty_1"><span class="id" title="lemma">union_empty_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:182" class="idref" href="#s:182"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:182"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.empty"><span class="id" title="definition">empty</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:182"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_empty_2" class="idref" href="#Make.union_empty_2"><span class="id" title="lemma">union_empty_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:183" class="idref" href="#s:183"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:183"><span class="id" title="variable">s</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:183"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.empty"><span class="id" title="definition">empty</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_comm" class="idref" href="#Make.union_comm"><span class="id" title="lemma">union_comm</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:184" class="idref" href="#s1:184"><span class="id" title="binder">s1</span></a> <a id="s2:185" class="idref" href="#s2:185"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:184"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:185"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:185"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:184"><span class="id" title="variable">s1</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_assoc" class="idref" href="#Make.union_assoc"><span class="id" title="lemma">union_assoc</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:186" class="idref" href="#s1:186"><span class="id" title="binder">s1</span></a> <a id="s2:187" class="idref" href="#s2:187"><span class="id" title="binder">s2</span></a> <a id="s3:188" class="idref" href="#s3:188"><span class="id" title="binder">s3</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:186"><span class="id" title="variable">s1</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:187"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s3:188"><span class="id" title="variable">s3</span></a>)) (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:186"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:187"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s3:188"><span class="id" title="variable">s3</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.filter_union" class="idref" href="#Make.filter_union"><span class="id" title="lemma">filter_union</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:189" class="idref" href="#P:189"><span class="id" title="binder">P</span></a> <a id="P_dec:190" class="idref" href="#P_dec:190"><span class="id" title="binder">P_dec</span></a> <a id="s1:191" class="idref" href="#s1:191"><span class="id" title="binder">s1</span></a> <a id="s2:192" class="idref" href="#s2:192"><span class="id" title="binder">s2</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:189"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:190"><span class="id" title="variable">P_dec</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:191"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:192"><span class="id" title="variable">s2</span></a>))  (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:189"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:190"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:191"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P:189"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.list_set.html#P_dec:190"><span class="id" title="variable">P_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:192"><span class="id" title="variable">s2</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_filter" class="idref" href="#Make.subset_filter"><span class="id" title="lemma">subset_filter</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P1:193" class="idref" href="#P1:193"><span class="id" title="binder">P1</span></a> <a id="P2:194" class="idref" href="#P2:194"><span class="id" title="binder">P2</span></a> : <a class="idref" href="hydras.rpo.list_set.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="P1_dec:195" class="idref" href="#P1_dec:195"><span class="id" title="binder">P1_dec</span></a> <a id="P2_dec:196" class="idref" href="#P2_dec:196"><span class="id" title="binder">P2_dec</span></a> <a id="s1:197" class="idref" href="#s1:197"><span class="id" title="binder">s1</span></a> <a id="s2:198" class="idref" href="#s2:198"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:197"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:198"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:199" class="idref" href="#e:199"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#P1:193"><span class="id" title="variable">P1</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:199"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#P2:194"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:199"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P1:193"><span class="id" title="variable">P1</span></a> <a class="idref" href="hydras.rpo.list_set.html#P1_dec:195"><span class="id" title="variable">P1_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:197"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="hydras.rpo.list_set.html#P2:194"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.list_set.html#P2_dec:196"><span class="id" title="variable">P2_dec</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:198"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_compat_1" class="idref" href="#Make.subset_compat_1"><span class="id" title="lemma">subset_compat_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:200" class="idref" href="#s1:200"><span class="id" title="binder">s1</span></a> <a id="s1':201" class="idref" href="#s1':201"><span class="id" title="binder">s1'</span></a> <a id="s2:202" class="idref" href="#s2:202"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:200"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':201"><span class="id" title="variable">s1'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:200"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:202"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':201"><span class="id" title="variable">s1'</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:202"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_compat_2" class="idref" href="#Make.subset_compat_2"><span class="id" title="lemma">subset_compat_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:203" class="idref" href="#s1:203"><span class="id" title="binder">s1</span></a> <a id="s2:204" class="idref" href="#s2:204"><span class="id" title="binder">s2</span></a> <a id="s2':205" class="idref" href="#s2':205"><span class="id" title="binder">s2'</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:204"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':205"><span class="id" title="variable">s2'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:203"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:204"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:203"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':205"><span class="id" title="variable">s2'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_compat" class="idref" href="#Make.subset_compat"><span class="id" title="lemma">subset_compat</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:206" class="idref" href="#s1:206"><span class="id" title="binder">s1</span></a> <a id="s1':207" class="idref" href="#s1':207"><span class="id" title="binder">s1'</span></a> <a id="s2:208" class="idref" href="#s2:208"><span class="id" title="binder">s2</span></a> <a id="s2':209" class="idref" href="#s2':209"><span class="id" title="binder">s2'</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:206"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':207"><span class="id" title="variable">s1'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:208"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':209"><span class="id" title="variable">s2'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:206"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:208"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':207"><span class="id" title="variable">s1'</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':209"><span class="id" title="variable">s2'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_compat_subset_1" class="idref" href="#Make.union_compat_subset_1"><span class="id" title="lemma">union_compat_subset_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:210" class="idref" href="#s1:210"><span class="id" title="binder">s1</span></a> <a id="s2:211" class="idref" href="#s2:211"><span class="id" title="binder">s2</span></a> <a id="s:212" class="idref" href="#s:212"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:210"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:211"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:210"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:212"><span class="id" title="variable">s</span></a>)  (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:211"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:212"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_compat_subset_2" class="idref" href="#Make.union_compat_subset_2"><span class="id" title="lemma">union_compat_subset_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:213" class="idref" href="#s1:213"><span class="id" title="binder">s1</span></a> <a id="s2:214" class="idref" href="#s2:214"><span class="id" title="binder">s2</span></a> <a id="s:215" class="idref" href="#s:215"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:213"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:214"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:215"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:213"><span class="id" title="variable">s1</span></a>)  (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:215"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:214"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.union_compat_eq_set" class="idref" href="#Make.union_compat_eq_set"><span class="id" title="lemma">union_compat_eq_set</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:216" class="idref" href="#s1:216"><span class="id" title="binder">s1</span></a> <a id="s1':217" class="idref" href="#s1':217"><span class="id" title="binder">s1'</span></a> <a id="s2:218" class="idref" href="#s2:218"><span class="id" title="binder">s2</span></a> <a id="s2':219" class="idref" href="#s2':219"><span class="id" title="binder">s2'</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:216"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':217"><span class="id" title="variable">s1'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:218"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':219"><span class="id" title="variable">s2'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:216"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:218"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1':217"><span class="id" title="variable">s1'</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2':219"><span class="id" title="variable">s2'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Make.subset_subset_union" class="idref" href="#Make.subset_subset_union"><span class="id" title="lemma">subset_subset_union</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:220" class="idref" href="#s1:220"><span class="id" title="binder">s1</span></a> <a id="s2:221" class="idref" href="#s2:221"><span class="id" title="binder">s2</span></a> <a id="s:222" class="idref" href="#s:222"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:220"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:222"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:221"><span class="id" title="variable">s2</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:222"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:220"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:221"><span class="id" title="variable">s2</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s:222"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.cardinal" class="idref" href="#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a id="s:223" class="idref" href="#s:223"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#length"><span class="id" title="abbreviation">List.length</span></a> <a class="idref" href="hydras.rpo.list_set.html#s:223"><span class="id" title="variable">s</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_subset" class="idref" href="#Make.cardinal_subset"><span class="id" title="lemma">cardinal_subset</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:224" class="idref" href="#s1:224"><span class="id" title="binder">s1</span></a> <a id="s2:225" class="idref" href="#s2:225"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:224"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:225"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:224"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:225"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_union_1" class="idref" href="#Make.cardinal_union_1"><span class="id" title="lemma">cardinal_union_1</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:226" class="idref" href="#s1:226"><span class="id" title="binder">s1</span></a> <a id="s2:227" class="idref" href="#s2:227"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:226"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:226"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:227"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_union_2" class="idref" href="#Make.cardinal_union_2"><span class="id" title="lemma">cardinal_union_2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:228" class="idref" href="#s1:228"><span class="id" title="binder">s1</span></a> <a id="s2:229" class="idref" href="#s2:229"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:229"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:228"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:229"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_union_inter_12" class="idref" href="#Make.cardinal_union_inter_12"><span class="id" title="lemma">cardinal_union_inter_12</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:230" class="idref" href="#s1:230"><span class="id" title="binder">s1</span></a> <a id="s2:231" class="idref" href="#s2:231"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:230"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:231"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:230"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:231"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:230"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:231"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_union" class="idref" href="#Make.cardinal_union"><span class="id" title="lemma">cardinal_union</span></a>:<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:246" class="idref" href="#s1:246"><span class="id" title="binder">s1</span></a> <a id="s2:247" class="idref" href="#s2:247"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:246"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:247"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:246"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:247"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'-'_x"><span class="id" title="notation">-</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> (<a class="idref" href="hydras.rpo.list_set.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:246"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:247"><span class="id" title="variable">s2</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_eq_set" class="idref" href="#Make.cardinal_eq_set"><span class="id" title="lemma">cardinal_eq_set</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:248" class="idref" href="#s1:248"><span class="id" title="binder">s1</span></a> <a id="s2:249" class="idref" href="#s2:249"><span class="id" title="binder">s2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:248"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:249"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:248"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:249"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.subset_cardinal_not_eq_not_eq_set" class="idref" href="#Make.subset_cardinal_not_eq_not_eq_set"><span class="id" title="lemma">subset_cardinal_not_eq_not_eq_set</span></a>  :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:250" class="idref" href="#s1:250"><span class="id" title="binder">s1</span></a> <a id="s2:251" class="idref" href="#s2:251"><span class="id" title="binder">s2</span></a> <a id="e:252" class="idref" href="#e:252"><span class="id" title="binder">e</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:250"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:251"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:252"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:250"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="hydras.rpo.list_set.html#e:252"><span class="id" title="variable">e</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:251"><span class="id" title="variable">s2</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:250"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:251"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.eq_set_list_permut_support" class="idref" href="#Make.eq_set_list_permut_support"><span class="id" title="lemma">eq_set_list_permut_support</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s1:255" class="idref" href="#s1:255"><span class="id" title="binder">s1</span></a> <a id="s2:256" class="idref" href="#s2:256"><span class="id" title="binder">s2</span></a>,  <a class="idref" href="hydras.rpo.list_set.html#Make.eq_set"><span class="id" title="definition">eq_set</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:255"><span class="id" title="variable">s1</span></a> <a class="idref" href="hydras.rpo.list_set.html#s2:256"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make.list_permut"><span class="id" title="definition">LP.list_permut</span></a> <a class="idref" href="hydras.rpo.list_set.html#s1:255"><span class="id" title="variable">s1</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>) <a class="idref" href="hydras.rpo.list_set.html#s2:256"><span class="id" title="variable">s2</span></a>.(<a class="idref" href="hydras.rpo.list_set.html#Make.support"><span class="id" title="projection">support</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.without_red_permut" class="idref" href="#Make.without_red_permut"><span class="id" title="lemma">without_red_permut</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:257" class="idref" href="#l1:257"><span class="id" title="binder">l1</span></a> <a id="l2:258" class="idref" href="#l2:258"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:257"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.list_permut"><span class="id" title="definition">LP.list_permut</span></a> <a class="idref" href="hydras.rpo.list_set.html#l1:257"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:258"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.list_set.html#Make.without_red"><span class="id" title="definition">without_red</span></a> <a class="idref" href="hydras.rpo.list_set.html#l2:258"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.list_set.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.more_list.html b/theories/html/hydras.rpo.more_list.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.rpo.more_list.html
@@ -0,0 +1,442 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.more_list</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.more_list</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+by Evelyne Contejean, LRI 
+<div class="paragraph"> </div>
+
+<a id="lab131"></a><h1 class="section">Some additional properties for the Coq lists.</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab132"></a><h2 class="section">Relations between length, map, append, In and nth.</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="map_map" class="idref" href="#map_map"><span class="id" title="lemma">map_map</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> <a id="B:2" class="idref" href="#B:2"><span class="id" title="binder">B</span></a> <a id="C:3" class="idref" href="#C:3"><span class="id" title="binder">C</span></a> : <span class="id" title="keyword">Set</span>) (<a id="l:4" class="idref" href="#l:4"><span class="id" title="binder">l</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:1"><span class="id" title="variable">A</span></a>)) (<a id="f:5" class="idref" href="#f:5"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:2"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#C:3"><span class="id" title="variable">C</span></a>) (<a id="g:6" class="idref" href="#g:6"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><a class="idref" href="hydras.rpo.more_list.html#B:2"><span class="id" title="variable">B</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:5"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#g:6"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:4"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#f:5"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.more_list.html#g:6"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.more_list.html#x:7"><span class="id" title="variable">x</span></a>)) <a class="idref" href="hydras.rpo.more_list.html#l:4"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="list_app_length" class="idref" href="#list_app_length"><span class="id" title="lemma">list_app_length</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="A:8" class="idref" href="#A:8"><span class="id" title="binder">A</span></a>, <span class="id" title="keyword">∀</span> <a id="l1:9" class="idref" href="#l1:9"><span class="id" title="binder">l1</span></a> <a id="l2:10" class="idref" href="#l2:10"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:8"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="hydras.rpo.more_list.html#l1:9"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:10"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:9"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:10"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="length_map" class="idref" href="#length_map"><span class="id" title="lemma">length_map</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a> <a id="B:12" class="idref" href="#B:12"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:13" class="idref" href="#f:13"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:12"><span class="id" title="variable">B</span></a>) (<a id="l:14" class="idref" href="#l:14"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:11"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:13"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:14"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:14"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="map_app" class="idref" href="#map_app"><span class="id" title="lemma">map_app</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:15" class="idref" href="#A:15"><span class="id" title="binder">A</span></a> <a id="B:16" class="idref" href="#B:16"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:17" class="idref" href="#f:17"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:15"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:16"><span class="id" title="variable">B</span></a>) <a id="l1:18" class="idref" href="#l1:18"><span class="id" title="binder">l1</span></a> <a id="l2:19" class="idref" href="#l2:19"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:17"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.more_list.html#l1:18"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:19"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:18"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:17"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:19"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="in_in_map" class="idref" href="#in_in_map"><span class="id" title="lemma">in_in_map</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a> <a id="B:21" class="idref" href="#B:21"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:22" class="idref" href="#f:22"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:21"><span class="id" title="variable">B</span></a>) <a id="a:23" class="idref" href="#a:23"><span class="id" title="binder">a</span></a> <a id="l:24" class="idref" href="#l:24"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:23"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:24"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="hydras.rpo.more_list.html#f:22"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:23"><span class="id" title="variable">a</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:22"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:24"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="in_map_in" class="idref" href="#in_map_in"><span class="id" title="lemma">in_map_in</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:25" class="idref" href="#A:25"><span class="id" title="binder">A</span></a> <a id="B:26" class="idref" href="#B:26"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:27" class="idref" href="#f:27"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:25"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:26"><span class="id" title="variable">B</span></a>) <a id="b:28" class="idref" href="#b:28"><span class="id" title="binder">b</span></a> <a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:28"><span class="id" title="variable">b</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:29"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:30" class="idref" href="#a:30"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:29"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:27"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:30"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:28"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nth_error_map" class="idref" href="#nth_error_map"><span class="id" title="lemma">nth_error_map</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:31" class="idref" href="#A:31"><span class="id" title="binder">A</span></a> <a id="B:32" class="idref" href="#B:32"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:33" class="idref" href="#f:33"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:31"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:32"><span class="id" title="variable">B</span></a>) (<a id="l:34" class="idref" href="#l:34"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:31"><span class="id" title="variable">A</span></a>) <a id="i:35" class="idref" href="#i:35"><span class="id" title="binder">i</span></a>,<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:33"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:34"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.rpo.more_list.html#i:35"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_li</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:34"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#i:35"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">li</span> ⇒ <span class="id" title="var">f_li</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:33"><span class="id" title="variable">f</span></a> <span class="id" title="var">li</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:34"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#i:35"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">li</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab133"></a><h2 class="section">A measure on lists based on a measure on elements.</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="list_size" class="idref" href="#list_size"><span class="id" title="definition">list_size</span></a> (<a id="A:36" class="idref" href="#A:36"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size:37" class="idref" href="#size:37"><span class="id" title="binder">size</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:36"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:38" class="idref" href="#l:38"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:36"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:38"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#size:37"><span class="id" title="variable">size</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size:39"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:37"><span class="id" title="variable">size</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="list_size_tl_compat" class="idref" href="#list_size_tl_compat"><span class="id" title="lemma">list_size_tl_compat</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:41" class="idref" href="#A:41"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size:42" class="idref" href="#size:42"><span class="id" title="binder">size</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:41"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="a:43" class="idref" href="#a:43"><span class="id" title="binder">a</span></a> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a> <a id="l:45" class="idref" href="#l:45"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.more_list.html#size:42"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:42"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:42"><span class="id" title="variable">size</span></a> (<a class="idref" href="hydras.rpo.more_list.html#a:43"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:45"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:42"><span class="id" title="variable">size</span></a> (<a class="idref" href="hydras.rpo.more_list.html#b:44"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:45"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="list_size_app" class="idref" href="#list_size_app"><span class="id" title="lemma">list_size_app</span></a>:<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:46" class="idref" href="#A:46"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size:47" class="idref" href="#size:47"><span class="id" title="binder">size</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:46"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="l1:48" class="idref" href="#l1:48"><span class="id" title="binder">l1</span></a> <a id="l2:49" class="idref" href="#l2:49"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;<a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:47"><span class="id" title="variable">size</span></a> (<a class="idref" href="hydras.rpo.more_list.html#l1:48"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:49"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:47"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:48"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:47"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:49"><span class="id" title="variable">l2</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="list_size_fold" class="idref" href="#list_size_fold"><span class="id" title="lemma">list_size_fold</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:50" class="idref" href="#A:50"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size:51" class="idref" href="#size:51"><span class="id" title="binder">size</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:50"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="l:52" class="idref" href="#l:52"><span class="id" title="binder">l</span></a> <a id="n:53" class="idref" href="#n:53"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> (<a id="size_acc:54" class="idref" href="#size_acc:54"><span class="id" title="binder">size_acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="a:55" class="idref" href="#a:55"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:50"><span class="id" title="variable">A</span></a>) ⇒ <a class="idref" href="hydras.rpo.more_list.html#size_acc:54"><span class="id" title="variable">size_acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:51"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:55"><span class="id" title="variable">a</span></a>) <a class="idref" href="hydras.rpo.more_list.html#l:52"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#n:53"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:51"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:52"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="list_size_size_eq" class="idref" href="#list_size_size_eq"><span class="id" title="lemma">list_size_size_eq</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:56" class="idref" href="#A:56"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size1:57" class="idref" href="#size1:57"><span class="id" title="binder">size1</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:56"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="size2:58" class="idref" href="#size2:58"><span class="id" title="binder">size2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:56"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="l:59" class="idref" href="#l:59"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:60" class="idref" href="#a:60"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:59"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#size1:57"><span class="id" title="variable">size1</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:60"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#size2:58"><span class="id" title="variable">size2</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:60"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size1:57"><span class="id" title="variable">size1</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:59"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size2:58"><span class="id" title="variable">size2</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:59"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab134"></a><h2 class="section">Induction principles for list.</h2>
+
+ Induction on the length. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="list_rec2" class="idref" href="#list_rec2"><span class="id" title="definition">list_rec2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:63" class="idref" href="#A:63"><span class="id" title="binder">A</span></a>, <span class="id" title="keyword">∀</span> <a id="P:64" class="idref" href="#P:64"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:63"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="n:65" class="idref" href="#n:65"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:66" class="idref" href="#l:66"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:63"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:66"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.more_list.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:64"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:66"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:67" class="idref" href="#l:67"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:63"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.rpo.more_list.html#P:64"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:67"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="o_length" class="idref" href="#o_length"><span class="id" title="definition">o_length</span></a> (<a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="l1:69" class="idref" href="#l1:69"><span class="id" title="binder">l1</span></a> <a id="l2:70" class="idref" href="#l2:70"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:68"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:69"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:70"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="well_founded_length" class="idref" href="#well_founded_length"><span class="id" title="lemma">well_founded_length</span></a> : <span class="id" title="keyword">∀</span> <a id="A:71" class="idref" href="#A:71"><span class="id" title="binder">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="hydras.rpo.more_list.html#o_length"><span class="id" title="definition">o_length</span></a> (<span class="id" title="var">A</span> := <a class="idref" href="hydras.rpo.more_list.html#A:71"><span class="id" title="variable">A</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Induction on the the size. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="list_rec3" class="idref" href="#list_rec3"><span class="id" title="definition">list_rec3</span></a> (<a id="A:72" class="idref" href="#A:72"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="size:73" class="idref" href="#size:73"><span class="id" title="binder">size</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="P:74" class="idref" href="#P:74"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:72"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="n:75" class="idref" href="#n:75"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:76" class="idref" href="#l:76"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:72"><span class="id" title="variable">A</span></a>), <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.more_list.html#size:73"><span class="id" title="variable">size</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:76"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.more_list.html#n:75"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:74"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:76"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:77" class="idref" href="#l:77"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:72"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.rpo.more_list.html#P:74"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:77"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab135"></a><h2 class="section">How to remove an element in a list, whenever it is present.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="split_list" class="idref" href="#split_list"><span class="id" title="definition">split_list</span></a> (<a id="A:78" class="idref" href="#A:78"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>)<br/>
+&nbsp;&nbsp;(<a id="eqA:81" class="idref" href="#eqA:81"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:79" class="idref" href="#a1:79"><span class="id" title="binder">a1</span></a> <a id="a2:80" class="idref" href="#a2:80"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:78"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:79"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:80"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:79"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:80"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="l:82" class="idref" href="#l:82"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:78"><span class="id" title="variable">A</span></a>) (<a id="t:83" class="idref" href="#t:83"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:78"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:78"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:78"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:82"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.more_list.html#eqA:81"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#t:83"><span class="id" title="variable">t</span></a> <span class="id" title="var">a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">l'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <span class="id" title="keyword">let</span> (<a id="l1:86" class="idref" href="#l1:86"><span class="id" title="binder">l1</span></a>,<a id="l2:87" class="idref" href="#l2:87"><span class="id" title="binder">l2</span></a>) := <a class="idref" href="hydras.rpo.more_list.html#split_list:84"><span class="id" title="definition">split_list</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:81"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">l'</span> <a class="idref" href="hydras.rpo.more_list.html#t:83"><span class="id" title="variable">t</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:86"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:87"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="split_list_app_cons" class="idref" href="#split_list_app_cons"><span class="id" title="lemma">split_list_app_cons</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:88" class="idref" href="#A:88"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:91" class="idref" href="#eqA:91"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:89" class="idref" href="#a1:89"><span class="id" title="binder">a1</span></a> <a id="a2:90" class="idref" href="#a2:90"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:88"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:89"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:90"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:89"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:90"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <a id="t:92" class="idref" href="#t:92"><span class="id" title="binder">t</span></a> <a id="l:93" class="idref" href="#l:93"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#t:92"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:93"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">let</span> (<a id="l1:94" class="idref" href="#l1:94"><span class="id" title="binder">l1</span></a>, <a id="l2:95" class="idref" href="#l2:95"><span class="id" title="binder">l2</span></a>) := <a class="idref" href="hydras.rpo.more_list.html#split_list"><span class="id" title="definition">split_list</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:91"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:93"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#t:92"><span class="id" title="variable">t</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="hydras.rpo.more_list.html#l:93"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:94"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.more_list.html#t:92"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:95"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="remove" class="idref" href="#remove"><span class="id" title="definition">remove</span></a> (<a id="A:96" class="idref" href="#A:96"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:99" class="idref" href="#eqA:99"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:97" class="idref" href="#a1:97"><span class="id" title="binder">a1</span></a> <a id="a2:98" class="idref" href="#a2:98"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:96"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:97"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:98"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:97"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:98"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="a:100" class="idref" href="#a:100"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:96"><span class="id" title="variable">A</span></a>) (<a id="l:101" class="idref" href="#l:101"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:96"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:96"><span class="id" title="variable">A</span></a>)) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:101"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <br/>
+&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.more_list.html#eqA:99"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:100"><span class="id" title="variable">a</span></a> <span class="id" title="var">h</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#remove:102"><span class="id" title="definition">remove</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:99"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:100"><span class="id" title="variable">a</span></a> <span class="id" title="var">tl</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">rmv</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">rmv</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="in_remove" class="idref" href="#in_remove"><span class="id" title="lemma">in_remove</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:104" class="idref" href="#A:104"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:107" class="idref" href="#eqA:107"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:105" class="idref" href="#a1:105"><span class="id" title="binder">a1</span></a> <a id="a2:106" class="idref" href="#a2:106"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:104"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:105"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:106"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:105"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:106"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <a id="a:108" class="idref" href="#a:108"><span class="id" title="binder">a</span></a> <a id="l:109" class="idref" href="#l:109"><span class="id" title="binder">l</span></a>,  <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:107"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:108"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:109"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:108"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:109"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">l'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:108"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:109"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">let</span> (<a id="l1:110" class="idref" href="#l1:110"><span class="id" title="binder">l1</span></a>, <a id="l2:111" class="idref" href="#l2:111"><span class="id" title="binder">l2</span></a>) := <a class="idref" href="hydras.rpo.more_list.html#split_list"><span class="id" title="definition">split_list</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:107"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:109"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:108"><span class="id" title="variable">a</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">l'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:110"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:111"><span class="id" title="variable">l2</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="remove_list" class="idref" href="#remove_list"><span class="id" title="definition">remove_list</span></a> (<a id="A:112" class="idref" href="#A:112"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:115" class="idref" href="#eqA:115"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:113" class="idref" href="#a1:113"><span class="id" title="binder">a1</span></a> <a id="a2:114" class="idref" href="#a2:114"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:112"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:113"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:114"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:113"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:114"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+(<a id="la:116" class="idref" href="#la:116"><span class="id" title="binder">la</span></a> <a id="l:117" class="idref" href="#l:117"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:112"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:112"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#la:116"><span class="id" title="variable">la</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:117"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">la'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:117"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.more_list.html#eqA:115"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.more_list.html#remove_list:118"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:115"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">la'</span> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#remove_list:118"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:115"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#la:116"><span class="id" title="variable">la</span></a> <span class="id" title="var">l'</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">rmv</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">rmv</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab136"></a><h2 class="section">Iterators.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="fold_left2" class="idref" href="#fold_left2"><span class="id" title="definition">fold_left2</span></a> (<a id="A:121" class="idref" href="#A:121"><span class="id" title="binder">A</span></a> <a id="B:122" class="idref" href="#B:122"><span class="id" title="binder">B</span></a> <a id="C:123" class="idref" href="#C:123"><span class="id" title="binder">C</span></a> : <span class="id" title="keyword">Set</span>) (<a id="f:124" class="idref" href="#f:124"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:121"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:122"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#C:123"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:121"><span class="id" title="variable">A</span></a>) (<a id="a:125" class="idref" href="#a:125"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:121"><span class="id" title="variable">A</span></a>) (<a id="l1:126" class="idref" href="#l1:126"><span class="id" title="binder">l1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:122"><span class="id" title="variable">B</span></a>) (<a id="l2:127" class="idref" href="#l2:127"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#C:123"><span class="id" title="variable">C</span></a>)  <br/>
+&nbsp;&nbsp;{<span class="id" title="keyword">struct</span> <span class="id" title="var">l1</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:121"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l1:126"><span class="id" title="variable">l1</span></a>, <a class="idref" href="hydras.rpo.more_list.html#l2:127"><span class="id" title="variable">l2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:125"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;| <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">t1</span>, <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">t2</span> ⇒ <a class="idref" href="hydras.rpo.more_list.html#fold_left2:128"><span class="id" title="definition">fold_left2</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:124"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.more_list.html#f:124"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:125"><span class="id" title="variable">a</span></a> <span class="id" title="var">b</span> <span class="id" title="var">c</span>) <span class="id" title="var">t1</span> <span class="id" title="var">t2</span><br/>
+&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab137"></a><h2 class="section">more properties on the nth element.</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="nth_error_ok_in" class="idref" href="#nth_error_ok_in"><span class="id" title="lemma">nth_error_ok_in</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:131" class="idref" href="#A:131"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Set</span>) <a id="n:132" class="idref" href="#n:132"><span class="id" title="binder">n</span></a> (<a id="l:133" class="idref" href="#l:133"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:131"><span class="id" title="variable">A</span></a>) (<a id="a:134" class="idref" href="#a:134"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:131"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:133"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.more_list.html#n:132"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:134"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:134"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:133"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab138"></a><h2 class="section">Association lists.</h2>
+
+<a id="lab139"></a><h3 class="section">find.</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="find" class="idref" href="#find"><span class="id" title="definition">find</span></a> (<a id="A:135" class="idref" href="#A:135"><span class="id" title="binder">A</span></a> <a id="B:136" class="idref" href="#B:136"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:139" class="idref" href="#eqA:139"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:137" class="idref" href="#a1:137"><span class="id" title="binder">a1</span></a> <a id="a2:138" class="idref" href="#a2:138"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:135"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:137"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:138"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:137"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:138"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+(<a id="a:140" class="idref" href="#a:140"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:135"><span class="id" title="variable">A</span></a>) (<a id="l:141" class="idref" href="#l:141"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:135"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:136"><span class="id" title="variable">B</span></a>)) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="hydras.rpo.more_list.html#B:136"><span class="id" title="variable">B</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:141"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">b1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.more_list.html#eqA:139"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:140"><span class="id" title="variable">a</span></a> <span class="id" title="var">a1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">b1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.more_list.html#find:142"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:139"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:140"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:141"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="find_not_mem" class="idref" href="#find_not_mem"><span class="id" title="lemma">find_not_mem</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:144" class="idref" href="#A:144"><span class="id" title="binder">A</span></a> <a id="B:145" class="idref" href="#B:145"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:148" class="idref" href="#eqA:148"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:146" class="idref" href="#a1:146"><span class="id" title="binder">a1</span></a> <a id="a2:147" class="idref" href="#a2:147"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:144"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:146"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:147"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:146"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:147"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="a:149" class="idref" href="#a:149"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:144"><span class="id" title="variable">A</span></a>) (<a id="b:150" class="idref" href="#b:150"><span class="id" title="binder">b</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:145"><span class="id" title="variable">B</span></a>) (<a id="l:151" class="idref" href="#l:151"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:144"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:145"><span class="id" title="variable">B</span></a>)) (<a id="dom:152" class="idref" href="#dom:152"><span class="id" title="binder">dom</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:144"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:149"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#dom:152"><span class="id" title="variable">dom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a':153" class="idref" href="#a':153"><span class="id" title="binder">a'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a':153"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.rpo.more_list.html#dom:152"><span class="id" title="variable">dom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:148"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a':153"><span class="id" title="variable">a'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.more_list.html#a:149"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.more_list.html#b:150"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:151"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:148"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a':153"><span class="id" title="variable">a'</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:151"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab140"></a><h3 class="section">number of occurences of the first element of a pair.</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="nb_occ" class="idref" href="#nb_occ"><span class="id" title="definition">nb_occ</span></a> (<a id="A:154" class="idref" href="#A:154"><span class="id" title="binder">A</span></a> <a id="B:155" class="idref" href="#B:155"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:158" class="idref" href="#eqA:158"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:156" class="idref" href="#a1:156"><span class="id" title="binder">a1</span></a> <a id="a2:157" class="idref" href="#a2:157"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:154"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:156"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:157"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:156"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:157"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="a:159" class="idref" href="#a:159"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:154"><span class="id" title="variable">A</span></a>) (<a id="l:160" class="idref" href="#l:160"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:154"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:155"><span class="id" title="variable">B</span></a>)) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:160"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ 0<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">a'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="hydras.rpo.more_list.html#eqA:158"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:159"><span class="id" title="variable">a</span></a> <span class="id" title="var">a'</span>) <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="hydras.rpo.more_list.html#nb_occ:161"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:158"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:159"><span class="id" title="variable">a</span></a> <span class="id" title="var">tl</span>) <span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.more_list.html#nb_occ:161"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:158"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:159"><span class="id" title="variable">a</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="none_nb_occ_O" class="idref" href="#none_nb_occ_O"><span class="id" title="lemma">none_nb_occ_O</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:163" class="idref" href="#A:163"><span class="id" title="binder">A</span></a> <a id="B:164" class="idref" href="#B:164"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:167" class="idref" href="#eqA:167"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:165" class="idref" href="#a1:165"><span class="id" title="binder">a1</span></a> <a id="a2:166" class="idref" href="#a2:166"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:163"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:165"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:166"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:165"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:166"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="a:168" class="idref" href="#a:168"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:163"><span class="id" title="variable">A</span></a>) (<a id="l:169" class="idref" href="#l:169"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:163"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:164"><span class="id" title="variable">B</span></a>)),<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:167"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:168"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:169"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:167"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:168"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:169"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="some_nb_occ_Sn" class="idref" href="#some_nb_occ_Sn"><span class="id" title="lemma">some_nb_occ_Sn</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:170" class="idref" href="#A:170"><span class="id" title="binder">A</span></a> <a id="B:171" class="idref" href="#B:171"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:174" class="idref" href="#eqA:174"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:172" class="idref" href="#a1:172"><span class="id" title="binder">a1</span></a> <a id="a2:173" class="idref" href="#a2:173"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:170"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:172"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:173"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:172"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:173"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;(<a id="a:175" class="idref" href="#a:175"><span class="id" title="binder">a</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:170"><span class="id" title="variable">A</span></a>) (<a id="l:176" class="idref" href="#l:176"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:170"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:171"><span class="id" title="variable">B</span></a>)) <a id="b:177" class="idref" href="#b:177"><span class="id" title="binder">b</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:174"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:175"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:176"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:177"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:174"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:175"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:176"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="nb_occ_app" class="idref" href="#nb_occ_app"><span class="id" title="lemma">nb_occ_app</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:178" class="idref" href="#A:178"><span class="id" title="binder">A</span></a> <a id="B:179" class="idref" href="#B:179"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:182" class="idref" href="#eqA:182"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:180" class="idref" href="#a1:180"><span class="id" title="binder">a1</span></a> <a id="a2:181" class="idref" href="#a2:181"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:178"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:180"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:181"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:180"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:181"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>)<br/>
+&nbsp;&nbsp;<a id="a:183" class="idref" href="#a:183"><span class="id" title="binder">a</span></a> (<a id="l1:184" class="idref" href="#l1:184"><span class="id" title="binder">l1</span></a> <a id="l2:185" class="idref" href="#l2:185"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:178"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:179"><span class="id" title="variable">B</span></a>)), <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:182"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:183"><span class="id" title="variable">a</span></a> (<a class="idref" href="hydras.rpo.more_list.html#l1:184"><span class="id" title="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a><a class="idref" href="hydras.rpo.more_list.html#l2:185"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:182"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:183"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l1:184"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:182"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:183"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l2:185"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="reduce_assoc_list" class="idref" href="#reduce_assoc_list"><span class="id" title="lemma">reduce_assoc_list</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:186" class="idref" href="#A:186"><span class="id" title="binder">A</span></a> <a id="B:187" class="idref" href="#B:187"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:190" class="idref" href="#eqA:190"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:188" class="idref" href="#a1:188"><span class="id" title="binder">a1</span></a> <a id="a2:189" class="idref" href="#a2:189"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:186"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:188"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:189"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:188"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:189"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>),<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:191" class="idref" href="#l:191"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.more_list.html#A:186"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:187"><span class="id" title="variable">B</span></a>)), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l':192" class="idref" href="#l':192"><span class="id" title="binder">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:193" class="idref" href="#a:193"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.rpo.more_list.html#nb_occ"><span class="id" title="definition">nb_occ</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:190"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:193"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l':192"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:194" class="idref" href="#a:194"><span class="id" title="binder">a</span></a>, <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:190"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:194"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:191"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:190"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:194"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l':192"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+map_without_repetition applies a function to the elements of a list,
+but only a single time when there are several consecutive occurences of the
+same element. Moreover, the function is supposed to return an option as a result,
+in order to simulate exceptions, and the abnormal results are discarted.
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="map_without_repetition" class="idref" href="#map_without_repetition"><span class="id" title="definition">map_without_repetition</span></a> (<a id="A:201" class="idref" href="#A:201"><span class="id" title="binder">A</span></a> <a id="B:202" class="idref" href="#B:202"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) <br/>
+&nbsp;&nbsp;(<a id="eqA:205" class="idref" href="#eqA:205"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:203" class="idref" href="#a1:203"><span class="id" title="binder">a1</span></a> <a id="a2:204" class="idref" href="#a2:204"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:201"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:203"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:204"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:203"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:204"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="f:206" class="idref" href="#f:206"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:201"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:202"><span class="id" title="variable">B</span></a>) (<a id="l:207" class="idref" href="#l:207"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:201"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:202"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:207"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:202"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:206"><span class="id" title="variable">f</span></a> <span class="id" title="var">h</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h</span> ⇒ <span class="id" title="var">f_h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a>(<span class="id" title="var">h2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="hydras.rpo.more_list.html#eqA:205"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">h1</span> <span class="id" title="var">h2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.more_list.html#map_without_repetition:208"><span class="id" title="definition">map_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:205"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:206"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:206"><span class="id" title="variable">f</span></a> <span class="id" title="var">h1</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#map_without_repetition:208"><span class="id" title="definition">map_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:205"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:206"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.more_list.html#map_without_repetition:208"><span class="id" title="definition">map_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:205"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:206"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="prop_map_without_repetition" class="idref" href="#prop_map_without_repetition"><span class="id" title="lemma">prop_map_without_repetition</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:210" class="idref" href="#A:210"><span class="id" title="binder">A</span></a> <a id="B:211" class="idref" href="#B:211"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:214" class="idref" href="#eqA:214"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:212" class="idref" href="#a1:212"><span class="id" title="binder">a1</span></a> <a id="a2:213" class="idref" href="#a2:213"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:210"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:212"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:213"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:212"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:213"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="P:215" class="idref" href="#P:215"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:211"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="f:216" class="idref" href="#f:216"><span class="id" title="binder">f</span></a> <a id="l:217" class="idref" href="#l:217"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:218" class="idref" href="#a:218"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:218"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:217"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:216"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:218"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a> <br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_a</span> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:215"><span class="id" title="variable">P</span></a> <span class="id" title="var">f_a</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:219" class="idref" href="#b:219"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:219"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.rpo.more_list.html#map_without_repetition"><span class="id" title="definition">map_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:214"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:216"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:217"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:215"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:219"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exists_map_without_repetition" class="idref" href="#exists_map_without_repetition"><span class="id" title="lemma">exists_map_without_repetition</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:220" class="idref" href="#A:220"><span class="id" title="binder">A</span></a> <a id="B:221" class="idref" href="#B:221"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:224" class="idref" href="#eqA:224"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:222" class="idref" href="#a1:222"><span class="id" title="binder">a1</span></a> <a id="a2:223" class="idref" href="#a2:223"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:220"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:222"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:223"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:222"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:223"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="P:225" class="idref" href="#P:225"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:221"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="f:226" class="idref" href="#f:226"><span class="id" title="binder">f</span></a> <a id="l:227" class="idref" href="#l:227"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:228" class="idref" href="#a:228"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:228"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:227"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:226"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:228"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_a</span> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:225"><span class="id" title="variable">P</span></a> <span class="id" title="var">f_a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:229" class="idref" href="#b:229"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:229"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.rpo.more_list.html#map_without_repetition"><span class="id" title="definition">map_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:224"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:226"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:227"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:225"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:229"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+map12_without_repetition is similar to map_without_repetition, but the 
+applied function returns two optional results instead of one.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="map12_without_repetition" class="idref" href="#map12_without_repetition"><span class="id" title="definition">map12_without_repetition</span></a> (<a id="A:238" class="idref" href="#A:238"><span class="id" title="binder">A</span></a> <a id="B:239" class="idref" href="#B:239"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) <br/>
+&nbsp;&nbsp;(<a id="eqA:242" class="idref" href="#eqA:242"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:240" class="idref" href="#a1:240"><span class="id" title="binder">a1</span></a> <a id="a2:241" class="idref" href="#a2:241"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:238"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:240"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:241"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:240"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:241"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="f:243" class="idref" href="#f:243"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:238"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:239"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:239"><span class="id" title="variable">B</span></a>) (<a id="l:244" class="idref" href="#l:244"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#A:238"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:239"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#l:244"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.more_list.html#B:239"><span class="id" title="variable">B</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">h</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">f_h2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a>(<span class="id" title="var">h'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a>) <span class="id" title="keyword">as</span> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> (<a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">h</span> <span class="id" title="var">h'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition:245"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">h</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition:245"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition:245"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition:245"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f_h2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <span class="id" title="var">f_h2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">f_h1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition:245"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:242"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:243"><span class="id" title="variable">f</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="prop_map12_without_repetition" class="idref" href="#prop_map12_without_repetition"><span class="id" title="lemma">prop_map12_without_repetition</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:247" class="idref" href="#A:247"><span class="id" title="binder">A</span></a> <a id="B:248" class="idref" href="#B:248"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:251" class="idref" href="#eqA:251"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:249" class="idref" href="#a1:249"><span class="id" title="binder">a1</span></a> <a id="a2:250" class="idref" href="#a2:250"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:247"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:249"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:250"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:249"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:250"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="P:252" class="idref" href="#P:252"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:248"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="f:253" class="idref" href="#f:253"><span class="id" title="binder">f</span></a> <a id="l:254" class="idref" href="#l:254"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="a:255" class="idref" href="#a:255"><span class="id" title="binder">a</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:255"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:254"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:253"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:255"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a> <br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f1_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:252"><span class="id" title="variable">P</span></a> <span class="id" title="var">f1_a</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f2_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:252"><span class="id" title="variable">P</span></a> <span class="id" title="var">f2_a</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f1_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f2_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:252"><span class="id" title="variable">P</span></a> <span class="id" title="var">f1_a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:252"><span class="id" title="variable">P</span></a> <span class="id" title="var">f2_a</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:256" class="idref" href="#b:256"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:256"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:251"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:253"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:254"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:252"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:256"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="exists_map12_without_repetition" class="idref" href="#exists_map12_without_repetition"><span class="id" title="lemma">exists_map12_without_repetition</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:259" class="idref" href="#A:259"><span class="id" title="binder">A</span></a> <a id="B:260" class="idref" href="#B:260"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) (<a id="eqA:263" class="idref" href="#eqA:263"><span class="id" title="binder">eqA</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:261" class="idref" href="#a1:261"><span class="id" title="binder">a1</span></a> <a id="a2:262" class="idref" href="#a2:262"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.more_list.html#A:259"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:261"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:262"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.more_list.html#a1:261"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.more_list.html#a2:262"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="P:264" class="idref" href="#P:264"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.rpo.more_list.html#B:260"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="f:265" class="idref" href="#f:265"><span class="id" title="binder">f</span></a> <a id="l:266" class="idref" href="#l:266"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a:267" class="idref" href="#a:267"><span class="id" title="binder">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:267"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:266"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#f:265"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#a:267"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f2_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:264"><span class="id" title="variable">P</span></a> <span class="id" title="var">f2_a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f1_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:264"><span class="id" title="variable">P</span></a> <span class="id" title="var">f1_a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f1_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">f2_a</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#P:264"><span class="id" title="variable">P</span></a> <span class="id" title="var">f1_a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:264"><span class="id" title="variable">P</span></a> <span class="id" title="var">f2_a</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:268" class="idref" href="#b:268"><span class="id" title="binder">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:268"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.rpo.more_list.html#map12_without_repetition"><span class="id" title="definition">map12_without_repetition</span></a> <a class="idref" href="hydras.rpo.more_list.html#eqA:263"><span class="id" title="variable">eqA</span></a> <a class="idref" href="hydras.rpo.more_list.html#f:265"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.more_list.html#l:266"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.more_list.html#P:264"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.more_list.html#b:268"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>).<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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\ No newline at end of file
diff --git a/theories/html/hydras.rpo.rpo.html b/theories/html/hydras.rpo.rpo.html
new file mode 100644
index 00000000..5466e69d
--- /dev/null
+++ b/theories/html/hydras.rpo.rpo.html
@@ -0,0 +1,676 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.rpo</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.rpo</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+by Evelyne Contejean, LRI 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Wellfounded.Wellfounded.html#"><span class="id" title="library">Wellfounded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Wf_nat.html#"><span class="id" title="library">Wf_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.more_list.html#"><span class="id" title="library">more_list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#"><span class="id" title="library">list_permut</span></a> <a class="idref" href="hydras.rpo.dickson.html#"><span class="id" title="library">dickson</span></a> <a class="idref" href="hydras.rpo.term.html#"><span class="id" title="library">term</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A non-dependent version of lexicographic extension. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lex" class="idref" href="#lex"><span class="id" title="definition">lex</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a> <a id="B:2" class="idref" href="#B:2"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) <br/>
+&nbsp;&nbsp;(<a id="eq_A_dec:5" class="idref" href="#eq_A_dec:5"><span class="id" title="binder">eq_A_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:3" class="idref" href="#a1:3"><span class="id" title="binder">a1</span></a> <a id="a2:4" class="idref" href="#a2:4"><span class="id" title="binder">a2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.rpo.html#a1:3"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="hydras.rpo.rpo.html#a2:4"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="hydras.rpo.rpo.html#a1:3"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a><a class="idref" href="hydras.rpo.rpo.html#a2:4"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>) <br/>
+&nbsp;&nbsp;(<a id="o1:6" class="idref" href="#o1:6"><span class="id" title="binder">o1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.rpo.html#A:1"><span class="id" title="variable">A</span></a>) (<a id="o2:7" class="idref" href="#o2:7"><span class="id" title="binder">o2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.rpo.html#B:2"><span class="id" title="variable">B</span></a>) (<a id="s:8" class="idref" href="#s:8"><span class="id" title="binder">s</span></a> <a id="t:9" class="idref" href="#t:9"><span class="id" title="binder">t</span></a> : <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <span class="id" title="var">_</span>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#s:8"><span class="id" title="variable">s</span></a>, <a class="idref" href="hydras.rpo.rpo.html#t:9"><span class="id" title="variable">t</span></a>  <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">t1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">t2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.rpo.html#eq_A_dec:5"><span class="id" title="variable">eq_A_dec</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">t1</span> <span class="id" title="keyword">then</span> <a class="idref" href="hydras.rpo.rpo.html#o2:7"><span class="id" title="variable">o2</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">t2</span> <span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.rpo.html#o1:6"><span class="id" title="variable">o1</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">t1</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Transitivity of  lexicographic extension. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lex_trans" class="idref" href="#lex_trans"><span class="id" title="lemma">lex_trans</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> (<a id="A:12" class="idref" href="#A:12"><span class="id" title="binder">A</span></a> <a id="B:13" class="idref" href="#B:13"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Set</span>) <a id="eq_A_dec:14" class="idref" href="#eq_A_dec:14"><span class="id" title="binder">eq_A_dec</span></a> <a id="o1:15" class="idref" href="#o1:15"><span class="id" title="binder">o1</span></a> <a id="o2:16" class="idref" href="#o2:16"><span class="id" title="binder">o2</span></a>, <br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> <a class="idref" href="hydras.rpo.rpo.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.rpo.rpo.html#o1:15"><span class="id" title="variable">o1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="hydras.rpo.rpo.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.rpo.rpo.html#o1:15"><span class="id" title="variable">o1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="hydras.rpo.rpo.html#B:13"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.rpo.rpo.html#o2:16"><span class="id" title="variable">o2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <span class="id" title="var">_</span> (<a class="idref" href="hydras.rpo.rpo.html#lex"><span class="id" title="definition">lex</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.rpo.html#eq_A_dec:14"><span class="id" title="variable">eq_A_dec</span></a> <a class="idref" href="hydras.rpo.rpo.html#o1:15"><span class="id" title="variable">o1</span></a> <a class="idref" href="hydras.rpo.rpo.html#o2:16"><span class="id" title="variable">o2</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Well-foundedness of  lexicographic extension. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="wf_lex" class="idref" href="#wf_lex"><span class="id" title="lemma">wf_lex</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="A:17" class="idref" href="#A:17"><span class="id" title="binder">A</span></a> <a id="B:18" class="idref" href="#B:18"><span class="id" title="binder">B</span></a> <a id="eq_A_dec:19" class="idref" href="#eq_A_dec:19"><span class="id" title="binder">eq_A_dec</span></a> <a id="o1:20" class="idref" href="#o1:20"><span class="id" title="binder">o1</span></a> <a id="o2:21" class="idref" href="#o2:21"><span class="id" title="binder">o2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#o1:20"><span class="id" title="variable">o1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#o2:21"><span class="id" title="variable">o2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> (<a class="idref" href="hydras.rpo.rpo.html#lex"><span class="id" title="definition">lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#A:17"><span class="id" title="variable">A</span></a> <a class="idref" href="hydras.rpo.rpo.html#B:18"><span class="id" title="variable">B</span></a> <a class="idref" href="hydras.rpo.rpo.html#eq_A_dec:19"><span class="id" title="variable">eq_A_dec</span></a> <a class="idref" href="hydras.rpo.rpo.html#o1:20"><span class="id" title="variable">o1</span></a> <a class="idref" href="hydras.rpo.rpo.html#o2:21"><span class="id" title="variable">o2</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab141"></a><h2 class="section">Module Type Precedence,</h2>
+
+<a id="lab142"></a><h2 class="section">Definition of a precedence.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="Precedence" class="idref" href="#Precedence"><span class="id" title="module">Precedence</span></a>.<br/>
+<span class="id" title="keyword">Parameter</span> <a id="Precedence.A" class="idref" href="#Precedence.A"><span class="id" title="axiom">A</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+<span class="id" title="keyword">Parameter</span> <a id="Precedence.prec" class="idref" href="#Precedence.prec"><span class="id" title="axiom">prec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.rpo.rpo.html#Precedence.A"><span class="id" title="axiom">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Precedence.status_type" class="idref" href="#Precedence.status_type"><span class="id" title="inductive">status_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Precedence.Lex" class="idref" href="#Precedence.Lex"><span class="id" title="constructor">Lex</span></a> : <a class="idref" href="hydras.rpo.rpo.html#status_type:24"><span class="id" title="inductive">status_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Precedence.Mul" class="idref" href="#Precedence.Mul"><span class="id" title="constructor">Mul</span></a> : <a class="idref" href="hydras.rpo.rpo.html#status_type:24"><span class="id" title="inductive">status_type</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Parameter</span> <a id="Precedence.status" class="idref" href="#Precedence.status"><span class="id" title="axiom">status</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Precedence.A"><span class="id" title="axiom">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Precedence.status_type"><span class="id" title="inductive">status_type</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="Precedence.prec_dec" class="idref" href="#Precedence.prec_dec"><span class="id" title="axiom">prec_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:27" class="idref" href="#a1:27"><span class="id" title="binder">a1</span></a> <a id="a2:28" class="idref" href="#a2:28"><span class="id" title="binder">a2</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Precedence.A"><span class="id" title="axiom">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.rpo.html#Precedence.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="hydras.rpo.rpo.html#a1:27"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.rpo.rpo.html#a2:28"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="hydras.rpo.rpo.html#Precedence.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="hydras.rpo.rpo.html#a1:27"><span class="id" title="variable">a1</span></a> <a class="idref" href="hydras.rpo.rpo.html#a2:28"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Axiom</span> <a id="Precedence.prec_antisym" class="idref" href="#Precedence.prec_antisym"><span class="id" title="axiom">prec_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="s:30" class="idref" href="#s:30"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Precedence.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:30"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:30"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+<span class="id" title="keyword">Axiom</span> <a id="Precedence.prec_transitive" class="idref" href="#Precedence.prec_transitive"><span class="id" title="axiom">prec_transitive</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="hydras.rpo.rpo.html#Precedence.A"><span class="id" title="axiom">A</span></a> <a class="idref" href="hydras.rpo.rpo.html#Precedence.prec"><span class="id" title="axiom">prec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.rpo.html#Precedence"><span class="id" title="module">Precedence</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab143"></a><h2 class="section">Module Type RPO,</h2>
+
+<a id="lab144"></a><h2 class="section">Definition of RPO from a precedence on symbols.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="RPO" class="idref" href="#RPO"><span class="id" title="module">RPO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Declare Module</span> <a id="RPO.T" class="idref" href="#RPO.T"><span class="id" title="module">T</span></a> : <a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">term.Term</span></a>.<br/>
+<span class="id" title="keyword">Declare Module</span> <a id="RPO.P" class="idref" href="#RPO.P"><span class="id" title="module">P</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Precedence"><span class="id" title="module">Precedence</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">A</span>:= <a class="idref" href="hydras.rpo.rpo.html#RPO.symbol"><span class="id" title="definition">T.symbol</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#RPO.T"><span class="id" title="module">T</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#RPO.P"><span class="id" title="module">P</span></a>.<br/>
+<span class="id" title="keyword">Declare Module</span> <a id="RPO.LP" class="idref" href="#RPO.LP"><span class="id" title="module">LP</span></a> : <a class="idref" href="hydras.rpo.list_permut.html#Permut"><span class="id" title="module">list_permut.Permut</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">DS.A</span>:=<a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#RPO.LP"><span class="id" title="module">LP</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab145"></a><h2 class="section">Definition of rpo.</h2>
+
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Inductive</span> <a id="RPO.rpo" class="idref" href="#RPO.rpo"><span class="id" title="inductive">rpo</span></a> : <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="RPO.Subterm" class="idref" href="#RPO.Subterm"><span class="id" title="constructor">Subterm</span></a> : <span class="id" title="keyword">∀</span> <a id="f:41" class="idref" href="#f:41"><span class="id" title="binder">f</span></a> <a id="l:42" class="idref" href="#l:42"><span class="id" title="binder">l</span></a> <a id="t:43" class="idref" href="#t:43"><span class="id" title="binder">t</span></a> <a id="s:44" class="idref" href="#s:44"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:44"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:42"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:34"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:43"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:44"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:43"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:42"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a id="RPO.Top_gt" class="idref" href="#RPO.Top_gt"><span class="id" title="constructor">Top_gt</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:45" class="idref" href="#f:45"><span class="id" title="binder">f</span></a> <a id="g:46" class="idref" href="#g:46"><span class="id" title="binder">g</span></a> <a id="l:47" class="idref" href="#l:47"><span class="id" title="binder">l</span></a> <a id="l':48" class="idref" href="#l':48"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="hydras.rpo.rpo.html#g:46"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="s':49" class="idref" href="#s':49"><span class="id" title="binder">s'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':49"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':48"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':49"><span class="id" title="variable">s'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:47"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#g:46"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':48"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:45"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:47"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a id="RPO.Top_eq_lex" class="idref" href="#RPO.Top_eq_lex"><span class="id" title="constructor">Top_eq_lex</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:50" class="idref" href="#f:50"><span class="id" title="binder">f</span></a> <a id="l:51" class="idref" href="#l:51"><span class="id" title="binder">l</span></a> <a id="l':52" class="idref" href="#l':52"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.status"><span class="id" title="axiom">status</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.Lex"><span class="id" title="constructor">Lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_lex:35"><span class="id" title="inductive">rpo_lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':52"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:51"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="s':53" class="idref" href="#s':53"><span class="id" title="binder">s'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':53"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':52"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':53"><span class="id" title="variable">s'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:51"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':52"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:50"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:51"><span class="id" title="variable">l</span></a>)<br/>
+<br/>
+&nbsp;&nbsp;| <a id="RPO.Top_eq_mul" class="idref" href="#RPO.Top_eq_mul"><span class="id" title="constructor">Top_eq_mul</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:54" class="idref" href="#f:54"><span class="id" title="binder">f</span></a> <a id="l:55" class="idref" href="#l:55"><span class="id" title="binder">l</span></a> <a id="l':56" class="idref" href="#l':56"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.status"><span class="id" title="axiom">status</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:54"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.Mul"><span class="id" title="constructor">Mul</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_mul:36"><span class="id" title="inductive">rpo_mul</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':56"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:55"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:54"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':56"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#RPO.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:54"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:55"><span class="id" title="variable">l</span></a>)<br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="RPO.rpo_eq" class="idref" href="#RPO.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> : <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="RPO.Eq" class="idref" href="#RPO.Eq"><span class="id" title="constructor">Eq</span></a> : <span class="id" title="keyword">∀</span> <a id="t:57" class="idref" href="#t:57"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:34"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:57"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:57"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;| <a id="RPO.Lt" class="idref" href="#RPO.Lt"><span class="id" title="constructor">Lt</span></a> : <span class="id" title="keyword">∀</span> <a id="s:58" class="idref" href="#s:58"><span class="id" title="binder">s</span></a> <a id="t:59" class="idref" href="#t:59"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:58"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:59"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:34"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:58"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:59"><span class="id" title="variable">t</span></a><br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="RPO.rpo_lex" class="idref" href="#RPO.rpo_lex"><span class="id" title="inductive">rpo_lex</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="RPO.List_gt" class="idref" href="#RPO.List_gt"><span class="id" title="constructor">List_gt</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> <a id="t:61" class="idref" href="#t:61"><span class="id" title="binder">t</span></a> <a id="l:62" class="idref" href="#l:62"><span class="id" title="binder">l</span></a> <a id="l':63" class="idref" href="#l':63"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:60"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:61"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:62"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':63"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_lex:35"><span class="id" title="inductive">rpo_lex</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:60"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:62"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#t:61"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':63"><span class="id" title="variable">l'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="RPO.List_eq" class="idref" href="#RPO.List_eq"><span class="id" title="constructor">List_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="s:64" class="idref" href="#s:64"><span class="id" title="binder">s</span></a> <a id="l:65" class="idref" href="#l:65"><span class="id" title="binder">l</span></a> <a id="l':66" class="idref" href="#l':66"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo_lex:35"><span class="id" title="inductive">rpo_lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:65"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':66"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_lex:35"><span class="id" title="inductive">rpo_lex</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:64"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:65"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#s:64"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':66"><span class="id" title="variable">l'</span></a>)<br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="RPO.rpo_mul" class="idref" href="#RPO.rpo_mul"><span class="id" title="inductive">rpo_mul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="RPO.List_mul" class="idref" href="#RPO.List_mul"><span class="id" title="constructor">List_mul</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:67" class="idref" href="#a:67"><span class="id" title="binder">a</span></a> <a id="lg:68" class="idref" href="#lg:68"><span class="id" title="binder">lg</span></a> <a id="ls:69" class="idref" href="#ls:69"><span class="id" title="binder">ls</span></a> <a id="lc:70" class="idref" href="#lc:70"><span class="id" title="binder">lc</span></a> <a id="l:71" class="idref" href="#l:71"><span class="id" title="binder">l</span></a> <a id="l':72" class="idref" href="#l':72"><span class="id" title="binder">l'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#RPO.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':72"><span class="id" title="variable">l'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#ls:69"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:70"><span class="id" title="variable">lc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#RPO.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:71"><span class="id" title="variable">l</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:68"><span class="id" title="variable">lg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:70"><span class="id" title="variable">lc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:73" class="idref" href="#b:73"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:73"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#ls:69"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a':74" class="idref" href="#a':74"><span class="id" title="binder">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':74"><span class="id" title="variable">a'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:67"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:68"><span class="id" title="variable">lg</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:33"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:73"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':74"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_mul:36"><span class="id" title="inductive">rpo_mul</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':72"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:71"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab146"></a><h2 class="section">rpo is a preorder, and its reflexive closure is an ordering.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="RPO.rpo_closure" class="idref" href="#RPO.rpo_closure"><span class="id" title="axiom">rpo_closure</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:75" class="idref" href="#s:75"><span class="id" title="binder">s</span></a> <a id="t:76" class="idref" href="#t:76"><span class="id" title="binder">t</span></a> <a id="u:77" class="idref" href="#u:77"><span class="id" title="binder">u</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:77"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:77"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#RPO.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:75"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:76"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="RPO.rpo_trans" class="idref" href="#RPO.rpo_trans"><span class="id" title="axiom">rpo_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="s:79" class="idref" href="#s:79"><span class="id" title="binder">s</span></a> <a id="t:80" class="idref" href="#t:80"><span class="id" title="binder">t</span></a> <a id="u:81" class="idref" href="#u:81"><span class="id" title="binder">u</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:80"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:79"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:81"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:80"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:81"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:79"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab147"></a><h2 class="section">Main theorem: when the precedence is well-founded, so is the rpo.</h2>
+
+
+<div class="paragraph"> </div>
+
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Axiom</span> <a id="RPO.wf_rpo" class="idref" href="#RPO.wf_rpo"><span class="id" title="axiom">wf_rpo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab148"></a><h2 class="section">RPO is compatible with the instanciation by a substitution.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="RPO.rpo_subst" class="idref" href="#RPO.rpo_subst"><span class="id" title="axiom">rpo_subst</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:84" class="idref" href="#t:84"><span class="id" title="binder">t</span></a> <a id="s:85" class="idref" href="#s:85"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:85"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:84"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma:86" class="idref" href="#sigma:86"><span class="id" title="binder">sigma</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.rpo.html#sigma:86"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:85"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#RPO.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.rpo.html#sigma:86"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:84"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab149"></a><h2 class="section">RPO is compatible with adding context.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Axiom</span> <a id="RPO.rpo_add_context" class="idref" href="#RPO.rpo_add_context"><span class="id" title="axiom">rpo_add_context</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="p:88" class="idref" href="#p:88"><span class="id" title="binder">p</span></a> <a id="ctx:89" class="idref" href="#ctx:89"><span class="id" title="binder">ctx</span></a> <a id="s:90" class="idref" href="#s:90"><span class="id" title="binder">s</span></a> <a id="t:91" class="idref" href="#t:91"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:90"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:91"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#RPO.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:89"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:88"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#RPO.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#RPO.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:89"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:90"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:88"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#RPO.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:89"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:91"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:88"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.rpo.html#RPO"><span class="id" title="module">RPO</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">T1</span>: <a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">term.Term</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P1</span> : <a class="idref" href="hydras.rpo.rpo.html#Precedence"><span class="id" title="module">Precedence</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">A</span> := <a class="idref" href="hydras.rpo.rpo.html#T1.symbol"><span class="id" title="definition">T1.symbol</span></a>)<br/>
+&lt;: <a class="idref" href="hydras.rpo.rpo.html#RPO"><span class="id" title="module">RPO</span></a>. 
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.T" class="idref" href="#Make.T"><span class="id" title="module">T</span></a> := <a class="idref" href="hydras.rpo.rpo.html#T1"><span class="id" title="module">T1</span></a>.<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.P" class="idref" href="#Make.P"><span class="id" title="module">P</span></a> := <a class="idref" href="hydras.rpo.rpo.html#P1"><span class="id" title="module">P1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#Make.T"><span class="id" title="module">T</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#Make.P"><span class="id" title="module">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make.LP" class="idref" href="#Make.LP"><span class="id" title="module">LP</span></a> := <a class="idref" href="hydras.rpo.list_permut.html#Make"><span class="id" title="module">list_permut.Make</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.T.Term_eq_dec"><span class="id" title="module">Term_eq_dec</span></a>).<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.rpo.html#Make.LP"><span class="id" title="module">LP</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab150"></a><h2 class="section">Definition of size-based well-founded orderings for induction.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.o_size" class="idref" href="#Make.o_size"><span class="id" title="definition">o_size</span></a> <a id="s:93" class="idref" href="#s:93"><span class="id" title="binder">s</span></a> <a id="t:94" class="idref" href="#t:94"><span class="id" title="binder">t</span></a> := <a class="idref" href="hydras.rpo.rpo.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:93"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:94"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_size" class="idref" href="#Make.wf_size"><span class="id" title="lemma">wf_size</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size"><span class="id" title="definition">o_size</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.size2" class="idref" href="#Make.size2"><span class="id" title="definition">size2</span></a> <a id="s:95" class="idref" href="#s:95"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#s:95"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">end</span>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.o_size2" class="idref" href="#Make.o_size2"><span class="id" title="definition">o_size2</span></a> <a id="s:97" class="idref" href="#s:97"><span class="id" title="binder">s</span></a> <a id="t:98" class="idref" href="#t:98"><span class="id" title="binder">t</span></a> := <a class="idref" href="hydras.rpo.rpo.html#lex"><span class="id" title="definition">lex</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.size2"><span class="id" title="definition">size2</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:97"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.size2"><span class="id" title="definition">size2</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:98"><span class="id" title="variable">t</span></a>).<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_size2" class="idref" href="#Make.wf_size2"><span class="id" title="lemma">wf_size2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size2"><span class="id" title="definition">o_size2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.size3" class="idref" href="#Make.size3"><span class="id" title="definition">size3</span></a> <a id="s:99" class="idref" href="#s:99"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#s:99"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.size2"><span class="id" title="definition">size2</span></a> <span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">end</span>.<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.o_size3" class="idref" href="#Make.o_size3"><span class="id" title="definition">o_size3</span></a> <a id="s:101" class="idref" href="#s:101"><span class="id" title="binder">s</span></a> <a id="t:102" class="idref" href="#t:102"><span class="id" title="binder">t</span></a> := <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#lex"><span class="id" title="definition">lex</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#lex"><span class="id" title="definition">lex</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Peano_dec.html#eq_nat_dec"><span class="id" title="abbreviation">eq_nat_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.size3"><span class="id" title="definition">size3</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:101"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.size3"><span class="id" title="definition">size3</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:102"><span class="id" title="variable">t</span></a>).<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_size3" class="idref" href="#Make.wf_size3"><span class="id" title="lemma">wf_size3</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.lex1" class="idref" href="#Make.lex1"><span class="id" title="lemma">lex1</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="s:103" class="idref" href="#s:103"><span class="id" title="binder">s</span></a> <a id="f:104" class="idref" href="#f:104"><span class="id" title="binder">f</span></a> <a id="l:105" class="idref" href="#l:105"><span class="id" title="binder">l</span></a> <a id="t1:106" class="idref" href="#t1:106"><span class="id" title="binder">t1</span></a> <a id="u1:107" class="idref" href="#u1:107"><span class="id" title="binder">u1</span></a> <a id="t2:108" class="idref" href="#t2:108"><span class="id" title="binder">t2</span></a> <a id="u2:109" class="idref" href="#u2:109"><span class="id" title="binder">u2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:103"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:105"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#s:103"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t1:106"><span class="id" title="variable">t1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u1:107"><span class="id" title="variable">u1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:104"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:105"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t2:108"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u2:109"><span class="id" title="variable">u2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.lex1_bis" class="idref" href="#Make.lex1_bis"><span class="id" title="lemma">lex1_bis</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:110" class="idref" href="#a:110"><span class="id" title="binder">a</span></a> <a id="f:111" class="idref" href="#f:111"><span class="id" title="binder">f</span></a> <a id="l:112" class="idref" href="#l:112"><span class="id" title="binder">l</span></a> <a id="t1:113" class="idref" href="#t1:113"><span class="id" title="binder">t1</span></a> <a id="u1:114" class="idref" href="#u1:114"><span class="id" title="binder">u1</span></a> <a id="t2:115" class="idref" href="#t2:115"><span class="id" title="binder">t2</span></a> <a id="u2:116" class="idref" href="#u2:116"><span class="id" title="binder">u2</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:111"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:112"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t1:113"><span class="id" title="variable">t1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u1:114"><span class="id" title="variable">u1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:111"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:110"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.rpo.rpo.html#l:112"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t2:115"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u2:116"><span class="id" title="variable">u2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.lex2" class="idref" href="#Make.lex2"><span class="id" title="lemma">lex2</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:117" class="idref" href="#t:117"><span class="id" title="binder">t</span></a> <a id="f:118" class="idref" href="#f:118"><span class="id" title="binder">f</span></a> <a id="l:119" class="idref" href="#l:119"><span class="id" title="binder">l</span></a> <a id="s:120" class="idref" href="#s:120"><span class="id" title="binder">s</span></a> <a id="u1:121" class="idref" href="#u1:121"><span class="id" title="binder">u1</span></a> <a id="u2:122" class="idref" href="#u2:122"><span class="id" title="binder">u2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:117"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:119"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#s:120"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t:117"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u1:121"><span class="id" title="variable">u1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#s:120"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:118"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:119"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.rpo.html#u2:122"><span class="id" title="variable">u2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.lex3" class="idref" href="#Make.lex3"><span class="id" title="lemma">lex3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="u:123" class="idref" href="#u:123"><span class="id" title="binder">u</span></a> <a id="f:124" class="idref" href="#f:124"><span class="id" title="binder">f</span></a> <a id="l:125" class="idref" href="#l:125"><span class="id" title="binder">l</span></a> <a id="s:126" class="idref" href="#s:126"><span class="id" title="binder">s</span></a> <a id="t:127" class="idref" href="#t:127"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:123"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:125"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#s:126"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t:127"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#u:123"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#s:126"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,(</span></a><a class="idref" href="hydras.rpo.rpo.html#t:127"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:124"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:125"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.o_size3_trans" class="idref" href="#Make.o_size3_trans"><span class="id" title="lemma">o_size3_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#transitive"><span class="id" title="definition">transitive</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.rpo.html#Make.o_size3"><span class="id" title="definition">o_size3</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab151"></a><h2 class="section">Definition of rpo.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Make.rpo" class="idref" href="#Make.rpo"><span class="id" title="inductive">rpo</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.Subterm" class="idref" href="#Make.Subterm"><span class="id" title="constructor">Subterm</span></a> : <span class="id" title="keyword">∀</span> <a id="f:136" class="idref" href="#f:136"><span class="id" title="binder">f</span></a> <a id="l:137" class="idref" href="#l:137"><span class="id" title="binder">l</span></a> <a id="t:138" class="idref" href="#t:138"><span class="id" title="binder">t</span></a> <a id="s:139" class="idref" href="#s:139"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:139"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:137"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:129"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:138"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:139"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:138"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:136"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:137"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a id="Make.Top_gt" class="idref" href="#Make.Top_gt"><span class="id" title="constructor">Top_gt</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:140" class="idref" href="#f:140"><span class="id" title="binder">f</span></a> <a id="g:141" class="idref" href="#g:141"><span class="id" title="binder">g</span></a> <a id="l:142" class="idref" href="#l:142"><span class="id" title="binder">l</span></a> <a id="l':143" class="idref" href="#l':143"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="hydras.rpo.rpo.html#g:141"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:140"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="s':144" class="idref" href="#s':144"><span class="id" title="binder">s'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':144"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':143"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':144"><span class="id" title="variable">s'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:140"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:142"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#g:141"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':143"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:140"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:142"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a id="Make.Top_eq_lex" class="idref" href="#Make.Top_eq_lex"><span class="id" title="constructor">Top_eq_lex</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:145" class="idref" href="#f:145"><span class="id" title="binder">f</span></a> <a id="l:146" class="idref" href="#l:146"><span class="id" title="binder">l</span></a> <a id="l':147" class="idref" href="#l':147"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.status"><span class="id" title="axiom">status</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:145"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.Lex"><span class="id" title="constructor">Lex</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_lex:130"><span class="id" title="inductive">rpo_lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':147"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:146"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="s':148" class="idref" href="#s':148"><span class="id" title="binder">s'</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':148"><span class="id" title="variable">s'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':147"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s':148"><span class="id" title="variable">s'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:145"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:146"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:145"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':147"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:145"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:146"><span class="id" title="variable">l</span></a>)<br/>
+<br/>
+&nbsp;&nbsp;| <a id="Make.Top_eq_mul" class="idref" href="#Make.Top_eq_mul"><span class="id" title="constructor">Top_eq_mul</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:149" class="idref" href="#f:149"><span class="id" title="binder">f</span></a> <a id="l:150" class="idref" href="#l:150"><span class="id" title="binder">l</span></a> <a id="l':151" class="idref" href="#l':151"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.status"><span class="id" title="axiom">status</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:149"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.Mul"><span class="id" title="constructor">Mul</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_mul:131"><span class="id" title="inductive">rpo_mul</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':151"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:150"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:149"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':151"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:149"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:150"><span class="id" title="variable">l</span></a>)<br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="Make.rpo_eq" class="idref" href="#Make.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.Eq" class="idref" href="#Make.Eq"><span class="id" title="constructor">Eq</span></a> : <span class="id" title="keyword">∀</span> <a id="t:152" class="idref" href="#t:152"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:129"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:152"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:152"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;| <a id="Make.Lt" class="idref" href="#Make.Lt"><span class="id" title="constructor">Lt</span></a> : <span class="id" title="keyword">∀</span> <a id="s:153" class="idref" href="#s:153"><span class="id" title="binder">s</span></a> <a id="t:154" class="idref" href="#t:154"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:153"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:154"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_eq:129"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:153"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:154"><span class="id" title="variable">t</span></a><br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="Make.rpo_lex" class="idref" href="#Make.rpo_lex"><span class="id" title="inductive">rpo_lex</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.List_gt" class="idref" href="#Make.List_gt"><span class="id" title="constructor">List_gt</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:155" class="idref" href="#s:155"><span class="id" title="binder">s</span></a> <a id="t:156" class="idref" href="#t:156"><span class="id" title="binder">t</span></a> <a id="l:157" class="idref" href="#l:157"><span class="id" title="binder">l</span></a> <a id="l':158" class="idref" href="#l':158"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:156"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:157"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':158"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_lex:130"><span class="id" title="inductive">rpo_lex</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:155"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:157"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#t:156"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':158"><span class="id" title="variable">l'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="Make.List_eq" class="idref" href="#Make.List_eq"><span class="id" title="constructor">List_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="s:159" class="idref" href="#s:159"><span class="id" title="binder">s</span></a> <a id="l:160" class="idref" href="#l:160"><span class="id" title="binder">l</span></a> <a id="l':161" class="idref" href="#l':161"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#rpo_lex:130"><span class="id" title="inductive">rpo_lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:160"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':161"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_lex:130"><span class="id" title="inductive">rpo_lex</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:159"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:160"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#s:159"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':161"><span class="id" title="variable">l'</span></a>)<br/>
+<br/>
+<span class="id" title="keyword">with</span> <a id="Make.rpo_mul" class="idref" href="#Make.rpo_mul"><span class="id" title="inductive">rpo_mul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.List_mul" class="idref" href="#Make.List_mul"><span class="id" title="constructor">List_mul</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:162" class="idref" href="#a:162"><span class="id" title="binder">a</span></a> <a id="lg:163" class="idref" href="#lg:163"><span class="id" title="binder">lg</span></a> <a id="ls:164" class="idref" href="#ls:164"><span class="id" title="binder">ls</span></a> <a id="lc:165" class="idref" href="#lc:165"><span class="id" title="binder">lc</span></a> <a id="l:166" class="idref" href="#l:166"><span class="id" title="binder">l</span></a> <a id="l':167" class="idref" href="#l':167"><span class="id" title="binder">l'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':167"><span class="id" title="variable">l'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#ls:164"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:165"><span class="id" title="variable">lc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:166"><span class="id" title="variable">l</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:163"><span class="id" title="variable">lg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:165"><span class="id" title="variable">lc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:168" class="idref" href="#b:168"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:168"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#ls:164"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a':169" class="idref" href="#a':169"><span class="id" title="binder">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':169"><span class="id" title="variable">a'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:162"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:163"><span class="id" title="variable">lg</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo:128"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:168"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':169"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_mul:131"><span class="id" title="inductive">rpo_mul</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':167"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:166"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_lex_same_length" class="idref" href="#Make.rpo_lex_same_length"><span class="id" title="lemma">rpo_lex_same_length</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:170" class="idref" href="#l:170"><span class="id" title="binder">l</span></a> <a id="l':171" class="idref" href="#l':171"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex"><span class="id" title="inductive">rpo_lex</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:170"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':171"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:170"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':171"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_subterm" class="idref" href="#Make.rpo_subterm"><span class="id" title="lemma">rpo_subterm</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="s:172" class="idref" href="#s:172"><span class="id" title="binder">s</span></a> <a id="t:173" class="idref" href="#t:173"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:173"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:172"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="tj:174" class="idref" href="#tj:174"><span class="id" title="binder">tj</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.direct_subterm"><span class="id" title="definition">direct_subterm</span></a> <a class="idref" href="hydras.rpo.rpo.html#tj:174"><span class="id" title="variable">tj</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:173"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#tj:174"><span class="id" title="variable">tj</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:172"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab152"></a><h2 class="section">rpo is a preorder, and its reflexive closure is an ordering.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_closure" class="idref" href="#Make.rpo_closure"><span class="id" title="lemma">rpo_closure</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:181" class="idref" href="#s:181"><span class="id" title="binder">s</span></a> <a id="t:182" class="idref" href="#t:182"><span class="id" title="binder">t</span></a> <a id="u:183" class="idref" href="#u:183"><span class="id" title="binder">u</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:183"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:183"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.rpo.html#Make.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_eq"><span class="id" title="inductive">rpo_eq</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:182"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_trans" class="idref" href="#Make.rpo_trans"><span class="id" title="lemma">rpo_trans</span></a> : <span class="id" title="keyword">∀</span> <a id="s:194" class="idref" href="#s:194"><span class="id" title="binder">s</span></a> <a id="t:195" class="idref" href="#t:195"><span class="id" title="binder">t</span></a> <a id="u:196" class="idref" href="#u:196"><span class="id" title="binder">u</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:195"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:194"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:196"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:195"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#u:196"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:194"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a id="Make.SN_term" class="idref" href="#Make.SN_term"><span class="id" title="record">SN_term</span></a> : <span class="id" title="keyword">Set</span> := <br/>
+&nbsp;&nbsp;<a id="Make.mk_sn" class="idref" href="#Make.mk_sn"><span class="id" title="constructor">mk_sn</span></a> <br/>
+&nbsp;&nbsp;{<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Make.tt" class="idref" href="#Make.tt"><span class="id" title="projection">tt</span></a> : <a class="idref" href="hydras.rpo.rpo.html#Make.term"><span class="id" title="inductive">term</span></a>; <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="Make.sn" class="idref" href="#Make.sn"><span class="id" title="projection">sn</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#tt:198"><span class="id" title="method">tt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab153"></a><h2 class="section">Well-foundedness of rpo.</h2>
+
+How to build a built a list of pairs (terms, proof of accessibility) from
+a global of accessibility on the list. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.build_list_of_SN_terms" class="idref" href="#Make.build_list_of_SN_terms"><span class="id" title="definition">build_list_of_SN_terms</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l:200" class="idref" href="#l:200"><span class="id" title="binder">l</span></a> (<a id="proof:202" class="idref" href="#proof:202"><span class="id" title="binder">proof</span></a> : <span class="id" title="keyword">∀</span> <a id="t:201" class="idref" href="#t:201"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:201"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:200"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:201"><span class="id" title="variable">t</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Projection on the first element of the pairs after building the
+pairs as above is the identity. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.projection_list_of_SN_terms" class="idref" href="#Make.projection_list_of_SN_terms"><span class="id" title="lemma">projection_list_of_SN_terms</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:205" class="idref" href="#l:205"><span class="id" title="binder">l</span></a> <a id="proof:206" class="idref" href="#proof:206"><span class="id" title="binder">proof</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.build_list_of_SN_terms"><span class="id" title="definition">build_list_of_SN_terms</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:205"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#proof:206"><span class="id" title="variable">proof</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:205"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.in_sn_sn" class="idref" href="#Make.in_sn_sn"><span class="id" title="lemma">in_sn_sn</span></a> : <br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l:209" class="idref" href="#l:209"><span class="id" title="binder">l</span></a> <a id="s:210" class="idref" href="#s:210"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:210"><span class="id" title="variable">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:209"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:210"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition of rpo on accessible terms. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.rpo_rest" class="idref" href="#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a> := <span class="id" title="keyword">fun</span> <a id="s:211" class="idref" href="#s:211"><span class="id" title="binder">s</span></a> <a id="t:212" class="idref" href="#t:212"><span class="id" title="binder">t</span></a> ⇒ <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:211"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:212"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Extension of <span class="inlinecode"><span class="id" title="var">rpo_lex</span></span> to the accessible terms. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Make.rpo_lex_rest" class="idref" href="#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.List_gt_rest" class="idref" href="#Make.List_gt_rest"><span class="id" title="constructor">List_gt_rest</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:215" class="idref" href="#s:215"><span class="id" title="binder">s</span></a> <a id="t:216" class="idref" href="#t:216"><span class="id" title="binder">t</span></a> <a id="l:217" class="idref" href="#l:217"><span class="id" title="binder">l</span></a> <a id="l':218" class="idref" href="#l':218"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:215"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:216"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:217"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':218"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_lex_rest:213"><span class="id" title="inductive">rpo_lex_rest</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:215"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:217"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#t:216"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':218"><span class="id" title="variable">l'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="Make.List_eq_rest" class="idref" href="#Make.List_eq_rest"><span class="id" title="constructor">List_eq_rest</span></a> : <span class="id" title="keyword">∀</span> <a id="s:219" class="idref" href="#s:219"><span class="id" title="binder">s</span></a> <a id="t:220" class="idref" href="#t:220"><span class="id" title="binder">t</span></a> <a id="l:221" class="idref" href="#l:221"><span class="id" title="binder">l</span></a> <a id="l':222" class="idref" href="#l':222"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:220"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#rpo_lex_rest:213"><span class="id" title="inductive">rpo_lex_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:221"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':222"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_lex_rest:213"><span class="id" title="inductive">rpo_lex_rest</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:221"><span class="id" title="variable">l</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#t:220"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':222"><span class="id" title="variable">l'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+A triviality: rpo on accessible terms is well-founded.
+
+<div class="paragraph"> </div>
+
+ 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_on_rest" class="idref" href="#Make.wf_on_rest"><span class="id" title="lemma">wf_on_rest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_lex_rest_same_length" class="idref" href="#Make.rpo_lex_rest_same_length"><span class="id" title="lemma">rpo_lex_rest_same_length</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:223" class="idref" href="#l:223"><span class="id" title="binder">l</span></a> <a id="l':224" class="idref" href="#l':224"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:223"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':224"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:223"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':224"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Proof of accessibility does not actually matter, provided at 
+  least one exists. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.acc_lex_drop_proof" class="idref" href="#Make.acc_lex_drop_proof"><span class="id" title="lemma">acc_lex_drop_proof</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:225" class="idref" href="#s:225"><span class="id" title="binder">s</span></a> <a id="t:226" class="idref" href="#t:226"><span class="id" title="binder">t</span></a> <a id="l:227" class="idref" href="#l:227"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:225"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:226"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a> (<a class="idref" href="hydras.rpo.rpo.html#s:225"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.rpo.rpo.html#l:227"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a> (<a class="idref" href="hydras.rpo.rpo.html#t:226"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.rpo.rpo.html#l:227"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Lexicographic extension of rpo on accessible terms lists is well-founded. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_on_lex_rest" class="idref" href="#Make.wf_on_lex_rest"><span class="id" title="lemma">wf_on_lex_rest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Extension of <span class="inlinecode"><span class="id" title="var">rpo_mul</span></span> to the accessible terms. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Make.rpo_mul_rest" class="idref" href="#Make.rpo_mul_rest"><span class="id" title="inductive">rpo_mul_rest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.List_mul_rest" class="idref" href="#Make.List_mul_rest"><span class="id" title="constructor">List_mul_rest</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:230" class="idref" href="#a:230"><span class="id" title="binder">a</span></a> <a id="lg:231" class="idref" href="#lg:231"><span class="id" title="binder">lg</span></a> <a id="ls:232" class="idref" href="#ls:232"><span class="id" title="binder">ls</span></a> <a id="lc:233" class="idref" href="#lc:233"><span class="id" title="binder">lc</span></a> <a id="l:234" class="idref" href="#l:234"><span class="id" title="binder">l</span></a> <a id="l':235" class="idref" href="#l':235"><span class="id" title="binder">l'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':235"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#ls:232"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:233"><span class="id" title="variable">lc</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:234"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:230"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:231"><span class="id" title="variable">lg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:233"><span class="id" title="variable">lc</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:236" class="idref" href="#b:236"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:236"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#ls:232"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="a':237" class="idref" href="#a':237"><span class="id" title="binder">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':237"><span class="id" title="variable">a'</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:230"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lg:231"><span class="id" title="variable">lg</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:236"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#a':237"><span class="id" title="variable">a'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_mul_rest:228"><span class="id" title="inductive">rpo_mul_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':235"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:234"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition of a finer grain for multiset extension. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Inductive</span> <a id="Make.rpo_mul_rest_step" class="idref" href="#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.List_mul_rest_step" class="idref" href="#Make.List_mul_rest_step"><span class="id" title="constructor">List_mul_rest_step</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="a:240" class="idref" href="#a:240"><span class="id" title="binder">a</span></a> <a id="ls:241" class="idref" href="#ls:241"><span class="id" title="binder">ls</span></a> <a id="lc:242" class="idref" href="#lc:242"><span class="id" title="binder">lc</span></a> <a id="l:243" class="idref" href="#l:243"><span class="id" title="binder">l</span></a> <a id="l':244" class="idref" href="#l':244"><span class="id" title="binder">l'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':244"><span class="id" title="variable">l'</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#ls:241"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:242"><span class="id" title="variable">lc</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:243"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:240"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#lc:242"><span class="id" title="variable">lc</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:245" class="idref" href="#b:245"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:245"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#ls:241"><span class="id" title="variable">ls</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:245"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#a:240"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#rpo_mul_rest_step:238"><span class="id" title="inductive">rpo_mul_rest_step</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':244"><span class="id" title="variable">l'</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:243"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The plain multiset extension is in the transitive closure of
+the finer grain extension. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_mul_trans_clos" class="idref" href="#Make.rpo_mul_trans_clos"><span class="id" title="lemma">rpo_mul_trans_clos</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#inclusion"><span class="id" title="definition">inclusion</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest"><span class="id" title="inductive">rpo_mul_rest</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Operators.html#clos_trans"><span class="id" title="inductive">clos_trans</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Splitting in two disjoint cases. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.two_cases_rpo" class="idref" href="#Make.two_cases_rpo"><span class="id" title="lemma">two_cases_rpo</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="a:256" class="idref" href="#a:256"><span class="id" title="binder">a</span></a> <a id="m:257" class="idref" href="#m:257"><span class="id" title="binder">m</span></a> <a id="n:258" class="idref" href="#n:258"><span class="id" title="binder">n</span></a>, <br/>
+&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a> <a class="idref" href="hydras.rpo.rpo.html#n:258"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:256"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#m:257"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="n':259" class="idref" href="#n':259"><span class="id" title="binder">n'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#n:258"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#a:256"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.rpo.html#n':259"><span class="id" title="variable">n'</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a> <a class="idref" href="hydras.rpo.rpo.html#n':259"><span class="id" title="variable">n'</span></a> <a class="idref" href="hydras.rpo.rpo.html#m:257"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="k:260" class="idref" href="#k:260"><span class="id" title="binder">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:261" class="idref" href="#b:261"><span class="id" title="binder">b</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:261"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#k:260"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_rest"><span class="id" title="definition">rpo_rest</span></a> <a class="idref" href="hydras.rpo.rpo.html#b:261"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.rpo.rpo.html#a:256"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#n:258"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> (<a class="idref" href="hydras.rpo.rpo.html#k:260"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.rpo.html#m:257"><span class="id" title="variable">m</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.list_permut_map_acc" class="idref" href="#Make.list_permut_map_acc"><span class="id" title="lemma">list_permut_map_acc</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="l:266" class="idref" href="#l:266"><span class="id" title="binder">l</span></a> <a id="l':267" class="idref" href="#l':267"><span class="id" title="binder">l'</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.list_permut"><span class="id" title="definition">list_permut</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:266"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':267"><span class="id" title="variable">l'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a> <a class="idref" href="hydras.rpo.rpo.html#l:266"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step"><span class="id" title="inductive">rpo_mul_rest_step</span></a> <a class="idref" href="hydras.rpo.rpo.html#l':267"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Multiset extension of rpo on accessible terms lists is well-founded. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_on_mul_rest" class="idref" href="#Make.wf_on_mul_rest"><span class="id" title="lemma">wf_on_mul_rest</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest"><span class="id" title="inductive">rpo_mul_rest</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Another definition of rpo, only on scheme of accessible terms. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Make.rpo_term" class="idref" href="#Make.rpo_term"><span class="id" title="definition">rpo_term</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.symbol"><span class="id" title="definition">symbol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.SN_term"><span class="id" title="record">SN_term</span></a>) :=<br/>
+&nbsp;<span class="id" title="keyword">fun</span> <a id="f_l:274" class="idref" href="#f_l:274"><span class="id" title="binder">f_l</span></a> <a id="g_l':275" class="idref" href="#g_l':275"><span class="id" title="binder">g_l'</span></a> ⇒ <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#f_l:274"><span class="id" title="variable">f_l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">l</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#g_l':275"><span class="id" title="variable">g_l'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">g</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">l'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="hydras.rpo.rpo.html#Make.eq_symbol_dec"><span class="id" title="axiom">F.eq_symbol_dec</span></a> <span class="id" title="var">f</span> <span class="id" title="var">g</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.rpo.html#Make.status"><span class="id" title="axiom">status</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.rpo.html#Make.Lex"><span class="id" title="constructor">Lex</span></a> ⇒ <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_lex_rest"><span class="id" title="inductive">rpo_lex_rest</span></a> <span class="id" title="var">l</span> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.rpo.html#Make.Mul"><span class="id" title="constructor">Mul</span></a> ⇒ <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_mul_rest"><span class="id" title="inductive">rpo_mul_rest</span></a> <span class="id" title="var">l</span> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="hydras.rpo.rpo.html#Make.prec"><span class="id" title="axiom">prec</span></a> <span class="id" title="var">f</span> <span class="id" title="var">g</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span>  <a id="Make.wf_rpo_term" class="idref" href="#Make.wf_rpo_term"><span class="id" title="lemma">wf_rpo_term</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo_term"><span class="id" title="definition">rpo_term</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.acc_build" class="idref" href="#Make.acc_build"><span class="id" title="lemma">acc_build</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="f:280" class="idref" href="#f:280"><span class="id" title="binder">f</span></a> <a id="l:281" class="idref" href="#l:281"><span class="id" title="binder">l</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.rpo.html#f:280"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <a id="sn_tt:282" class="idref" href="#sn_tt:282"><span class="id" title="binder">sn_tt</span></a> ⇒ <a class="idref" href="hydras.rpo.rpo.html#Make.tt"><span class="id" title="projection">tt</span></a> <a class="idref" href="hydras.rpo.rpo.html#sn_tt:282"><span class="id" title="variable">sn_tt</span></a>) <a class="idref" href="hydras.rpo.rpo.html#l:281"><span class="id" title="variable">l</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab154"></a><h2 class="section">Main theorem: when the precedence is well-founded, so is the rpo.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.wf_rpo" class="idref" href="#Make.wf_rpo"><span class="id" title="lemma">wf_rpo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.prec"><span class="id" title="axiom">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab155"></a><h2 class="section">RPO is compatible with the instanciation by a substitution.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_subst" class="idref" href="#Make.rpo_subst"><span class="id" title="lemma">rpo_subst</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:313" class="idref" href="#t:313"><span class="id" title="binder">t</span></a> <a id="s:314" class="idref" href="#s:314"><span class="id" title="binder">s</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:314"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:313"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma:315" class="idref" href="#sigma:315"><span class="id" title="binder">sigma</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.rpo.html#sigma:315"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:314"><span class="id" title="variable">s</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.rpo.html#sigma:315"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:313"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab156"></a><h2 class="section">RPO is compatible with adding context.</h2>
+
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Make.rpo_add_context" class="idref" href="#Make.rpo_add_context"><span class="id" title="lemma">rpo_add_context</span></a> :<br/>
+&nbsp;<span class="id" title="keyword">∀</span> <a id="p:316" class="idref" href="#p:316"><span class="id" title="binder">p</span></a> <a id="ctx:317" class="idref" href="#ctx:317"><span class="id" title="binder">ctx</span></a> <a id="s:318" class="idref" href="#s:318"><span class="id" title="binder">s</span></a> <a id="t:319" class="idref" href="#t:319"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:318"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:319"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.rpo.html#Make.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:317"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:316"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.rpo.html#Make.rpo"><span class="id" title="inductive">rpo</span></a> (<a class="idref" href="hydras.rpo.rpo.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:317"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#s:318"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:316"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.rpo.rpo.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.rpo.html#ctx:317"><span class="id" title="variable">ctx</span></a> <a class="idref" href="hydras.rpo.rpo.html#t:319"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.rpo.html#p:316"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.rpo.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.rpo.term.html b/theories/html/hydras.rpo.term.html
new file mode 100644
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--- /dev/null
+++ b/theories/html/hydras.rpo.term.html
@@ -0,0 +1,916 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.rpo.term</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.rpo.term</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+by Evelyne Contejean, LRI 
+<div class="paragraph"> </div>
+
+<a id="lab157"></a><h1 class="section">Term algebra defined as functor from a Module Signature and a Module Variable.</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.more_list.html#"><span class="id" title="library">more_list</span></a> <a class="idref" href="hydras.rpo.list_permut.html#"><span class="id" title="library">list_permut</span></a> <a class="idref" href="hydras.rpo.list_set.html#"><span class="id" title="library">list_set</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab158"></a><h1 class="section">Module Type Signature.</h1>
+
+ There are almost no assumptions, except a decidable equality 
+and an arity function. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="Signature" class="idref" href="#Signature"><span class="id" title="module">Signature</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="Signature.symb" class="idref" href="#Signature.symb"><span class="id" title="axiom">symb</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Signature.eq_symbol_dec" class="idref" href="#Signature.eq_symbol_dec"><span class="id" title="axiom">eq_symbol_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="f1:2" class="idref" href="#f1:2"><span class="id" title="binder">f1</span></a> <a id="f2:3" class="idref" href="#f2:3"><span class="id" title="binder">f2</span></a> : <a class="idref" href="hydras.rpo.term.html#Signature.symb"><span class="id" title="axiom">symb</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#f1:2"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#f2:3"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#f1:2"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.term.html#f2:3"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The arity of a symbol contains also the information about built-in theories as in CiME 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Signature.arity_type" class="idref" href="#Signature.arity_type"><span class="id" title="inductive">arity_type</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Signature.AC" class="idref" href="#Signature.AC"><span class="id" title="constructor">AC</span></a> : <a class="idref" href="hydras.rpo.term.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Signature.C" class="idref" href="#Signature.C"><span class="id" title="constructor">C</span></a> : <a class="idref" href="hydras.rpo.term.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a><br/>
+&nbsp;&nbsp;| <a id="Signature.Free" class="idref" href="#Signature.Free"><span class="id" title="constructor">Free</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#arity_type:5"><span class="id" title="inductive">arity_type</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="Signature.arity" class="idref" href="#Signature.arity"><span class="id" title="axiom">arity</span></a> : <a class="idref" href="hydras.rpo.term.html#Signature.symb"><span class="id" title="axiom">symb</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Signature.arity_type"><span class="id" title="inductive">arity_type</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Signature"><span class="id" title="module">Signature</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab159"></a><h1 class="section">Module Type Variables.</h1>
+
+ There are almost no assumptions, except a decidable equality. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="Variables" class="idref" href="#Variables"><span class="id" title="module">Variables</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="Variables.var" class="idref" href="#Variables.var"><span class="id" title="axiom">var</span></a> : <span class="id" title="keyword">Set</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Variables.eq_variable_dec" class="idref" href="#Variables.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="v1:9" class="idref" href="#v1:9"><span class="id" title="binder">v1</span></a> <a id="v2:10" class="idref" href="#v2:10"><span class="id" title="binder">v2</span></a> : <a class="idref" href="hydras.rpo.term.html#Variables.var"><span class="id" title="axiom">var</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#v1:9"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#v2:10"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#v1:9"><span class="id" title="variable">v1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.term.html#v2:10"><span class="id" title="variable">v2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Variables"><span class="id" title="module">Variables</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab160"></a><h1 class="section">Module Type Term built from a signature and a set of variables.</h1>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="Term" class="idref" href="#Term"><span class="id" title="module">Term</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="Term.F" class="idref" href="#Term.F"><span class="id" title="module">F</span></a> : <a class="idref" href="hydras.rpo.term.html#Signature"><span class="id" title="module">Signature</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="Term.X" class="idref" href="#Term.X"><span class="id" title="module">X</span></a> : <a class="idref" href="hydras.rpo.term.html#Variables"><span class="id" title="module">Variables</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.symbol" class="idref" href="#Term.symbol"><span class="id" title="definition">symbol</span></a> := <a class="idref" href="hydras.rpo.term.html#Term.symb"><span class="id" title="axiom">F.symb</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.variable" class="idref" href="#Term.variable"><span class="id" title="definition">variable</span></a> := <a class="idref" href="hydras.rpo.term.html#Term.var"><span class="id" title="axiom">X.var</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.term.html#Term.F"><span class="id" title="module">F</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.term.html#Term.X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">destruct_arity</span> <span class="id" title="var">f</span> <span class="id" title="var">n</span> <span class="id" title="var">Af</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span>)); <span class="id" title="tactic">pattern</span> <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> 1; <span class="id" title="tactic">destruct</span> (<a class="idref" href="hydras.rpo.term.html#Term.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span>) <span class="id" title="keyword">as</span> [ |  | <span class="id" title="var">n</span>]; <span class="id" title="tactic">intro</span> <span class="id" title="var">Af</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition of terms. 
+Arity is not taken into account, and terms may be hill-formed. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Term.term" class="idref" href="#Term.term"><span class="id" title="inductive">term</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Term.Var" class="idref" href="#Term.Var"><span class="id" title="constructor">Var</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#term:12"><span class="id" title="inductive">term</span></a><br/>
+&nbsp;&nbsp;| <a id="Term.Term" class="idref" href="#Term.Term"><span class="id" title="constructor">Term</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.symbol"><span class="id" title="definition">symbol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#term:12"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#term:12"><span class="id" title="inductive">term</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.direct_subterm" class="idref" href="#Term.direct_subterm"><span class="id" title="definition">direct_subterm</span></a> <a id="t1:14" class="idref" href="#t1:14"><span class="id" title="binder">t1</span></a> <a id="t2:15" class="idref" href="#t2:15"><span class="id" title="binder">t2</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t2:15"><span class="id" title="variable">t2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:14"><span class="id" title="variable">t1</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition and a few properties of the size of a term.
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.size" class="idref" href="#Term.size"><span class="id" title="definition">size</span></a> (<a id="t:17" class="idref" href="#t:17"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:17"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">v</span> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> <a id="size_acc:20" class="idref" href="#size_acc:20"><span class="id" title="binder">size_acc</span></a> <a id="e:21" class="idref" href="#e:21"><span class="id" title="binder">e</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#size_acc:20"><span class="id" title="variable">size_acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.term.html#size:18"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#e:21"><span class="id" title="variable">e</span></a>) <span class="id" title="var">l</span> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.size_unfold" class="idref" href="#Term.size_unfold"><span class="id" title="axiom">size_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:22" class="idref" href="#t:22"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:22"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:22"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.size_ge_one" class="idref" href="#Term.size_ge_one"><span class="id" title="axiom">size_ge_one</span></a> : <span class="id" title="keyword">∀</span> <a id="t:25" class="idref" href="#t:25"><span class="id" title="binder">t</span></a>, 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:25"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.size_direct_subterm" class="idref" href="#Term.size_direct_subterm"><span class="id" title="axiom">size_direct_subterm</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:27" class="idref" href="#t1:27"><span class="id" title="binder">t1</span></a> <a id="t2:28" class="idref" href="#t2:28"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.direct_subterm"><span class="id" title="definition">direct_subterm</span></a> <a class="idref" href="hydras.rpo.term.html#t1:27"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:28"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t1:27"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t2:28"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab161"></a><h2 class="section">Recursion on terms.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Term.Recursion" class="idref" href="#Term.Recursion"><span class="id" title="section">Recursion</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Term.Recursion.P" class="idref" href="#Term.Recursion.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Term.Recursion.Pl" class="idref" href="#Term.Recursion.Pl"><span class="id" title="variable">Pl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.term_rec2" class="idref" href="#Term.term_rec2"><span class="id" title="axiom">term_rec2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> <a id="t:33" class="idref" href="#t:33"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:33"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.term.html#n:32"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:33"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:34" class="idref" href="#t:34"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:34"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.term_rec3" class="idref" href="#Term.term_rec3"><span class="id" title="axiom">term_rec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:36" class="idref" href="#v:36"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v:36"><span class="id" title="variable">v</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:37" class="idref" href="#f:37"><span class="id" title="binder">f</span></a> <a id="l:38" class="idref" href="#l:38"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t:39" class="idref" href="#t:39"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t:39"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#l:38"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:39"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:37"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:38"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:40" class="idref" href="#t:40"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:40"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.term_rec4" class="idref" href="#Term.term_rec4"><span class="id" title="axiom">term_rec4</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:42" class="idref" href="#v:42"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v:42"><span class="id" title="variable">v</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:43" class="idref" href="#f:43"><span class="id" title="binder">f</span></a> <a id="l:44" class="idref" href="#l:44"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.Pl"><span class="id" title="variable">Pl</span></a> <a class="idref" href="hydras.rpo.term.html#l:44"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:43"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:44"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l:45" class="idref" href="#l:45"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t:46" class="idref" href="#t:46"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t:46"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#l:45"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:46"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Recursion.Pl"><span class="id" title="variable">Pl</span></a> <a class="idref" href="hydras.rpo.term.html#l:45"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:47" class="idref" href="#t:47"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:47"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Term.Recursion"><span class="id" title="section">Recursion</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab162"></a><h2 class="section">Double recursion on terms.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Term.DoubleRecursion" class="idref" href="#Term.DoubleRecursion"><span class="id" title="section">DoubleRecursion</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Term.DoubleRecursion.P2" class="idref" href="#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Term.DoubleRecursion.Pl2" class="idref" href="#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.term_rec7" class="idref" href="#Term.term_rec7"><span class="id" title="axiom">term_rec7</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v1:51" class="idref" href="#v1:51"><span class="id" title="binder">v1</span></a> <a id="t2:52" class="idref" href="#t2:52"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v1:51"><span class="id" title="variable">v1</span></a>) <a class="idref" href="hydras.rpo.term.html#t2:52"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:53" class="idref" href="#t1:53"><span class="id" title="binder">t1</span></a> <a id="v2:54" class="idref" href="#v2:54"><span class="id" title="binder">v2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:53"><span class="id" title="variable">t1</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v2:54"><span class="id" title="variable">v2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f1:55" class="idref" href="#f1:55"><span class="id" title="binder">f1</span></a> <a id="f2:56" class="idref" href="#f2:56"><span class="id" title="binder">f2</span></a> <a id="l1:57" class="idref" href="#l1:57"><span class="id" title="binder">l1</span></a> <a id="l2:58" class="idref" href="#l2:58"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:57"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:58"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f1:55"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:57"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f2:56"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:58"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l1:59" class="idref" href="#l1:59"><span class="id" title="binder">l1</span></a> <a id="l2:60" class="idref" href="#l2:60"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:61" class="idref" href="#t1:61"><span class="id" title="binder">t1</span></a> <a id="t2:62" class="idref" href="#t2:62"><span class="id" title="binder">t2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:61"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:59"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t2:62"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:60"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:61"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:62"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:59"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:60"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:63" class="idref" href="#t1:63"><span class="id" title="binder">t1</span></a> <a id="t2:64" class="idref" href="#t2:64"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:63"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:64"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.term_rec8" class="idref" href="#Term.term_rec8"><span class="id" title="axiom">term_rec8</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v1:66" class="idref" href="#v1:66"><span class="id" title="binder">v1</span></a> <a id="t2:67" class="idref" href="#t2:67"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v1:66"><span class="id" title="variable">v1</span></a>) <a class="idref" href="hydras.rpo.term.html#t2:67"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:68" class="idref" href="#t1:68"><span class="id" title="binder">t1</span></a> <a id="v2:69" class="idref" href="#v2:69"><span class="id" title="binder">v2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:68"><span class="id" title="variable">t1</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v2:69"><span class="id" title="variable">v2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f1:70" class="idref" href="#f1:70"><span class="id" title="binder">f1</span></a> <a id="f2:71" class="idref" href="#f2:71"><span class="id" title="binder">f2</span></a> <a id="l1:72" class="idref" href="#l1:72"><span class="id" title="binder">l1</span></a> <a id="l2:73" class="idref" href="#l2:73"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:72"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:73"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f1:70"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:72"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f2:71"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:73"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l1:74" class="idref" href="#l1:74"><span class="id" title="binder">l1</span></a> <a id="l2:75" class="idref" href="#l2:75"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:76" class="idref" href="#t1:76"><span class="id" title="binder">t1</span></a> <a id="t2:77" class="idref" href="#t2:77"><span class="id" title="binder">t2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:76"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:74"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t2:77"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:75"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:76"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:77"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:74"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:75"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:78" class="idref" href="#l1:78"><span class="id" title="binder">l1</span></a> <a id="l2:79" class="idref" href="#l2:79"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:78"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:79"><span class="id" title="variable">l2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Term.DoubleRecursion"><span class="id" title="section">DoubleRecursion</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab163"></a><h2 class="section">Equality on terms is decidable.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.eq_term_dec" class="idref" href="#Term.eq_term_dec"><span class="id" title="axiom">eq_term_dec</span></a> :  <span class="id" title="keyword">∀</span> <a id="t1:81" class="idref" href="#t1:81"><span class="id" title="binder">t1</span></a> <a id="t2:82" class="idref" href="#t2:82"><span class="id" title="binder">t2</span></a>:<a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#t1:81"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t2:82"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#t1:81"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.term.html#t2:82"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="Term.Term_eq_dec" class="idref" href="#Term.Term_eq_dec"><span class="id" title="module">Term_eq_dec</span></a> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">A</span>:= <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">eq_A_dec</span> := <a class="idref" href="hydras.rpo.term.html#Term.eq_term_dec"><span class="id" title="axiom">eq_term_dec</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab164"></a><h2 class="section">Well-formedness of terms, according to the arity.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.well_formed" class="idref" href="#Term.well_formed"><span class="id" title="definition">well_formed</span></a> (<a id="t:84" class="idref" href="#t:84"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:84"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="well_formed_list:90" class="idref" href="#well_formed_list:90"><span class="id" title="binder">well_formed_list</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fix</span> <span class="id" title="var">well_formed_list</span> (<a id="l:87" class="idref" href="#l:87"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:87"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#well_formed:85"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.term.html#well_formed_list:88"><span class="id" title="variable">well_formed_list</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#well_formed_list:90"><span class="id" title="variable">well_formed_list</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Term.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.well_formed_unfold" class="idref" href="#Term.well_formed_unfold"><span class="id" title="axiom">well_formed_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:91" class="idref" href="#t:91"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:91"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:91"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="u:93" class="idref" href="#u:93"><span class="id" title="binder">u</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#u:93"><span class="id" title="variable">u</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#u:93"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Term.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.well_formed_fold" class="idref" href="#Term.well_formed_fold"><span class="id" title="axiom">well_formed_fold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:95" class="idref" href="#t:95"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:95"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="u:97" class="idref" href="#u:97"><span class="id" title="binder">u</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#u:97"><span class="id" title="variable">u</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#u:97"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Term.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:95"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.well_formed_list" class="idref" href="#Term.well_formed_list"><span class="id" title="definition">well_formed_list</span></a> (<a id="l:99" class="idref" href="#l:99"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:99"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.term.html#well_formed_list:100"><span class="id" title="definition">well_formed_list</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab165"></a><h2 class="section">Substitutions.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.substitution" class="idref" href="#Term.substitution"><span class="id" title="definition">substitution</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.apply_subst" class="idref" href="#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> (<a id="sigma:102" class="idref" href="#sigma:102"><span class="id" title="binder">sigma</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.substitution"><span class="id" title="definition">substitution</span></a>) (<a id="t:103" class="idref" href="#t:103"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t</span>} : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:103"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">v</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Term.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <span class="id" title="var">v</span> <a class="idref" href="hydras.rpo.term.html#sigma:102"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#t:103"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v_sigma</span> ⇒ <span class="id" title="var">v_sigma</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="hydras.rpo.term.html#apply_subst:104"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:102"><span class="id" title="variable">sigma</span></a>) <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.empty_subst_is_id" class="idref" href="#Term.empty_subst_is_id"><span class="id" title="axiom">empty_subst_is_id</span></a> : <span class="id" title="keyword">∀</span> <a id="t:106" class="idref" href="#t:106"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.rpo.term.html#t:106"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t:106"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.empty_subst_is_id_list" class="idref" href="#Term.empty_subst_is_id_list"><span class="id" title="axiom">empty_subst_is_id_list</span></a> : <span class="id" title="keyword">∀</span> <a id="l:108" class="idref" href="#l:108"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.rpo.term.html#l:108"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#l:108"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Composition of substitutions. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.map_subst" class="idref" href="#Term.map_subst"><span class="id" title="definition">map_subst</span></a> (<a id="f:110" class="idref" href="#f:110"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) <a id="sigma:111" class="idref" href="#sigma:111"><span class="id" title="binder">sigma</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <a id="x:112" class="idref" href="#x:112"><span class="id" title="binder">x</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.rpo.term.html#x:112"><span class="id" title="variable">x</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">v_sigma</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.term.html#f:110"><span class="id" title="variable">f</span></a> <span class="id" title="var">v</span> <span class="id" title="var">v_sigma</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#sigma:111"><span class="id" title="variable">sigma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.subst_comp" class="idref" href="#Term.subst_comp"><span class="id" title="definition">subst_comp</span></a> <a id="sigma1:113" class="idref" href="#sigma1:113"><span class="id" title="binder">sigma1</span></a> <a id="sigma2:114" class="idref" href="#sigma2:114"><span class="id" title="binder">sigma2</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Term.map_subst"><span class="id" title="definition">map_subst</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="t:115" class="idref" href="#t:115"><span class="id" title="binder">t</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:113"><span class="id" title="variable">sigma1</span></a> <a class="idref" href="hydras.rpo.term.html#t:115"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.rpo.term.html#sigma2:114"><span class="id" title="variable">sigma2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Term.map_subst"><span class="id" title="definition">map_subst</span></a> (<span class="id" title="keyword">fun</span> <a id="v:116" class="idref" href="#v:116"><span class="id" title="binder">v</span></a> <a id="t:117" class="idref" href="#t:117"><span class="id" title="binder">t</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Term.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:116"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:114"><span class="id" title="variable">sigma2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#t:117"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v_sigma2</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:113"><span class="id" title="variable">sigma1</span></a> <span class="id" title="var">v_sigma2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#sigma1:113"><span class="id" title="variable">sigma1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.subst_comp_is_subst_comp_aux1" class="idref" href="#Term.subst_comp_is_subst_comp_aux1"><span class="id" title="axiom">subst_comp_is_subst_comp_aux1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:118" class="idref" href="#v:118"><span class="id" title="binder">v</span></a> <a id="sigma:119" class="idref" href="#sigma:119"><span class="id" title="binder">sigma</span></a> <a id="f:120" class="idref" href="#f:120"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Term.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:118"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.map_subst"><span class="id" title="definition">map_subst</span></a> <a class="idref" href="hydras.rpo.term.html#f:120"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:119"><span class="id" title="variable">sigma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Term.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:118"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:119"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.rpo.term.html#f:120"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#v:118"><span class="id" title="variable">v</span></a> <span class="id" title="var">t</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.subst_comp_is_subst_comp" class="idref" href="#Term.subst_comp_is_subst_comp"><span class="id" title="axiom">subst_comp_is_subst_comp</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma1:122" class="idref" href="#sigma1:122"><span class="id" title="binder">sigma1</span></a> <a id="sigma2:123" class="idref" href="#sigma2:123"><span class="id" title="binder">sigma2</span></a> <a id="t:124" class="idref" href="#t:124"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.subst_comp"><span class="id" title="definition">subst_comp</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:122"><span class="id" title="variable">sigma1</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:123"><span class="id" title="variable">sigma2</span></a>) <a class="idref" href="hydras.rpo.term.html#t:124"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:122"><span class="id" title="variable">sigma1</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:123"><span class="id" title="variable">sigma2</span></a> <a class="idref" href="hydras.rpo.term.html#t:124"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Well-formed substitutions. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Term.well_formed_subst" class="idref" href="#Term.well_formed_subst"><span class="id" title="definition">well_formed_subst</span></a> <a id="sigma:126" class="idref" href="#sigma:126"><span class="id" title="binder">sigma</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:127" class="idref" href="#v:127"><span class="id" title="binder">v</span></a>, <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Term.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:127"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:126"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">t</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.well_formed_apply_subst" class="idref" href="#Term.well_formed_apply_subst"><span class="id" title="axiom">well_formed_apply_subst</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma:128" class="idref" href="#sigma:128"><span class="id" title="binder">sigma</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.well_formed_subst"><span class="id" title="definition">well_formed_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:128"><span class="id" title="variable">sigma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:129" class="idref" href="#t:129"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:129"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.well_formed"><span class="id" title="definition">well_formed</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:128"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.term.html#t:129"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab166"></a><h2 class="section">Positions in a term.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.is_a_pos" class="idref" href="#Term.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> (<a id="t:131" class="idref" href="#t:131"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) (<a id="p:132" class="idref" href="#p:132"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:132"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:131"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <span class="id" title="var">l</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">ti</span> ⇒ <a class="idref" href="hydras.rpo.term.html#is_a_pos:133"><span class="id" title="definition">is_a_pos</span></a> <span class="id" title="var">ti</span> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.subterm_at_pos" class="idref" href="#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a id="t:136" class="idref" href="#t:136"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) (<a id="p:137" class="idref" href="#p:137"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:137"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#t:136"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:136"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <span class="id" title="var">l</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">ti</span> ⇒ <a class="idref" href="hydras.rpo.term.html#subterm_at_pos:138"><span class="id" title="definition">subterm_at_pos</span></a> <span class="id" title="var">ti</span> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.size_subterm_at_pos" class="idref" href="#Term.size_subterm_at_pos"><span class="id" title="axiom">size_subterm_at_pos</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:141" class="idref" href="#t:141"><span class="id" title="binder">t</span></a> <a id="i:142" class="idref" href="#i:142"><span class="id" title="binder">i</span></a> <a id="p:143" class="idref" href="#p:143"><span class="id" title="binder">p</span></a>, <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:141"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.rpo.term.html#i:142"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#p:143"><span class="id" title="variable">p</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">u</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">u</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.term.html#Term.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:141"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.is_a_pos_exists_subtem" class="idref" href="#Term.is_a_pos_exists_subtem"><span class="id" title="axiom">is_a_pos_exists_subtem</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:145" class="idref" href="#t:145"><span class="id" title="binder">t</span></a> <a id="p:146" class="idref" href="#p:146"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:145"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:146"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u:147" class="idref" href="#u:147"><span class="id" title="binder">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:145"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:146"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:147"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.replace_at_pos" class="idref" href="#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> (<a id="t:149" class="idref" href="#t:149"><span class="id" title="binder">t</span></a> <a id="u:150" class="idref" href="#u:150"><span class="id" title="binder">u</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) (<a id="p:151" class="idref" href="#p:151"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>} : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:151"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#u:150"><span class="id" title="variable">u</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:149"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.rpo.term.html#t:149"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="replace_at_pos_list:160" class="idref" href="#replace_at_pos_list:160"><span class="id" title="binder">replace_at_pos_list</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fix</span> <span class="id" title="var">replace_at_pos_list</span> <a id="j:155" class="idref" href="#j:155"><span class="id" title="binder">j</span></a> (<a id="l:156" class="idref" href="#l:156"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:156"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#j:155"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos:152"><span class="id" title="definition">replace_at_pos</span></a> <span class="id" title="var">h</span> <a class="idref" href="hydras.rpo.term.html#u:150"><span class="id" title="variable">u</span></a> <span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">k</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:157"><span class="id" title="variable">replace_at_pos_list</span></a> <span class="id" title="var">k</span> <span class="id" title="var">tl</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:160"><span class="id" title="variable">replace_at_pos_list</span></a> <span class="id" title="var">i</span> <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Term.replace_at_pos_list" class="idref" href="#Term.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> (<a id="l:161" class="idref" href="#l:161"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) (<a id="u:162" class="idref" href="#u:162"><span class="id" title="binder">u</span></a> : <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a>) (<a id="i:163" class="idref" href="#i:163"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:164" class="idref" href="#p:164"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Term.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:161"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#i:163"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <span class="id" title="var">h</span> <a class="idref" href="hydras.rpo.term.html#u:162"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:164"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:165"><span class="id" title="definition">replace_at_pos_list</span></a> <span class="id" title="var">tl</span>  <a class="idref" href="hydras.rpo.term.html#u:162"><span class="id" title="variable">u</span></a> <span class="id" title="var">j</span> <a class="idref" href="hydras.rpo.term.html#p:164"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.replace_at_pos_unfold" class="idref" href="#Term.replace_at_pos_unfold"><span class="id" title="axiom">replace_at_pos_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:168" class="idref" href="#f:168"><span class="id" title="binder">f</span></a> <a id="l:169" class="idref" href="#l:169"><span class="id" title="binder">l</span></a> <a id="u:170" class="idref" href="#u:170"><span class="id" title="binder">u</span></a> <a id="i:171" class="idref" href="#i:171"><span class="id" title="binder">i</span></a> <a id="q:172" class="idref" href="#q:172"><span class="id" title="binder">q</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:168"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:169"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.rpo.term.html#u:170"><span class="id" title="variable">u</span></a> (<a class="idref" href="hydras.rpo.term.html#i:171"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#q:172"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#Term.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:168"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> <a class="idref" href="hydras.rpo.term.html#l:169"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.term.html#u:170"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#i:171"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.rpo.term.html#q:172"><span class="id" title="variable">q</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.replace_at_pos_is_replace_at_pos1" class="idref" href="#Term.replace_at_pos_is_replace_at_pos1"><span class="id" title="axiom">replace_at_pos_is_replace_at_pos1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:174" class="idref" href="#t:174"><span class="id" title="binder">t</span></a> <a id="u:175" class="idref" href="#u:175"><span class="id" title="binder">u</span></a> <a id="p:176" class="idref" href="#p:176"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:174"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:176"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:174"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#u:175"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:176"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.rpo.term.html#p:176"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:175"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.replace_at_pos_is_replace_at_pos2" class="idref" href="#Term.replace_at_pos_is_replace_at_pos2"><span class="id" title="axiom">replace_at_pos_is_replace_at_pos2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:178" class="idref" href="#t:178"><span class="id" title="binder">t</span></a> <a id="p:179" class="idref" href="#p:179"><span class="id" title="binder">p</span></a> <a id="u:180" class="idref" href="#u:180"><span class="id" title="binder">u</span></a>, <a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:178"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:179"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:180"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:178"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#u:180"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:179"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t:178"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos" class="idref" href="#Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos"><span class="id" title="axiom">subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:182" class="idref" href="#t:182"><span class="id" title="binder">t</span></a> <a id="p:183" class="idref" href="#p:183"><span class="id" title="binder">p</span></a> <a id="sigma:184" class="idref" href="#sigma:184"><span class="id" title="binder">sigma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:182"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:183"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">u</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:184"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.term.html#t:182"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.rpo.term.html#p:183"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.rpo.term.html#Term.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:184"><span class="id" title="variable">sigma</span></a> <span class="id" title="var">u</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="Term.replace_at_pos_list_replace_at_pos_in_subterm" class="idref" href="#Term.replace_at_pos_list_replace_at_pos_in_subterm"><span class="id" title="axiom">replace_at_pos_list_replace_at_pos_in_subterm</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:186" class="idref" href="#l1:186"><span class="id" title="binder">l1</span></a> <a id="ui:187" class="idref" href="#ui:187"><span class="id" title="binder">ui</span></a> <a id="l2:188" class="idref" href="#l2:188"><span class="id" title="binder">l2</span></a> <a id="u:189" class="idref" href="#u:189"><span class="id" title="binder">u</span></a> <a id="i:190" class="idref" href="#i:190"><span class="id" title="binder">i</span></a> <a id="p:191" class="idref" href="#p:191"><span class="id" title="binder">p</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.term.html#l1:186"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#i:190"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> (<a class="idref" href="hydras.rpo.term.html#l1:186"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.term.html#ui:187"><span class="id" title="variable">ui</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#l2:188"><span class="id" title="variable">l2</span></a>) <a class="idref" href="hydras.rpo.term.html#u:189"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#i:190"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.rpo.term.html#p:191"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#l1:186"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Term.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#ui:187"><span class="id" title="variable">ui</span></a> <a class="idref" href="hydras.rpo.term.html#u:189"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:191"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#l2:188"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">Term</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab167"></a><h1 class="section">A functor building a Term Module from a Signature and a set of Variables.</h1>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">F1</span> : <a class="idref" href="hydras.rpo.term.html#Signature"><span class="id" title="module">Signature</span></a>) (<span class="id" title="var">X1</span> : <a class="idref" href="hydras.rpo.term.html#Variables"><span class="id" title="module">Variables</span></a>) &lt;: <br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Term"><span class="id" title="module">Term</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">F</span> := <a class="idref" href="hydras.rpo.term.html#F1"><span class="id" title="module">F1</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">X</span> := <a class="idref" href="hydras.rpo.term.html#X1"><span class="id" title="module">X1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Make.F" class="idref" href="#Make.F"><span class="id" title="module">F</span></a> := <a class="idref" href="hydras.rpo.term.html#F1"><span class="id" title="module">F1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Make.X" class="idref" href="#Make.X"><span class="id" title="module">X</span></a> := <a class="idref" href="hydras.rpo.term.html#X1"><span class="id" title="module">X1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.symbol" class="idref" href="#Make.symbol"><span class="id" title="definition">symbol</span></a> := <a class="idref" href="hydras.rpo.term.html#Make.symb"><span class="id" title="axiom">F.symb</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.variable" class="idref" href="#Make.variable"><span class="id" title="definition">variable</span></a> := <a class="idref" href="hydras.rpo.term.html#Make.var"><span class="id" title="axiom">X.var</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.term.html#Make.F"><span class="id" title="module">F</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.rpo.term.html#Make.X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Make.DecVar" class="idref" href="#Make.DecVar"><span class="id" title="module">DecVar</span></a> &lt;: <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.DecVar.A" class="idref" href="#Make.DecVar.A"><span class="id" title="definition">A</span></a> := <a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.DecVar.eq_A_dec" class="idref" href="#Make.DecVar.eq_A_dec"><span class="id" title="lemma">eq_A_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:193" class="idref" href="#x:193"><span class="id" title="binder">x</span></a> <a id="y:194" class="idref" href="#y:194"><span class="id" title="binder">y</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.DecVar.A"><span class="id" title="definition">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a> <a class="idref" href="hydras.rpo.term.html#x:193"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#y:194"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a> <a class="idref" href="hydras.rpo.term.html#x:193"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.term.html#y:194"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Make.DecVar"><span class="id" title="module">DecVar</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Make.VSet" class="idref" href="#Make.VSet"><span class="id" title="module">VSet</span></a> &lt;: <a class="idref" href="hydras.rpo.list_set.html#S"><span class="id" title="module">list_set.S</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">DS.A</span> := <a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.list_set.html#Make"><span class="id" title="module">list_set.Make</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.DecVar"><span class="id" title="module">DecVar</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition of terms. 
+Arity is not taken into account, and terms may be ill-formed. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Make.term" class="idref" href="#Make.term"><span class="id" title="inductive">term</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
+&nbsp;&nbsp;| <a id="Make.Var" class="idref" href="#Make.Var"><span class="id" title="constructor">Var</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#term:195"><span class="id" title="inductive">term</span></a><br/>
+&nbsp;&nbsp;| <a id="Make.Term" class="idref" href="#Make.Term"><span class="id" title="constructor">Term</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.symbol"><span class="id" title="definition">symbol</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#term:195"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#term:195"><span class="id" title="inductive">term</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.direct_subterm" class="idref" href="#Make.direct_subterm"><span class="id" title="definition">direct_subterm</span></a> <a id="t1:197" class="idref" href="#t1:197"><span class="id" title="binder">t1</span></a> <a id="t2:198" class="idref" href="#t2:198"><span class="id" title="binder">t2</span></a> : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t2:198"><span class="id" title="variable">t2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:197"><span class="id" title="variable">t1</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Definition and a few properties of the size of a term.
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.size" class="idref" href="#Make.size"><span class="id" title="definition">size</span></a> (<a id="t:200" class="idref" href="#t:200"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:200"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">v</span> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> <a id="size_acc:203" class="idref" href="#size_acc:203"><span class="id" title="binder">size_acc</span></a> <a id="e:204" class="idref" href="#e:204"><span class="id" title="binder">e</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#size_acc:203"><span class="id" title="variable">size_acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.term.html#size:201"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#e:204"><span class="id" title="variable">e</span></a>) <span class="id" title="var">l</span> 0<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.size_unfold" class="idref" href="#Make.size_unfold"><span class="id" title="lemma">size_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:205" class="idref" href="#t:205"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:205"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:205"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.rpo.more_list.html#list_size"><span class="id" title="definition">list_size</span></a> <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.size_ge_one" class="idref" href="#Make.size_ge_one"><span class="id" title="lemma">size_ge_one</span></a> : <span class="id" title="keyword">∀</span> <a id="t:207" class="idref" href="#t:207"><span class="id" title="binder">t</span></a>, 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:207"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.size_direct_subterm" class="idref" href="#Make.size_direct_subterm"><span class="id" title="lemma">size_direct_subterm</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:208" class="idref" href="#t1:208"><span class="id" title="binder">t1</span></a> <a id="t2:209" class="idref" href="#t2:209"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.direct_subterm"><span class="id" title="definition">direct_subterm</span></a> <a class="idref" href="hydras.rpo.term.html#t1:208"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:209"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t1:208"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t2:209"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab168"></a><h2 class="section">Recursion on terms.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Make.Recursion" class="idref" href="#Make.Recursion"><span class="id" title="section">Recursion</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Make.Recursion.P" class="idref" href="#Make.Recursion.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Make.Recursion.Pl" class="idref" href="#Make.Recursion.Pl"><span class="id" title="variable">Pl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.term_rec2" class="idref" href="#Make.term_rec2"><span class="id" title="definition">term_rec2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:212" class="idref" href="#n:212"><span class="id" title="binder">n</span></a> <a id="t:213" class="idref" href="#t:213"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:213"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.rpo.term.html#n:212"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:213"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:214" class="idref" href="#t:214"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:214"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.term_rec3" class="idref" href="#Make.term_rec3"><span class="id" title="definition">term_rec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:215" class="idref" href="#v:215"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v:215"><span class="id" title="variable">v</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:216" class="idref" href="#f:216"><span class="id" title="binder">f</span></a> <a id="l:217" class="idref" href="#l:217"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t:218" class="idref" href="#t:218"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t:218"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#l:217"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:218"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:216"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:217"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:219" class="idref" href="#t:219"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:219"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.term_rec4" class="idref" href="#Make.term_rec4"><span class="id" title="definition">term_rec4</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v:220" class="idref" href="#v:220"><span class="id" title="binder">v</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v:220"><span class="id" title="variable">v</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f:221" class="idref" href="#f:221"><span class="id" title="binder">f</span></a> <a id="l:222" class="idref" href="#l:222"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.Pl"><span class="id" title="variable">Pl</span></a> <a class="idref" href="hydras.rpo.term.html#l:222"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:221"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:222"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l:223" class="idref" href="#l:223"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t:224" class="idref" href="#t:224"><span class="id" title="binder">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t:224"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#l:223"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:224"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Recursion.Pl"><span class="id" title="variable">Pl</span></a> <a class="idref" href="hydras.rpo.term.html#l:223"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="t:225" class="idref" href="#t:225"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.Recursion.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.rpo.term.html#t:225"><span class="id" title="variable">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Make.Recursion"><span class="id" title="section">Recursion</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab169"></a><h2 class="section">Double recursion on terms.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Make.DoubleRecursion" class="idref" href="#Make.DoubleRecursion"><span class="id" title="section">DoubleRecursion</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Make.DoubleRecursion.P2" class="idref" href="#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Make.DoubleRecursion.Pl2" class="idref" href="#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.term_rec7" class="idref" href="#Make.term_rec7"><span class="id" title="definition">term_rec7</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v1:228" class="idref" href="#v1:228"><span class="id" title="binder">v1</span></a> <a id="t2:229" class="idref" href="#t2:229"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v1:228"><span class="id" title="variable">v1</span></a>) <a class="idref" href="hydras.rpo.term.html#t2:229"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:230" class="idref" href="#t1:230"><span class="id" title="binder">t1</span></a> <a id="v2:231" class="idref" href="#v2:231"><span class="id" title="binder">v2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:230"><span class="id" title="variable">t1</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v2:231"><span class="id" title="variable">v2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f1:232" class="idref" href="#f1:232"><span class="id" title="binder">f1</span></a> <a id="f2:233" class="idref" href="#f2:233"><span class="id" title="binder">f2</span></a> <a id="l1:234" class="idref" href="#l1:234"><span class="id" title="binder">l1</span></a> <a id="l2:235" class="idref" href="#l2:235"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:234"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:235"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f1:232"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:234"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f2:233"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:235"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l1:236" class="idref" href="#l1:236"><span class="id" title="binder">l1</span></a> <a id="l2:237" class="idref" href="#l2:237"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:238" class="idref" href="#t1:238"><span class="id" title="binder">t1</span></a> <a id="t2:239" class="idref" href="#t2:239"><span class="id" title="binder">t2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:238"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:236"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t2:239"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:237"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:238"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:239"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:236"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:237"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:240" class="idref" href="#t1:240"><span class="id" title="binder">t1</span></a> <a id="t2:241" class="idref" href="#t2:241"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:240"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:241"><span class="id" title="variable">t2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.term_rec8" class="idref" href="#Make.term_rec8"><span class="id" title="definition">term_rec8</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="v1:242" class="idref" href="#v1:242"><span class="id" title="binder">v1</span></a> <a id="t2:243" class="idref" href="#t2:243"><span class="id" title="binder">t2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v1:242"><span class="id" title="variable">v1</span></a>) <a class="idref" href="hydras.rpo.term.html#t2:243"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:244" class="idref" href="#t1:244"><span class="id" title="binder">t1</span></a> <a id="v2:245" class="idref" href="#v2:245"><span class="id" title="binder">v2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:244"><span class="id" title="variable">t1</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <a class="idref" href="hydras.rpo.term.html#v2:245"><span class="id" title="variable">v2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="f1:246" class="idref" href="#f1:246"><span class="id" title="binder">f1</span></a> <a id="f2:247" class="idref" href="#f2:247"><span class="id" title="binder">f2</span></a> <a id="l1:248" class="idref" href="#l1:248"><span class="id" title="binder">l1</span></a> <a id="l2:249" class="idref" href="#l2:249"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:248"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:249"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f1:246"><span class="id" title="variable">f1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:248"><span class="id" title="variable">l1</span></a>) (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f2:247"><span class="id" title="variable">f2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:249"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l1:250" class="idref" href="#l1:250"><span class="id" title="binder">l1</span></a> <a id="l2:251" class="idref" href="#l2:251"><span class="id" title="binder">l2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="t1:252" class="idref" href="#t1:252"><span class="id" title="binder">t1</span></a> <a id="t2:253" class="idref" href="#t2:253"><span class="id" title="binder">t2</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t1:252"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#l1:250"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#t2:253"><span class="id" title="variable">t2</span></a> <a class="idref" href="hydras.rpo.term.html#l2:251"><span class="id" title="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.P2"><span class="id" title="variable">P2</span></a> <a class="idref" href="hydras.rpo.term.html#t1:252"><span class="id" title="variable">t1</span></a> <a class="idref" href="hydras.rpo.term.html#t2:253"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:250"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:251"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:254" class="idref" href="#l1:254"><span class="id" title="binder">l1</span></a> <a id="l2:255" class="idref" href="#l2:255"><span class="id" title="binder">l2</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2"><span class="id" title="variable">Pl2</span></a> <a class="idref" href="hydras.rpo.term.html#l1:254"><span class="id" title="variable">l1</span></a> <a class="idref" href="hydras.rpo.term.html#l2:255"><span class="id" title="variable">l2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Make.DoubleRecursion"><span class="id" title="section">DoubleRecursion</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab170"></a><h2 class="section">Equality on terms is decidable.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Make.eq_term_dec" class="idref" href="#Make.eq_term_dec"><span class="id" title="lemma">eq_term_dec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t1:256" class="idref" href="#t1:256"><span class="id" title="binder">t1</span></a> <a id="t2:257" class="idref" href="#t2:257"><span class="id" title="binder">t2</span></a>:<a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#t1:256"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t2:257"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="hydras.rpo.term.html#t1:256"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.rpo.term.html#t2:257"><span class="id" title="variable">t2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Make.Term_eq_dec" class="idref" href="#Make.Term_eq_dec"><span class="id" title="module">Term_eq_dec</span></a> : <a class="idref" href="hydras.rpo.decidable_set.html#S"><span class="id" title="module">decidable_set.S</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">A</span>:= <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">with</span> <span class="id" title="keyword">Definition</span> <span class="id" title="var">eq_A_dec</span> := <a class="idref" href="hydras.rpo.term.html#Make.eq_term_dec"><span class="id" title="lemma">eq_term_dec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Term_eq_dec.A" class="idref" href="#Make.Term_eq_dec.A"><span class="id" title="definition">A</span></a> := <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Term_eq_dec.eq_A_dec" class="idref" href="#Make.Term_eq_dec.eq_A_dec"><span class="id" title="definition">eq_A_dec</span></a> := <a class="idref" href="hydras.rpo.term.html#Make.eq_term_dec"><span class="id" title="lemma">eq_term_dec</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Make.Term_eq_dec"><span class="id" title="module">Term_eq_dec</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab171"></a><h2 class="section">Well-formedness of terms, according to the arity.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.well_formed" class="idref" href="#Make.well_formed"><span class="id" title="definition">well_formed</span></a> (<a id="t:264" class="idref" href="#t:264"><span class="id" title="binder">t</span></a>:<a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:264"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="well_formed_list:270" class="idref" href="#well_formed_list:270"><span class="id" title="binder">well_formed_list</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fix</span> <span class="id" title="var">well_formed_list</span> (<a id="l:267" class="idref" href="#l:267"><span class="id" title="binder">l</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:267"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#well_formed:265"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.term.html#well_formed_list:268"><span class="id" title="variable">well_formed_list</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#well_formed_list:270"><span class="id" title="variable">well_formed_list</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Make.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.well_formed_unfold" class="idref" href="#Make.well_formed_unfold"><span class="id" title="lemma">well_formed_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:271" class="idref" href="#t:271"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:271"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:271"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="u:273" class="idref" href="#u:273"><span class="id" title="binder">u</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#u:273"><span class="id" title="variable">u</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#u:273"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Make.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.well_formed_fold" class="idref" href="#Make.well_formed_fold"><span class="id" title="lemma">well_formed_fold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:274" class="idref" href="#t:274"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:274"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="u:276" class="idref" href="#u:276"><span class="id" title="binder">u</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="hydras.rpo.term.html#u:276"><span class="id" title="variable">u</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#u:276"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Make.arity"><span class="id" title="axiom">arity</span></a> <span class="id" title="var">f</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Free"><span class="id" title="constructor">Free</span></a> <span class="id" title="var">n</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">n</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:274"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.well_formed_list" class="idref" href="#Make.well_formed_list"><span class="id" title="definition">well_formed_list</span></a> (<a id="l:277" class="idref" href="#l:277"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:277"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.rpo.term.html#well_formed_list:278"><span class="id" title="definition">well_formed_list</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab172"></a><h2 class="section">Substitutions.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.substitution" class="idref" href="#Make.substitution"><span class="id" title="definition">substitution</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.apply_subst" class="idref" href="#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> (<a id="sigma:280" class="idref" href="#sigma:280"><span class="id" title="binder">sigma</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.substitution"><span class="id" title="definition">substitution</span></a>) (<a id="t:281" class="idref" href="#t:281"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t</span>} : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:281"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">v</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <span class="id" title="var">v</span> <a class="idref" href="hydras.rpo.term.html#sigma:280"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#t:281"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v_sigma</span> ⇒ <span class="id" title="var">v_sigma</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="hydras.rpo.term.html#apply_subst:282"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:280"><span class="id" title="variable">sigma</span></a>) <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.empty_subst_is_id" class="idref" href="#Make.empty_subst_is_id"><span class="id" title="lemma">empty_subst_is_id</span></a> : <span class="id" title="keyword">∀</span> <a id="t:284" class="idref" href="#t:284"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="hydras.rpo.term.html#t:284"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t:284"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.empty_subst_is_id_list" class="idref" href="#Make.empty_subst_is_id_list"><span class="id" title="lemma">empty_subst_is_id_list</span></a> : <span class="id" title="keyword">∀</span> <a id="l:287" class="idref" href="#l:287"><span class="id" title="binder">l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="hydras.rpo.term.html#l:287"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#l:287"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Composition of substitutions. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.map_subst" class="idref" href="#Make.map_subst"><span class="id" title="definition">map_subst</span></a> (<a id="f:288" class="idref" href="#f:288"><span class="id" title="binder">f</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) <a id="sigma:289" class="idref" href="#sigma:289"><span class="id" title="binder">sigma</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <a id="x:290" class="idref" href="#x:290"><span class="id" title="binder">x</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="hydras.rpo.term.html#x:290"><span class="id" title="variable">x</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.variable"><span class="id" title="definition">variable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">v_sigma</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.term.html#f:288"><span class="id" title="variable">f</span></a> <span class="id" title="var">v</span> <span class="id" title="var">v_sigma</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#sigma:289"><span class="id" title="variable">sigma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.subst_comp" class="idref" href="#Make.subst_comp"><span class="id" title="definition">subst_comp</span></a> <a id="sigma1:291" class="idref" href="#sigma1:291"><span class="id" title="binder">sigma1</span></a> <a id="sigma2:292" class="idref" href="#sigma2:292"><span class="id" title="binder">sigma2</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Make.map_subst"><span class="id" title="definition">map_subst</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="t:293" class="idref" href="#t:293"><span class="id" title="binder">t</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:291"><span class="id" title="variable">sigma1</span></a> <a class="idref" href="hydras.rpo.term.html#t:293"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.rpo.term.html#sigma2:292"><span class="id" title="variable">sigma2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Make.map_subst"><span class="id" title="definition">map_subst</span></a> (<span class="id" title="keyword">fun</span> <a id="v:294" class="idref" href="#v:294"><span class="id" title="binder">v</span></a> <a id="t:295" class="idref" href="#t:295"><span class="id" title="binder">t</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:294"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:292"><span class="id" title="variable">sigma2</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#t:295"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v_sigma2</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:291"><span class="id" title="variable">sigma1</span></a> <span class="id" title="var">v_sigma2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#sigma1:291"><span class="id" title="variable">sigma1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.subst_comp_is_subst_comp_aux1" class="idref" href="#Make.subst_comp_is_subst_comp_aux1"><span class="id" title="lemma">subst_comp_is_subst_comp_aux1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:296" class="idref" href="#v:296"><span class="id" title="binder">v</span></a> <a id="sigma:297" class="idref" href="#sigma:297"><span class="id" title="binder">sigma</span></a> <a id="f:298" class="idref" href="#f:298"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:296"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.map_subst"><span class="id" title="definition">map_subst</span></a> <a class="idref" href="hydras.rpo.term.html#f:298"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:297"><span class="id" title="variable">sigma</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:296"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:297"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.rpo.term.html#f:298"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#v:296"><span class="id" title="variable">v</span></a> <span class="id" title="var">t</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.subst_comp_is_subst_comp_aux2" class="idref" href="#Make.subst_comp_is_subst_comp_aux2"><span class="id" title="lemma">subst_comp_is_subst_comp_aux2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:299" class="idref" href="#v:299"><span class="id" title="binder">v</span></a> <a id="sigma1:300" class="idref" href="#sigma1:300"><span class="id" title="binder">sigma1</span></a> <a id="sigma2:301" class="idref" href="#sigma2:301"><span class="id" title="binder">sigma2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> (<span class="id" title="var">B</span>:= <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:299"><span class="id" title="variable">v</span></a> (<a class="idref" href="hydras.rpo.term.html#sigma1:300"><span class="id" title="variable">sigma1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:301"><span class="id" title="variable">sigma2</span></a>)  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:299"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:300"><span class="id" title="variable">sigma1</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:299"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:300"><span class="id" title="variable">sigma1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:299"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:301"><span class="id" title="variable">sigma2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Make.subst_comp_is_subst_comp" class="idref" href="#Make.subst_comp_is_subst_comp"><span class="id" title="lemma">subst_comp_is_subst_comp</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma1:302" class="idref" href="#sigma1:302"><span class="id" title="binder">sigma1</span></a> <a id="sigma2:303" class="idref" href="#sigma2:303"><span class="id" title="binder">sigma2</span></a> <a id="t:304" class="idref" href="#t:304"><span class="id" title="binder">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.subst_comp"><span class="id" title="definition">subst_comp</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:302"><span class="id" title="variable">sigma1</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:303"><span class="id" title="variable">sigma2</span></a>) <a class="idref" href="hydras.rpo.term.html#t:304"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma1:302"><span class="id" title="variable">sigma1</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma2:303"><span class="id" title="variable">sigma2</span></a> <a class="idref" href="hydras.rpo.term.html#t:304"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Well-formed substitutions. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.well_formed_subst" class="idref" href="#Make.well_formed_subst"><span class="id" title="definition">well_formed_subst</span></a> <a id="sigma:307" class="idref" href="#sigma:307"><span class="id" title="binder">sigma</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="v:308" class="idref" href="#v:308"><span class="id" title="binder">v</span></a>, <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.more_list.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="hydras.rpo.term.html#Make.eq_variable_dec"><span class="id" title="axiom">eq_variable_dec</span></a> <a class="idref" href="hydras.rpo.term.html#v:308"><span class="id" title="variable">v</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:307"><span class="id" title="variable">sigma</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <span class="id" title="var">t</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Make.well_formed_apply_subst" class="idref" href="#Make.well_formed_apply_subst"><span class="id" title="lemma">well_formed_apply_subst</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="sigma:309" class="idref" href="#sigma:309"><span class="id" title="binder">sigma</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.well_formed_subst"><span class="id" title="definition">well_formed_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:309"><span class="id" title="variable">sigma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:310" class="idref" href="#t:310"><span class="id" title="binder">t</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> <a class="idref" href="hydras.rpo.term.html#t:310"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.well_formed"><span class="id" title="definition">well_formed</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:309"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.term.html#t:310"><span class="id" title="variable">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab173"></a><h2 class="section">Positions in a term.</h2>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.is_a_pos" class="idref" href="#Make.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> (<a id="t:311" class="idref" href="#t:311"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) (<a id="p:312" class="idref" href="#p:312"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:312"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:311"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <span class="id" title="var">l</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">ti</span> ⇒ <a class="idref" href="hydras.rpo.term.html#is_a_pos:313"><span class="id" title="definition">is_a_pos</span></a> <span class="id" title="var">ti</span> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.subterm_at_pos" class="idref" href="#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a id="t:316" class="idref" href="#t:316"><span class="id" title="binder">t</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) (<a id="p:317" class="idref" href="#p:317"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:317"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#t:316"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:316"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nth_error"><span class="id" title="definition">nth_error</span></a> <span class="id" title="var">l</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">ti</span> ⇒ <a class="idref" href="hydras.rpo.term.html#subterm_at_pos:318"><span class="id" title="definition">subterm_at_pos</span></a> <span class="id" title="var">ti</span> <span class="id" title="var">q</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.size_subterm_at_pos" class="idref" href="#Make.size_subterm_at_pos"><span class="id" title="lemma">size_subterm_at_pos</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:321" class="idref" href="#t:321"><span class="id" title="binder">t</span></a> <a id="i:322" class="idref" href="#i:322"><span class="id" title="binder">i</span></a> <a id="p:323" class="idref" href="#p:323"><span class="id" title="binder">p</span></a>, <span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:321"><span class="id" title="variable">t</span></a> (<a class="idref" href="hydras.rpo.term.html#i:322"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#p:323"><span class="id" title="variable">p</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">u</span> ⇒ <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <span class="id" title="var">u</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.rpo.term.html#Make.size"><span class="id" title="definition">size</span></a> <a class="idref" href="hydras.rpo.term.html#t:321"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.is_a_pos_exists_subtem" class="idref" href="#Make.is_a_pos_exists_subtem"><span class="id" title="lemma">is_a_pos_exists_subtem</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:324" class="idref" href="#t:324"><span class="id" title="binder">t</span></a> <a id="p:325" class="idref" href="#p:325"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:324"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:325"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="u:326" class="idref" href="#u:326"><span class="id" title="binder">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:324"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:325"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:326"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.replace_at_pos" class="idref" href="#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> (<a id="t:327" class="idref" href="#t:327"><span class="id" title="binder">t</span></a> <a id="u:328" class="idref" href="#u:328"><span class="id" title="binder">u</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) (<a id="p:329" class="idref" href="#p:329"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">p</span>} : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#p:329"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="hydras.rpo.term.html#u:328"><span class="id" title="variable">u</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">q</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#t:327"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Var"><span class="id" title="constructor">Var</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.rpo.term.html#t:327"><span class="id" title="variable">t</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="replace_at_pos_list:338" class="idref" href="#replace_at_pos_list:338"><span class="id" title="binder">replace_at_pos_list</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fix</span> <span class="id" title="var">replace_at_pos_list</span> <a id="j:333" class="idref" href="#j:333"><span class="id" title="binder">j</span></a> (<a id="l:334" class="idref" href="#l:334"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:334"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#j:333"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos:330"><span class="id" title="definition">replace_at_pos</span></a> <span class="id" title="var">h</span> <a class="idref" href="hydras.rpo.term.html#u:328"><span class="id" title="variable">u</span></a> <span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">k</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:335"><span class="id" title="variable">replace_at_pos_list</span></a> <span class="id" title="var">k</span> <span class="id" title="var">tl</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <span class="id" title="var">f</span> (<a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:338"><span class="id" title="variable">replace_at_pos_list</span></a> <span class="id" title="var">i</span> <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Make.replace_at_pos_list" class="idref" href="#Make.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> (<a id="l:339" class="idref" href="#l:339"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) (<a id="u:340" class="idref" href="#u:340"><span class="id" title="binder">u</span></a> : <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a>) (<a id="i:341" class="idref" href="#i:341"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="p:342" class="idref" href="#p:342"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{<span class="id" title="keyword">struct</span> <span class="id" title="var">l</span>}: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="hydras.rpo.term.html#Make.term"><span class="id" title="inductive">term</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#l:339"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#tl"><span class="id" title="definition">tl</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#i:341"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <span class="id" title="var">h</span> <a class="idref" href="hydras.rpo.term.html#u:340"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:342"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">tl</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#replace_at_pos_list:343"><span class="id" title="definition">replace_at_pos_list</span></a> <span class="id" title="var">tl</span>  <a class="idref" href="hydras.rpo.term.html#u:340"><span class="id" title="variable">u</span></a> <span class="id" title="var">j</span> <a class="idref" href="hydras.rpo.term.html#p:342"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.replace_at_pos_unfold" class="idref" href="#Make.replace_at_pos_unfold"><span class="id" title="lemma">replace_at_pos_unfold</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:346" class="idref" href="#f:346"><span class="id" title="binder">f</span></a> <a id="l:347" class="idref" href="#l:347"><span class="id" title="binder">l</span></a> <a id="u:348" class="idref" href="#u:348"><span class="id" title="binder">u</span></a> <a id="i:349" class="idref" href="#i:349"><span class="id" title="binder">i</span></a> <a id="q:350" class="idref" href="#q:350"><span class="id" title="binder">q</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:346"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.rpo.term.html#l:347"><span class="id" title="variable">l</span></a>) <a class="idref" href="hydras.rpo.term.html#u:348"><span class="id" title="variable">u</span></a> (<a class="idref" href="hydras.rpo.term.html#i:349"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#q:350"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#Make.Term"><span class="id" title="constructor">Term</span></a> <a class="idref" href="hydras.rpo.term.html#f:346"><span class="id" title="variable">f</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> <a class="idref" href="hydras.rpo.term.html#l:347"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.rpo.term.html#u:348"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#i:349"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.rpo.term.html#q:350"><span class="id" title="variable">q</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.replace_at_pos_is_replace_at_pos1" class="idref" href="#Make.replace_at_pos_is_replace_at_pos1"><span class="id" title="lemma">replace_at_pos_is_replace_at_pos1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:353" class="idref" href="#t:353"><span class="id" title="binder">t</span></a> <a id="u:354" class="idref" href="#u:354"><span class="id" title="binder">u</span></a> <a id="p:355" class="idref" href="#p:355"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.is_a_pos"><span class="id" title="definition">is_a_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:353"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:355"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:353"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#u:354"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:355"><span class="id" title="variable">p</span></a>) <a class="idref" href="hydras.rpo.term.html#p:355"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:354"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.replace_at_pos_is_replace_at_pos2" class="idref" href="#Make.replace_at_pos_is_replace_at_pos2"><span class="id" title="lemma">replace_at_pos_is_replace_at_pos2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:356" class="idref" href="#t:356"><span class="id" title="binder">t</span></a> <a id="p:357" class="idref" href="#p:357"><span class="id" title="binder">p</span></a> <a id="u:358" class="idref" href="#u:358"><span class="id" title="binder">u</span></a>, <a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:356"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:357"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="hydras.rpo.term.html#u:358"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:356"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#u:358"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:357"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#t:356"><span class="id" title="variable">t</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos" class="idref" href="#Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos"><span class="id" title="lemma">subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="t:361" class="idref" href="#t:361"><span class="id" title="binder">t</span></a> <a id="p:362" class="idref" href="#p:362"><span class="id" title="binder">p</span></a> <a id="sigma:363" class="idref" href="#sigma:363"><span class="id" title="binder">sigma</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#t:361"><span class="id" title="variable">t</span></a> <a class="idref" href="hydras.rpo.term.html#p:362"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">u</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.subterm_at_pos"><span class="id" title="definition">subterm_at_pos</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:363"><span class="id" title="variable">sigma</span></a> <a class="idref" href="hydras.rpo.term.html#t:361"><span class="id" title="variable">t</span></a>) <a class="idref" href="hydras.rpo.term.html#p:362"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="hydras.rpo.term.html#Make.apply_subst"><span class="id" title="definition">apply_subst</span></a> <a class="idref" href="hydras.rpo.term.html#sigma:363"><span class="id" title="variable">sigma</span></a> <span class="id" title="var">u</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.replace_at_pos_list_replace_at_pos_in_subterm" class="idref" href="#Make.replace_at_pos_list_replace_at_pos_in_subterm"><span class="id" title="lemma">replace_at_pos_list_replace_at_pos_in_subterm</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l1:368" class="idref" href="#l1:368"><span class="id" title="binder">l1</span></a> <a id="ui:369" class="idref" href="#ui:369"><span class="id" title="binder">ui</span></a> <a id="l2:370" class="idref" href="#l2:370"><span class="id" title="binder">l2</span></a> <a id="u:371" class="idref" href="#u:371"><span class="id" title="binder">u</span></a> <a id="i:372" class="idref" href="#i:372"><span class="id" title="binder">i</span></a> <a id="p:373" class="idref" href="#p:373"><span class="id" title="binder">p</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="hydras.rpo.term.html#l1:368"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.rpo.term.html#i:372"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos_list"><span class="id" title="definition">replace_at_pos_list</span></a> (<a class="idref" href="hydras.rpo.term.html#l1:368"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="hydras.rpo.term.html#ui:369"><span class="id" title="variable">ui</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#l2:370"><span class="id" title="variable">l2</span></a>) <a class="idref" href="hydras.rpo.term.html#u:371"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#i:372"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.rpo.term.html#p:373"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.rpo.term.html#l1:368"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.rpo.term.html#Make.replace_at_pos"><span class="id" title="definition">replace_at_pos</span></a> <a class="idref" href="hydras.rpo.term.html#ui:369"><span class="id" title="variable">ui</span></a> <a class="idref" href="hydras.rpo.term.html#u:371"><span class="id" title="variable">u</span></a> <a class="idref" href="hydras.rpo.term.html#p:373"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.rpo.term.html#l2:370"><span class="id" title="variable">l2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.rpo.term.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.F_3.html b/theories/html/hydras.solutions_exercises.F_3.html
new file mode 100644
index 00000000..de8c8d5c
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.F_3.html
@@ -0,0 +1,200 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.F_3</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.F_3</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#"><span class="id" title="library">F_alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Max.html#"><span class="id" title="library">Max</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="module">Exp2</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Compat815.html#"><span class="id" title="library">Compat815</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="LF3" class="idref" href="#LF3"><span class="id" title="lemma">LF3</span></a> : <a class="idref" href="hydras.Prelude.Iterates.html#dominates_from"><span class="id" title="definition">dominates_from</span></a> 2 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 3) (<span class="id" title="keyword">fun</span>  <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:1"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab335"></a><h1 class="section">Proof that F_alpha  (S n) &gt; exp2 (F alpha n) for alpha &gt;= 3 and</h1>
+
+    n &gt;= 2  (by induction over alpha) 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="S1" class="idref" href="#S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="S1.P" class="idref" href="#S1.P"><span class="id" title="variable">P</span></a> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a> := <span class="id" title="keyword">∀</span> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:3"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:2"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:3"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="F_3_eqn" class="idref" href="#F_3_eqn"><span class="id" title="lemma">F_3_eqn</span></a> : <span class="id" title="keyword">∀</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 3  <a class="idref" href="hydras.solutions_exercises.F_3.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:4"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.solutions_exercises.F_3.html#n:4"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab336"></a><h2 class="section">Base case</h2>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="P_3" class="idref" href="#P_3"><span class="id" title="lemma">P_3</span></a> : <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.P"><span class="id" title="variable">P</span></a> 3.<br/>
+</div>
+
+<div class="doc">
+deprecated, not replaced yet 
+</div>
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+Successor case 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="S1.Successor" class="idref" href="#S1.Successor"><span class="id" title="section">Successor</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="S1.Successor.alpha" class="idref" href="#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>: <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.Successor.H_alpha" class="idref" href="#S1.Successor.H_alpha"><span class="id" title="variable">H_alpha</span></a> : <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a>: <span class="id" title="keyword">∀</span> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:10" class="idref" href="#p:10"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.solutions_exercises.F_3.html#n:9"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#p:10"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:9"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#p:10"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="S1.Successor.S2" class="idref" href="#S1.Successor.S2"><span class="id" title="section">S2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="S1.Successor.S2.n" class="idref" href="#S1.Successor.S2.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.Successor.S2.Hn" class="idref" href="#S1.Successor.S2.Hn"><span class="id" title="variable">Hn</span></a> : 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3'" class="idref" href="#R3'"><span class="id" title="lemma">R3'</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4" class="idref" href="#R4"><span class="id" title="lemma">R4</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.Iterates.html#iterate"><span class="id" title="definition">iterate</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>)  <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.S2"><span class="id" title="section">S2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L3" class="idref" href="#L3"><span class="id" title="lemma">L3</span></a> : <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.P"><span class="id" title="variable">P</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0_succ"><span class="id" title="instance">E0_succ</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha"><span class="id" title="variable">alpha</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Successor"><span class="id" title="section">Successor</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab337"></a><h2 class="section">Limit case</h2>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="S1.Limit" class="idref" href="#S1.Limit"><span class="id" title="section">Limit</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="S1.Limit.lambda" class="idref" href="#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.Limit.Hlambda" class="idref" href="#S1.Limit.Hlambda"><span class="id" title="variable">Hlambda</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.Limit.IHlambda" class="idref" href="#S1.Limit.IHlambda"><span class="id" title="variable">IHlambda</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="alpha:15" class="idref" href="#alpha:15"><span class="id" title="binder">alpha</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> 3 <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:15"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:15"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R5" class="idref" href="#R5"><span class="id" title="lemma">R5</span></a>: <span class="id" title="keyword">∀</span> <a id="beta:17" class="idref" href="#beta:17"><span class="id" title="binder">beta</span></a>, 3 <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:18" class="idref" href="#n:18"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="p:19" class="idref" href="#p:19"><span class="id" title="binder">p</span></a>, <a class="idref" href="hydras.solutions_exercises.F_3.html#n:18"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#p:19"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:18"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#p:19"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="S1.Limit.S3" class="idref" href="#S1.Limit.S3"><span class="id" title="section">S3</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="S1.Limit.S3.n" class="idref" href="#S1.Limit.S3.n"><span class="id" title="variable">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.Limit.S3.Hn" class="idref" href="#S1.Limit.S3.Hn"><span class="id" title="variable">Hn</span></a> : 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.S3.n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L04" class="idref" href="#L04"><span class="id" title="lemma">L04</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:22" class="idref" href="#beta:22"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:22"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">\</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">F</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:23"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a"><span class="id" title="notation">)</span></a>) (<a class="idref" href="hydras.Epsilon0.Canon.html#canon"><span class="id" title="definition">Canon.canon</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:22"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:23"><span class="id" title="variable">n</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L04'" class="idref" href="#L04'"><span class="id" title="lemma">L04'</span></a> : <span class="id" title="keyword">∀</span> <a id="beta:24" class="idref" href="#beta:24"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#E0limit"><span class="id" title="definition">E0limit</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:24"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:25"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Epsilon0.Canon.html#Canon"><span class="id" title="definition">Canon.Canon</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#beta:24"><span class="id" title="variable">beta</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:25"><span class="id" title="variable">n</span></a>)<a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L4" class="idref" href="#L4"><span class="id" title="lemma">L4</span></a> : <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.S3.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda"><span class="id" title="variable">lambda</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.S3.n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit.S3"><span class="id" title="section">S3</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.Limit"><span class="id" title="section">Limit</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L" class="idref" href="#L"><span class="id" title="lemma">L</span></a> <a id="alpha:26" class="idref" href="#alpha:26"><span class="id" title="binder">alpha</span></a> :  3 <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:26"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:26"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.F_3.html#S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="F_alpha_Sn" class="idref" href="#F_alpha_Sn"><span class="id" title="lemma">F_alpha_Sn</span></a> <a id="alpha:27" class="idref" href="#alpha:27"><span class="id" title="binder">alpha</span></a> : 3 <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="n:28" class="idref" href="#n:28"><span class="id" title="binder">n</span></a>, 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:28"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Prelude.Exp2.html#exp2"><span class="id" title="definition">exp2</span></a> (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:27"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:28"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#alpha:27"><span class="id" title="variable">alpha</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.F_3.html#n:28"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+What happens if alpha is less than 3, or n less than 2 ? 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Goal</span>  <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">F_</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">2</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">3</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">63</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span>  <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">F_</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">2</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">2</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">23</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Goal</span>  <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">F_</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">3</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">1</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm1" class="idref" href="#Unnamed_thm1"><span class="id" title="definition">2047</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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@@ -0,0 +1,207 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.FibonacciPR</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.FibonacciPR</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The famous Fibonacci function 
+</div>
+<div class="code">
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fib" class="idref" href="#fib"><span class="id" title="definition">fib</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;| 0 ⇒ 1<br/>
+&nbsp;&nbsp;| 1 ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> ((<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">q</span>) ⇒ <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib:2"><span class="id" title="definition">fib</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_FibIsPR" class="idref" href="#Proof_of_FibIsPR"><span class="id" title="section">Proof_of_FibIsPR</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+To do :  Some parts of this proof may be made more generic 
+<div class="paragraph"> </div>
+
+ let us consider another definition of fib, as an application of
+      <span class="inlinecode"><span class="id" title="var">nat_rec</span></span>
+   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_FibIsPR.fib_step" class="idref" href="#Proof_of_FibIsPR.fib_step"><span class="id" title="variable">fib_step</span></a> (<a id="p:4" class="idref" href="#p:4"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:4"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:4"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:4"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_FibIsPR.fib_iter" class="idref" href="#Proof_of_FibIsPR.fib_iter"><span class="id" title="variable">fib_iter</span></a> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> <a id="p:6" class="idref" href="#p:6"><span class="id" title="binder">p</span></a>:= (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)%<span class="id" title="keyword">type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:6"><span class="id" title="variable">p</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="p:7" class="idref" href="#p:7"><span class="id" title="binder">p</span></a> ⇒ <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step"><span class="id" title="variable">fib_step</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:7"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:5"><span class="id" title="variable">n</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="fib_alt" class="idref" href="#fib_alt"><span class="id" title="definition">fib_alt</span></a> <a id="n:8" class="idref" href="#n:8"><span class="id" title="binder">n</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter"><span class="id" title="variable">fib_iter</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:8"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The theory of primitive functions deals only with functions
+    of type <span class="inlinecode"><span class="id" title="var">naryFunc</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>.
+
+<div class="paragraph"> </div>
+
+   So, let us define a variant of <span class="inlinecode"><span class="id" title="var">fib_alt</span></span> 
+
+<div class="paragraph"> </div>
+
+   
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations"><span class="id" title="module">LispAbbreviations</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Print</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">car</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_FibIsPR.fib_step_cPair" class="idref" href="#Proof_of_FibIsPR.fib_step_cPair"><span class="id" title="variable">fib_step_cPair</span></a> <a id="p:9" class="idref" href="#p:9"><span class="id" title="binder">p</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:9"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:9"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.car"><span class="id" title="abbreviation">car</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:9"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_FibIsPR.fib_iter_cPair" class="idref" href="#Proof_of_FibIsPR.fib_iter_cPair"><span class="id" title="variable">fib_iter_cPair</span></a> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:11"><span class="id" title="variable">p</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="p:12" class="idref" href="#p:12"><span class="id" title="binder">p</span></a> ⇒ <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step_cPair"><span class="id" title="variable">fib_step_cPair</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:12"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:10"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="fibPR" class="idref" href="#fibPR"><span class="id" title="definition">fibPR</span></a> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a> := <a class="idref" href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr"><span class="id" title="abbreviation">cdr</span></a> (<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter_cPair"><span class="id" title="variable">fib_iter_cPair</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:13"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 1)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Let us prove that <span class="inlinecode"><span class="id" title="var">fibPR</span></span> is PR 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="fibPRIsPR" class="idref" href="#fibPRIsPR"><span class="id" title="instance">fibPRIsPR</span></a>: <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fibPR"><span class="id" title="definition">fibPR</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Ok, but we must prove that <span class="inlinecode"><span class="id" title="var">fibPR</span></span> is extensionaly equal to <span class="inlinecode"><span class="id" title="var">fib</span></span> 
+<div class="paragraph"> </div>
+
+ let us consider the following relation 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="inv" class="idref" href="#inv"><span class="id" title="definition">inv</span></a> (<a id="p:14" class="idref" href="#p:14"><span class="id" title="binder">p</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="c:15" class="idref" href="#c:15"><span class="id" title="binder">c</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=  <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:15"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:14"><span class="id" title="variable">p</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:14"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="inv_Pi" class="idref" href="#inv_Pi"><span class="id" title="lemma">inv_Pi</span></a> : <span class="id" title="keyword">∀</span> <a id="p:16" class="idref" href="#p:16"><span class="id" title="binder">p</span></a> <a id="c:17" class="idref" href="#c:17"><span class="id" title="binder">c</span></a>, <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:16"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:17"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:16"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:17"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L0" class="idref" href="#L0"><span class="id" title="lemma">L0</span></a>: <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPair"><span class="id" title="definition">cPair</span></a> 1 1).<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LS" class="idref" href="#LS"><span class="id" title="lemma">LS</span></a> : <span class="id" title="keyword">∀</span> <a id="p:18" class="idref" href="#p:18"><span class="id" title="binder">p</span></a> <a id="c:19" class="idref" href="#c:19"><span class="id" title="binder">c</span></a>,  <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:18"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:19"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> (<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step"><span class="id" title="variable">fib_step</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:18"><span class="id" title="variable">p</span></a>) (<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step_cPair"><span class="id" title="variable">fib_step_cPair</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:19"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a> : <span class="id" title="keyword">∀</span>  <a id="p:20" class="idref" href="#p:20"><span class="id" title="binder">p</span></a> <a id="c:21" class="idref" href="#c:21"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:20"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:21"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:22" class="idref" href="#n:22"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#inv"><span class="id" title="definition">inv</span></a> (<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter"><span class="id" title="variable">fib_iter</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#p:20"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter_cPair"><span class="id" title="variable">fib_iter_cPair</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:22"><span class="id" title="variable">n</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#c:21"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib_alt"><span class="id" title="definition">fib_alt</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fibPR"><span class="id" title="definition">fibPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="fib_altIsPR" class="idref" href="#fib_altIsPR"><span class="id" title="instance">fib_altIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib_alt"><span class="id" title="definition">fib_alt</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+It remains to prove that fib_alt is equivalent to the "classical" fib 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="fib_OK0" class="idref" href="#fib_OK0"><span class="id" title="lemma">fib_OK0</span></a> : <span class="id" title="keyword">∀</span> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter"><span class="id" title="variable">fib_iter</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:23"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib"><span class="id" title="definition">fib</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:23"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#n:23"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="fib_alt_Ok" class="idref" href="#fib_alt_Ok"><span class="id" title="lemma">fib_alt_Ok</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib"><span class="id" title="definition">fib</span></a> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib_alt"><span class="id" title="definition">fib_alt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="fibIsPR" class="idref" href="#fibIsPR"><span class="id" title="instance">fibIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#fib"><span class="id" title="definition">fib</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR"><span class="id" title="section">Proof_of_FibIsPR</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Too long !
+
+<div class="paragraph"> </div>
+
+Time Compute fibPR 3.
+ 
+</div>
+<div class="code">
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+++ b/theories/html/hydras.solutions_exercises.Limit_Infinity.html
@@ -0,0 +1,107 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.Limit_Infinity</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.Limit_Infinity</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Prove that, for any ordinal <span class="inlinecode">0</span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">epsilon0</span></span>, <span class="inlinecode"><span class="id" title="var">alpha</span></span> is a limit 
+  if and only if forall ordinal <span class="inlinecode"><span class="id" title="keyword">beta</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">alpha</span></span>, there exists an infinite 
+  number of ordinals betawee <span class="inlinecode"><span class="id" title="keyword">beta</span></span> and <span class="inlinecode"><span class="id" title="var">alpha</span></span>
+ 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.Epsilon0.html#"><span class="id" title="library">Epsilon0</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">t1_scope</span>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#"><span class="id" title="library">Ensembles</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#"><span class="id" title="library">Image</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="Infinite" class="idref" href="#Infinite"><span class="id" title="definition">Infinite</span></a> {<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>: <span class="id" title="keyword">Type</span>} (<a id="E:2" class="idref" href="#E:2"><span class="id" title="binder">E</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#Ensemble"><span class="id" title="definition">Ensemble</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#A:1"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s:3" class="idref" href="#s:3"><span class="id" title="binder">s</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#A:1"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s:3"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <a id="i:4" class="idref" href="#i:4"><span class="id" title="binder">i</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Ensembles.html#In"><span class="id" title="definition">In</span></a>  <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#E:2"><span class="id" title="variable">E</span></a> (<a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s:3"><span class="id" title="variable">s</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#i:4"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="On_alpha" class="idref" href="#On_alpha"><span class="id" title="section">On_alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="On_alpha.alpha" class="idref" href="#On_alpha.alpha"><span class="id" title="variable">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_alpha.Halpha" class="idref" href="#On_alpha.Halpha"><span class="id" title="variable">Halpha</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#nf"><span class="id" title="definition">nf</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_alpha.HnonZero" class="idref" href="#On_alpha.HnonZero"><span class="id" title="variable">HnonZero</span></a> : <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#zero"><span class="id" title="constructor">zero</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="On_alpha.S1" class="idref" href="#On_alpha.S1"><span class="id" title="section">S1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_alpha.S1.H" class="idref" href="#On_alpha.S1.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a>  <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="On_alpha.S1.beta" class="idref" href="#On_alpha.S1.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_alpha.S1.Hbeta" class="idref" href="#On_alpha.S1.Hbeta"><span class="id" title="variable">Hbeta</span></a> : <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="s" class="idref" href="#s"><span class="id" title="definition">s</span></a> (<a id="i:11" class="idref" href="#i:11"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#i:11"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a> (<a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s"><span class="id" title="definition">s</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#i:12"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Shows that <span class="inlinecode"><span class="id" title="var">s</span></span> is infinite 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <span class="id" title="keyword">∀</span> <a id="i:13" class="idref" href="#i:13"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s"><span class="id" title="definition">s</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#i:13"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s"><span class="id" title="definition">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#i:13"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L3" class="idref" href="#L3"><span class="id" title="lemma">L3</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sets.Image.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#s"><span class="id" title="definition">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="On_alpha.S2" class="idref" href="#On_alpha.S2"><span class="id" title="section">S2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="On_alpha.S2.H" class="idref" href="#On_alpha.S2.H"><span class="id" title="variable">H</span></a> :  <span class="id" title="keyword">∀</span> <a id="beta:18" class="idref" href="#beta:18"><span class="id" title="binder">beta</span></a>, <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#beta:18"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="gamma:19" class="idref" href="#gamma:19"><span class="id" title="binder">gamma</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#beta:18"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#gamma:19"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L4" class="idref" href="#L4"><span class="id" title="lemma">L4</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S2"><span class="id" title="section">S2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="Limit_Infinity" class="idref" href="#Limit_Infinity"><span class="id" title="lemma">Limit_Infinity</span></a> : <a class="idref" href="hydras.Epsilon0.T1.html#T1limit"><span class="id" title="definition">T1limit</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="beta:21" class="idref" href="#beta:21"><span class="id" title="binder">beta</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#beta:21"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#Infinite"><span class="id" title="definition">Infinite</span></a> (<span class="id" title="keyword">fun</span> <a id="gamma:22" class="idref" href="#gamma:22"><span class="id" title="binder">gamma</span></a> ⇒  <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#beta:21"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#gamma:22"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">t1</span></a><a class="idref" href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha"><span class="id" title="variable">alpha</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha"><span class="id" title="section">On_alpha</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.MinPR.html b/theories/html/hydras.solutions_exercises.MinPR.html
new file mode 100644
index 00000000..20ec2f3f
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.MinPR.html
@@ -0,0 +1,65 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.MinPR</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.MinPR</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Prove that Stdlib's function <span class="inlinecode"><span class="id" title="var">min</span></span> is primitive recursive 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">PeanoNat.Nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_MinIsPR" class="idref" href="#Proof_of_MinIsPR"><span class="id" title="section">Proof_of_MinIsPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_MinIsPR.min_alt" class="idref" href="#Proof_of_MinIsPR.min_alt"><span class="id" title="variable">min_alt</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a> <a id="b:2" class="idref" href="#b:2"><span class="id" title="binder">b</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#leBool"><span class="id" title="definition">leBool</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#b:2"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#a:1"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.Ackermann.extEqualNat.html#charFunction"><span class="id" title="definition">charFunction</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#ltBool"><span class="id" title="definition">ltBool</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#b:2"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#a:1"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR.html#b:2"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="min_alt_correct" class="idref" href="#min_alt_correct"><span class="id" title="lemma">min_alt_correct</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR.min_alt"><span class="id" title="variable">min_alt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#min"><span class="id" title="definition">Nat.min</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="minPR_PR" class="idref" href="#minPR_PR"><span class="id" title="instance">minPR_PR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR.min_alt"><span class="id" title="variable">min_alt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="minIsPR" class="idref" href="#minIsPR"><span class="id" title="instance">minIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Nat.html#min"><span class="id" title="definition">Nat.min</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR"><span class="id" title="section">Proof_of_MinIsPR</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.MinPR2.html b/theories/html/hydras.solutions_exercises.MinPR2.html
new file mode 100644
index 00000000..da281986
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.MinPR2.html
@@ -0,0 +1,88 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.MinPR2</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.MinPR2</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+Prove that Stdlib's function Min is primitive recursive  this is a variant of the exercise MinPR 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Define an n-ary if-then-else 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="naryIf" class="idref" href="#naryIf"><span class="id" title="definition">naryIf</span></a> (<a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:1"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;:=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:1"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">return</span> (<a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:3"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#n:3"><span class="id" title="variable">n</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ (<span class="id" title="keyword">fun</span> <a id="b:4" class="idref" href="#b:4"><span class="id" title="binder">b</span></a> <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> ⇒  <span class="id" title="keyword">if</span> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#b:4"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#x:5"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#y:6"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒ <span class="id" title="keyword">fun</span> (<a id="p':7" class="idref" href="#p':7"><span class="id" title="binder">p'</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) (<a id="g:8" class="idref" href="#g:8"><span class="id" title="binder">g</span></a> <a id="h:9" class="idref" href="#h:9"><span class="id" title="binder">h</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span>)) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="hydras.solutions_exercises.MinPR2.html#naryIf:2"><span class="id" title="definition">naryIf</span></a> <span class="id" title="var">m</span> (<a class="idref" href="hydras.solutions_exercises.MinPR2.html#p':7"><span class="id" title="variable">p'</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#x:10"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.solutions_exercises.MinPR2.html#g:8"><span class="id" title="variable">g</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#x:10"><span class="id" title="variable">x</span></a>) (<a class="idref" href="hydras.solutions_exercises.MinPR2.html#h:9"><span class="id" title="variable">h</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#x:10"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="If2IsPR" class="idref" href="#If2IsPR"><span class="id" title="instance">If2IsPR</span></a> (<a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a>: <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2)(<a id="f:12" class="idref" href="#f:12"><span class="id" title="binder">f</span></a> <a id="g:13" class="idref" href="#g:13"><span class="id" title="binder">g</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPRrel"><span class="id" title="class">isPRrel</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#f:12"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#g:13"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 (<a class="idref" href="hydras.solutions_exercises.MinPR2.html#naryIf"><span class="id" title="definition">naryIf</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#f:12"><span class="id" title="variable">f</span></a> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#g:13"><span class="id" title="variable">g</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_MinIsPR" class="idref" href="#Proof_of_MinIsPR"><span class="id" title="section">Proof_of_MinIsPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_MinIsPR.minPR" class="idref" href="#Proof_of_MinIsPR.minPR"><span class="id" title="variable">minPR</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryFunc"><span class="id" title="definition">naryFunc</span></a> 2 :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.MinPR2.html#naryIf"><span class="id" title="definition">naryIf</span></a> 2 <a class="idref" href="hydras.Ackermann.primRec.html#leBool"><span class="id" title="definition">leBool</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="x:18" class="idref" href="#x:18"><span class="id" title="binder">x</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="hydras.solutions_exercises.MinPR2.html#x:18"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="y:19" class="idref" href="#y:19"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="hydras.solutions_exercises.MinPR2.html#y:19"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="minPR_correct" class="idref" href="#minPR_correct"><span class="id" title="lemma">minPR_correct</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR.minPR"><span class="id" title="variable">minPR</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.min"><span class="id" title="definition">PeanoNat.Nat.min</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="minPR_PR" class="idref" href="#minPR_PR"><span class="id" title="instance">minPR_PR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR.minPR"><span class="id" title="variable">minPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="minIsPR" class="idref" href="#minIsPR"><span class="id" title="instance">minIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#min"><span class="id" title="abbreviation">min</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR"><span class="id" title="section">Proof_of_MinIsPR</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.MorePRExamples.html b/theories/html/hydras.solutions_exercises.MorePRExamples.html
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@@ -0,0 +1,123 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.MorePRExamples</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.MorePRExamples</h1>
+
+<div class="code">
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="module">extEqualNat</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab334"></a><h2 class="section">Solution to an exercise</h2>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="double" class="idref" href="#double"><span class="id" title="definition">double</span></a> <a id="n:1" class="idref" href="#n:1"><span class="id" title="binder">n</span></a> := <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:1"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> 2.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="doubleIsPR" class="idref" href="#doubleIsPR"><span class="id" title="lemma">doubleIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#double"><span class="id" title="definition">double</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="exp" class="idref" href="#exp"><span class="id" title="definition">exp</span></a> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> <a id="p:3" class="idref" href="#p:3"><span class="id" title="binder">p</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#p:3"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">m</span> ⇒  <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp:4"><span class="id" title="definition">exp</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:2"><span class="id" title="variable">n</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:2"><span class="id" title="variable">n</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="exp_alt" class="idref" href="#exp_alt"><span class="id" title="definition">exp_alt</span></a> := <span class="id" title="keyword">fun</span> <a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a> <a id="b:7" class="idref" href="#b:7"><span class="id" title="binder">b</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <a id="y:8" class="idref" href="#y:8"><span class="id" title="binder">y</span></a> ⇒  <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#y:8"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#a:6"><span class="id" title="variable">a</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#b:7"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span>  <a id="exp_alt_ok" class="idref" href="#exp_alt_ok"><span class="id" title="lemma">exp_alt_ok</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp_alt"><span class="id" title="definition">exp_alt</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp"><span class="id" title="definition">exp</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="exp_alt_PR" class="idref" href="#exp_alt_PR"><span class="id" title="instance">exp_alt_PR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp_alt"><span class="id" title="definition">exp_alt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="expIsPR" class="idref" href="#expIsPR"><span class="id" title="instance">expIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp"><span class="id" title="definition">exp</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="tower2" class="idref" href="#tower2"><span class="id" title="definition">tower2</span></a> <a id="n:9" class="idref" href="#n:9"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:9"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ 1<br/>
+&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span> ⇒ <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp"><span class="id" title="definition">exp</span></a> 2 (<a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#tower2:10"><span class="id" title="definition">tower2</span></a> <span class="id" title="var">p</span>)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="tower2_alt" class="idref" href="#tower2_alt"><span class="id" title="definition">tower2_alt</span></a> <a id="h:12" class="idref" href="#h:12"><span class="id" title="binder">h</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> (<span class="id" title="keyword">fun</span> <a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span>  <a id="y:13" class="idref" href="#y:13"><span class="id" title="binder">y</span></a> ⇒  <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#exp"><span class="id" title="definition">exp</span></a> 2 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#y:13"><span class="id" title="variable">y</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#h:12"><span class="id" title="variable">h</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="tower2_alt_ok" class="idref" href="#tower2_alt_ok"><span class="id" title="lemma">tower2_alt_ok</span></a>  : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt"><span class="id" title="definition">tower2_alt</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#tower2"><span class="id" title="definition">tower2</span></a>.<br/>
+ 
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="tower2_alt_PR" class="idref" href="#tower2_alt_PR"><span class="id" title="instance">tower2_alt_PR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt"><span class="id" title="definition">tower2_alt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="tower2IsPR" class="idref" href="#tower2IsPR"><span class="id" title="instance">tower2IsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#tower2"><span class="id" title="definition">tower2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a id="fact" class="idref" href="#fact"><span class="id" title="definition">fact</span></a> <a id="n:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:15"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span> 0 ⇒ 1<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;|   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">p</span>  ⇒ <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#n:15"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#fact:16"><span class="id" title="definition">fact</span></a> <span class="id" title="var">p</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="fact_alt" class="idref" href="#fact_alt"><span class="id" title="definition">fact_alt</span></a><br/>
+&nbsp;&nbsp;: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <a id="a:18" class="idref" href="#a:18"><span class="id" title="binder">a</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat_rec"><span class="id" title="definition">nat_rec</span></a> <span class="id" title="var">_</span> 1 (<span class="id" title="keyword">fun</span> <a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> <a id="y:20" class="idref" href="#y:20"><span class="id" title="binder">y</span></a> ⇒  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#y:20"><span class="id" title="variable">y</span></a>) <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#a:18"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Remark</span> <a id="fact_alt_ok" class="idref" href="#fact_alt_ok"><span class="id" title="lemma">fact_alt_ok</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#extEqual"><span class="id" title="definition">extEqual</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#fact_alt"><span class="id" title="definition">fact_alt</span></a> <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#fact"><span class="id" title="definition">fact</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="fact_altIsPR" class="idref" href="#fact_altIsPR"><span class="id" title="instance">fact_altIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#fact_alt"><span class="id" title="definition">fact_alt</span></a>.<br/>
+
+<br/>
+#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="factIsPR" class="idref" href="#factIsPR"><span class="id" title="instance">factIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.MorePRExamples.html#fact"><span class="id" title="definition">fact</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.MultisetWf</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.MultisetWf</h1>
+
+<div class="code">
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#"><span class="id" title="library">Sorted</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Restriction.html#"><span class="id" title="library">Restriction</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="t" class="idref" href="#t"><span class="id" title="definition">t</span></a> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:1"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="A_given" class="idref" href="#A_given"><span class="id" title="section">A_given</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="A_given.A" class="idref" href="#A_given.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="A_given.LtA" class="idref" href="#A_given.LtA"><span class="id" title="variable">LtA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A_given.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="lexpower" class="idref" href="#lexpower"><span class="id" title="inductive">lexpower</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A_given.A"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="lex1" class="idref" href="#lex1"><span class="id" title="constructor">lex1</span></a>: <span class="id" title="keyword">∀</span> <a id="a:6" class="idref" href="#a:6"><span class="id" title="binder">a</span></a> <a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> <a id="l:8" class="idref" href="#l:8"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower:4"><span class="id" title="inductive">lexpower</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:6"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:7"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:8"><span class="id" title="variable">l</span></a>)<br/>
+&nbsp;&nbsp;| <a id="lex2" class="idref" href="#lex2"><span class="id" title="constructor">lex2</span></a>: <span class="id" title="keyword">∀</span> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a> <a id="p:11" class="idref" href="#p:11"><span class="id" title="binder">p</span></a> <a id="l:12" class="idref" href="#l:12"><span class="id" title="binder">l</span></a> <a id="l':13" class="idref" href="#l':13"><span class="id" title="binder">l'</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:10"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:11"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower:4"><span class="id" title="inductive">lexpower</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:9"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:10"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:12"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:9"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:11"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':13"><span class="id" title="variable">l'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="lex3" class="idref" href="#lex3"><span class="id" title="constructor">lex3</span></a>: <span class="id" title="keyword">∀</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a> <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a> <a id="p:17" class="idref" href="#p:17"><span class="id" title="binder">p</span></a> <a id="l:18" class="idref" href="#l:18"><span class="id" title="binder">l</span></a> <a id="l':19" class="idref" href="#l':19"><span class="id" title="binder">l'</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A_given.LtA"><span class="id" title="variable">LtA</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower:4"><span class="id" title="inductive">lexpower</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:14"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:16"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:18"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:15"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:17"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':19"><span class="id" title="variable">l'</span></a>)<br/>
+&nbsp;&nbsp;| <a id="lex4" class="idref" href="#lex4"><span class="id" title="constructor">lex4</span></a>: <span class="id" title="keyword">∀</span> <a id="a:20" class="idref" href="#a:20"><span class="id" title="binder">a</span></a> <a id="n:21" class="idref" href="#n:21"><span class="id" title="binder">n</span></a> <a id="l:22" class="idref" href="#l:22"><span class="id" title="binder">l</span></a> <a id="l':23" class="idref" href="#l':23"><span class="id" title="binder">l'</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower:4"><span class="id" title="inductive">lexpower</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:22"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':23"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower:4"><span class="id" title="inductive">lexpower</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:20"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:21"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:22"><span class="id" title="variable">l</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:20"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:21"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':23"><span class="id" title="variable">l'</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A_given"><span class="id" title="section">A_given</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Counter_Example" class="idref" href="#Counter_Example"><span class="id" title="section">Counter_Example</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Counter_Example.R" class="idref" href="#Counter_Example.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower"><span class="id" title="inductive">lexpower</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Counter_Example.Hwf" class="idref" href="#Counter_Example.Hwf"><span class="id" title="variable">Hwf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="seq" class="idref" href="#seq"><span class="id" title="definition">seq</span></a> (<a id="n:25" class="idref" href="#n:25"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#repeat"><span class="id" title="definition">repeat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:25"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="decr_seq" class="idref" href="#decr_seq"><span class="id" title="lemma">decr_seq</span></a> : <span class="id" title="keyword">∀</span> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#seq"><span class="id" title="definition">seq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:26"><span class="id" title="variable">n</span></a>)) (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#seq"><span class="id" title="definition">seq</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:26"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="not_acc" class="idref" href="#not_acc"><span class="id" title="lemma">not_acc</span></a> : <span class="id" title="keyword">∀</span> <a id="a:27" class="idref" href="#a:27"><span class="id" title="binder">a</span></a> <a id="b:28" class="idref" href="#b:28"><span class="id" title="binder">b</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:28"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:27"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:28"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Counter_Example.is_in_seq" class="idref" href="#Counter_Example.is_in_seq"><span class="id" title="variable">is_in_seq</span></a> <a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:30" class="idref" href="#i:30"><span class="id" title="binder">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:29"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#seq"><span class="id" title="definition">seq</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#i:30"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="is_in_seq_not_Acc" class="idref" href="#is_in_seq_not_Acc"><span class="id" title="lemma">is_in_seq_not_Acc</span></a> : <span class="id" title="keyword">∀</span> <a id="x:31" class="idref" href="#x:31"><span class="id" title="binder">x</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.is_in_seq"><span class="id" title="variable">is_in_seq</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#x:31"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="contrad" class="idref" href="#contrad"><span class="id" title="lemma">contrad</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Counter_Example"><span class="id" title="section">Counter_Example</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Lists in normal form 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lexnf" class="idref" href="#lexnf"><span class="id" title="definition">lexnf</span></a> {<a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a>: <span class="id" title="keyword">Type</span>}(<a id="ltA:33" class="idref" href="#ltA:33"><span class="id" title="binder">ltA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:32"><span class="id" title="variable">A</span></a>) (<a id="l:34" class="idref" href="#l:34"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:32"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;:= <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Sorting.Sorted.html#LocallySorted"><span class="id" title="inductive">LocallySorted</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">Basics.flip</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:33"><span class="id" title="variable">ltA</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:34"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="lexlt" class="idref" href="#lexlt"><span class="id" title="definition">lexlt</span></a> {<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a>}(<a id="ltA:36" class="idref" href="#ltA:36"><span class="id" title="binder">ltA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:35"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.Restriction.html#restrict"><span class="id" title="definition">restrict</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexnf"><span class="id" title="definition">lexnf</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:36"><span class="id" title="variable">ltA</span></a>) (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower"><span class="id" title="inductive">lexpower</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:36"><span class="id" title="variable">ltA</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="ProofOfLexwf" class="idref" href="#ProofOfLexwf"><span class="id" title="section">ProofOfLexwf</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> (<a id="ProofOfLexwf.A" class="idref" href="#ProofOfLexwf.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="ProofOfLexwf.ltA" class="idref" href="#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:37"><span class="id" title="variable">A</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="ProofOfLexwf.HwfA" class="idref" href="#ProofOfLexwf.HwfA"><span class="id" title="variable">HwfA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="NF" class="idref" href="#NF"><span class="id" title="abbreviation">NF</span></a> := (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexnf"><span class="id" title="definition">lexnf</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a>).<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="LT" class="idref" href="#LT"><span class="id" title="abbreviation">LT</span></a> := (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexlt"><span class="id" title="definition">lexlt</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="lexwf" class="idref" href="#lexwf"><span class="id" title="lemma">lexwf</span></a>:<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:40" class="idref" href="#l:40"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:40"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:40"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="NF_inv1" class="idref" href="#NF_inv1"><span class="id" title="lemma">NF_inv1</span></a> : <span class="id" title="keyword">∀</span> <a id="a:41" class="idref" href="#a:41"><span class="id" title="binder">a</span></a> <a id="n:42" class="idref" href="#n:42"><span class="id" title="binder">n</span></a> <a id="l:43" class="idref" href="#l:43"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:41"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:42"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:43"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:43"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="NF_inv2" class="idref" href="#NF_inv2"><span class="id" title="lemma">NF_inv2</span></a> : <span class="id" title="keyword">∀</span> <a id="a:44" class="idref" href="#a:44"><span class="id" title="binder">a</span></a> <a id="n:45" class="idref" href="#n:45"><span class="id" title="binder">n</span></a> <a id="b:46" class="idref" href="#b:46"><span class="id" title="binder">b</span></a> <a id="p:47" class="idref" href="#p:47"><span class="id" title="binder">p</span></a> <a id="l:48" class="idref" href="#l:48"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a>(<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:44"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:45"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:46"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:47"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:48"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:46"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:44"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="LT_inv" class="idref" href="#LT_inv"><span class="id" title="lemma">LT_inv</span></a> : <span class="id" title="keyword">∀</span> <a id="a:49" class="idref" href="#a:49"><span class="id" title="binder">a</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a> <a id="l:51" class="idref" href="#l:51"><span class="id" title="binder">l</span></a> <a id="l':52" class="idref" href="#l':52"><span class="id" title="binder">l'</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':52"><span class="id" title="variable">l'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:49"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:50"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:51"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':52"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="b:53" class="idref" href="#b:53"><span class="id" title="binder">b</span></a> <a id="p:54" class="idref" href="#p:54"><span class="id" title="binder">p</span></a> <a id="l'':55" class="idref" href="#l'':55"><span class="id" title="binder">l''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':52"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:53"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:54"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l'':55"><span class="id" title="variable">l''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#b:53"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:49"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="l'':56" class="idref" href="#l'':56"><span class="id" title="binder">l''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a>  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':52"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:49"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:50"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l'':56"><span class="id" title="variable">l''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l'':56"><span class="id" title="variable">l''</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:51"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a>  <a id="p:57" class="idref" href="#p:57"><span class="id" title="binder">p</span></a> <a id="l'':58" class="idref" href="#l'':58"><span class="id" title="binder">l''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':52"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:49"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:57"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l'':58"><span class="id" title="variable">l''</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#p:57"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:50"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="ProofOfLexwf.Accs" class="idref" href="#ProofOfLexwf.Accs"><span class="id" title="variable">Accs</span></a> (<a id="a:59" class="idref" href="#a:59"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.A"><span class="id" title="variable">A</span></a>) := <span class="id" title="keyword">∀</span> <a id="n:60" class="idref" href="#n:60"><span class="id" title="binder">n</span></a> <a id="l:61" class="idref" href="#l:61"><span class="id" title="binder">l</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:59"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:60"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:61"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:59"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:60"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:61"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Acc_nil" class="idref" href="#Acc_nil"><span class="id" title="lemma">Acc_nil</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+ 
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Accs_all" class="idref" href="#Accs_all"><span class="id" title="lemma">Accs_all</span></a>: <span class="id" title="keyword">∀</span> <a id="a:62" class="idref" href="#a:62"><span class="id" title="binder">a</span></a>:<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.Accs"><span class="id" title="variable">Accs</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#a:62"><span class="id" title="variable">a</span></a>.  </div>
+
+<div class="doc">
+let us prepare an induction on l 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="NF_Acc" class="idref" href="#NF_Acc"><span class="id" title="lemma">NF_Acc</span></a> : <span class="id" title="keyword">∀</span> <a id="l:67" class="idref" href="#l:67"><span class="id" title="binder">l</span></a>: <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#NF"><span class="id" title="abbreviation">NF</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:67"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#LT"><span class="id" title="abbreviation">LT</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:67"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf"><span class="id" title="section">ProofOfLexwf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a id="lexwf" class="idref" href="#lexwf"><span class="id" title="lemma">lexwf</span></a> {<a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a>}( <a id="ltA:69" class="idref" href="#ltA:69"><span class="id" title="binder">ltA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#A:68"><span class="id" title="variable">A</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:69"><span class="id" title="variable">ltA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:70" class="idref" href="#l:70"><span class="id" title="binder">l</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexnf"><span class="id" title="definition">lexnf</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:69"><span class="id" title="variable">ltA</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:70"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexlt"><span class="id" title="definition">lexlt</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#ltA:69"><span class="id" title="variable">ltA</span></a>) <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:70"><span class="id" title="variable">l</span></a>.  
+<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex1" class="idref" href="#Ex1"><span class="id" title="definition">Ex1</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower"><span class="id" title="inductive">lexpower</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>7<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>3<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex2" class="idref" href="#Ex2"><span class="id" title="definition">Ex2</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexpower"><span class="id" title="inductive">lexpower</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a>  (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>7<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>33<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>7<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>6<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Example</span> <a id="Ex3" class="idref" href="#Ex3"><span class="id" title="definition">Ex3</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexnf"><span class="id" title="definition">lexnf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>7<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>33<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Impossibility1" class="idref" href="#Impossibility1"><span class="id" title="section">Impossibility1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Impossibility1.m" class="idref" href="#Impossibility1.m"><span class="id" title="variable">m</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Impossibility1.mDecr" class="idref" href="#Impossibility1.mDecr"><span class="id" title="variable">mDecr</span></a> : <span class="id" title="keyword">∀</span> <a id="l:72" class="idref" href="#l:72"><span class="id" title="binder">l</span></a> <a id="l':73" class="idref" href="#l':73"><span class="id" title="binder">l'</span></a>: <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#lexlt"><span class="id" title="definition">lexlt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:72"><span class="id" title="variable">l</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':73"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l:72"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a>  <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m"><span class="id" title="variable">m</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#l':73"><span class="id" title="variable">l'</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="iota" class="idref" href="#iota"><span class="id" title="definition">iota</span></a> (<a id="n:75" class="idref" href="#n:75"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#n:75"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility1.x" class="idref" href="#Impossibility1.x"><span class="id" title="variable">x</span></a> := <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Impossibility1.y" class="idref" href="#Impossibility1.y"><span class="id" title="variable">y</span></a> := <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m"><span class="id" title="variable">m</span></a> (<a class="idref" href="hydras.solutions_exercises.MultisetWf.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.x"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F1" class="idref" href="#F1"><span class="id" title="lemma">F1</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fact</span> <a id="F2" class="idref" href="#F2"><span class="id" title="lemma">F2</span></a> : <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.y"><span class="id" title="variable">y</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="impossible_nat" class="idref" href="#impossible_nat"><span class="id" title="lemma">impossible_nat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.MultisetWf.html#Impossibility1"><span class="id" title="section">Impossibility1</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.OnCodeList.html b/theories/html/hydras.solutions_exercises.OnCodeList.html
new file mode 100644
index 00000000..bca6224b
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.OnCodeList.html
@@ -0,0 +1,50 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
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+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras.Ackermann</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#"><span class="id" title="library">cPair</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="keyword">Equations</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <span class="id" title="library">Equations</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Equations</span> <a id="members" class="idref" href="#members"><span class="id" title="definition">members</span></a> (<a id="a:1" class="idref" href="#a:1"><span class="id" title="binder">a</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>): <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <span class="id" title="tactic">by</span> <span class="id" title="keyword">wf</span> <span class="id" title="var">a</span>:=<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.OnCodeList.html#members:3"><span class="id" title="definition">members</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#O"><span class="id" title="constructor">0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#nil"><span class="id" title="abbreviation">List.nil</span></a>;<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.OnCodeList.html#members:5"><span class="id" title="definition">members</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#z:4"><span class="id" title="variable">z</span></a>) := <a class="idref" href="hydras.Ackermann.cPair.html#cPairPi1"><span class="id" title="definition">cPairPi1</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#members:2"><span class="id" title="definition">members</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#cPairPi2"><span class="id" title="definition">cPairPi2</span></a> <span class="id" title="var">z</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="membersOk" class="idref" href="#membersOk"><span class="id" title="lemma">membersOk</span></a> <a id="n:6" class="idref" href="#n:6"><span class="id" title="binder">n</span></a> : <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#n:6"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> (<a class="idref" href="hydras.solutions_exercises.OnCodeList.html#members"><span class="id" title="definition">members</span></a> <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#n:6"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="membersOk'" class="idref" href="#membersOk'"><span class="id" title="lemma">membersOk'</span></a> <a id="l:7" class="idref" href="#l:7"><span class="id" title="binder">l</span></a> : <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#l:7"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#members"><span class="id" title="definition">members</span></a> (<a class="idref" href="hydras.Ackermann.cPair.html#codeList"><span class="id" title="definition">codeList</span></a> <a class="idref" href="hydras.solutions_exercises.OnCodeList.html#l:7"><span class="id" title="variable">l</span></a>).<br/>
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diff --git a/theories/html/hydras.solutions_exercises.T1_ltNotWf.html b/theories/html/hydras.solutions_exercises.T1_ltNotWf.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.T1_ltNotWf</title>
+</head>
+
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+
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+
+<h1 class="libtitle">Library hydras.solutions_exercises.T1_ltNotWf</h1>
+
+<div class="code">
+</div>
+
+<div class="doc">
+The order <span class="inlinecode"><span class="id" title="var">lt</span></span> on <span class="inlinecode"><span class="id" title="var">T</span></span> is not well-founded 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proof_of_lt_not_wf" class="idref" href="#Proof_of_lt_not_wf"><span class="id" title="section">Proof_of_lt_not_wf</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_lt_not_wf.lt_wf" class="idref" href="#Proof_of_lt_not_wf.lt_wf"><span class="id" title="variable">lt_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span>  <a id="s" class="idref" href="#s"><span class="id" title="definition">s</span></a> (<a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.Epsilon0.T1.html#T1"><span class="id" title="inductive">T1</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:2"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#phi0"><span class="id" title="abbreviation">phi0</span></a> 2<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <span class="id" title="var">j</span> ⇒ <a class="idref" href="hydras.Epsilon0.T1.html#cons"><span class="id" title="constructor">cons</span></a> 1 0 (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#s:3"><span class="id" title="definition">s</span></a> <span class="id" title="var">j</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="s_decr" class="idref" href="#s_decr"><span class="id" title="lemma">s_decr</span></a> : <span class="id" title="keyword">∀</span> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a>,  <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#s"><span class="id" title="definition">s</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:5"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#s"><span class="id" title="definition">s</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:5"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proof_of_lt_not_wf.seq_intro" class="idref" href="#Proof_of_lt_not_wf.seq_intro"><span class="id" title="section">seq_intro</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_lt_not_wf.seq_intro.A" class="idref" href="#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_lt_not_wf.seq_intro.seq" class="idref" href="#Proof_of_lt_not_wf.seq_intro.seq"><span class="id" title="variable">seq</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proof_of_lt_not_wf.seq_intro.R" class="idref" href="#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proof_of_lt_not_wf.seq_intro.Rwf" class="idref" href="#Proof_of_lt_not_wf.seq_intro.Rwf"><span class="id" title="variable">Rwf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Proof_of_lt_not_wf.seq_intro.is_in_seq" class="idref" href="#Proof_of_lt_not_wf.seq_intro.is_in_seq"><span class="id" title="variable">is_in_seq</span></a> (<a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a>:<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a>) :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="i:11" class="idref" href="#i:11"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.seq"><span class="id" title="variable">seq</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:11"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="not_acc" class="idref" href="#not_acc"><span class="id" title="lemma">not_acc</span></a> : <span class="id" title="keyword">∀</span> <a id="a:12" class="idref" href="#a:12"><span class="id" title="binder">a</span></a> <a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#a:12"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#b:13"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="acc_imp" class="idref" href="#acc_imp"><span class="id" title="lemma">acc_imp</span></a> : <span class="id" title="keyword">∀</span> <a id="a:14" class="idref" href="#a:14"><span class="id" title="binder">a</span></a> <a id="b:15" class="idref" href="#b:15"><span class="id" title="binder">b</span></a>:<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A"><span class="id" title="variable">A</span></a>, <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#a:14"><span class="id" title="variable">a</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#b:15"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#a:14"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="not_decreasing_aux" class="idref" href="#not_decreasing_aux"><span class="id" title="lemma">not_decreasing_aux</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:16" class="idref" href="#n:16"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.seq"><span class="id" title="variable">seq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#n:16"><span class="id" title="variable">n</span></a>)) (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.seq"><span class="id" title="variable">seq</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#n:16"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro"><span class="id" title="section">seq_intro</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="not_decreasing" class="idref" href="#not_decreasing"><span class="id" title="lemma">not_decreasing</span></a> (<a id="A:19" class="idref" href="#A:19"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)(<a id="R:20" class="idref" href="#R:20"><span class="id" title="binder">R</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#A:19"><span class="id" title="variable">A</span></a>) (<a id="Hwf:21" class="idref" href="#Hwf:21"><span class="id" title="binder">Hwf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#R:20"><span class="id" title="variable">R</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="seq:22" class="idref" href="#seq:22"><span class="id" title="binder">seq</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#A:19"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:23" class="idref" href="#i:23"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#R:20"><span class="id" title="variable">R</span></a> (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#seq:22"><span class="id" title="variable">seq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:23"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#seq:22"><span class="id" title="variable">seq</span></a> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#i:23"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="contrad" class="idref" href="#contrad"><span class="id" title="lemma">contrad</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf"><span class="id" title="section">Proof_of_lt_not_wf</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="lt_not_wf" class="idref" href="#lt_not_wf"><span class="id" title="lemma">lt_not_wf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Wf.html#well_founded"><span class="id" title="definition">well_founded</span></a> <a class="idref" href="hydras.Epsilon0.T1.html#lt"><span class="id" title="definition">lt</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.ge_omega_iff.html b/theories/html/hydras.solutions_exercises.ge_omega_iff.html
new file mode 100644
index 00000000..36c8e98d
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.ge_omega_iff.html
@@ -0,0 +1,43 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.ge_omega_iff</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.ge_omega_iff</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Epsilon0.T1.html#"><span class="id" title="library">T1</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">E0_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="ge_omega_iff" class="idref" href="#ge_omega_iff"><span class="id" title="lemma">ge_omega_iff</span></a> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> : <a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.ge_omega_iff.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="i:2" class="idref" href="#i:2"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="hydras.solutions_exercises.ge_omega_iff.html#i:2"><span class="id" title="variable">i</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c"><span class="id" title="notation">+</span></a> <a class="idref" href="hydras.solutions_exercises.ge_omega_iff.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.ge_omega_iff.html#alpha:1"><span class="id" title="variable">alpha</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.is_F_monotonous.html b/theories/html/hydras.solutions_exercises.is_F_monotonous.html
new file mode 100644
index 00000000..d4ebc210
--- /dev/null
+++ b/theories/html/hydras.solutions_exercises.is_F_monotonous.html
@@ -0,0 +1,73 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.is_F_monotonous</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.is_F_monotonous</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Iterates.html#"><span class="id" title="library">Iterates</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#"><span class="id" title="library">F_alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#"><span class="id" title="library">E0</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>   <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Max.html#"><span class="id" title="library">Max</span></a>.<br/>
+<span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Prelude.Exp2.html#"><span class="id" title="module">Exp2</span></a> <a class="idref" href="hydras.Epsilon0.Canon.html#"><span class="id" title="module">Canon</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Mult.html#"><span class="id" title="library">Mult</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="S1" class="idref" href="#S1"><span class="id" title="section">S1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="S1.H" class="idref" href="#S1.H"><span class="id" title="variable">H</span></a> : <span class="id" title="keyword">∀</span> (<a id="alpha:1" class="idref" href="#alpha:1"><span class="id" title="binder">alpha</span></a> <a id="beta:2" class="idref" href="#beta:2"><span class="id" title="binder">beta</span></a>:<a class="idref" href="hydras.Epsilon0.E0.html#E0"><span class="id" title="class">E0</span></a>), <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#beta:2"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="i:3" class="idref" href="#i:3"><span class="id" title="binder">i</span></a>,  2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#i:3"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#alpha:1"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#i:3"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#beta:2"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#i:3"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : 3 <a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">o</span></a><a class="idref" href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Let</span> <a id="S1.instance_H" class="idref" href="#S1.instance_H"><span class="id" title="variable">instance_H</span></a> := <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#S1.H"><span class="id" title="variable">H</span></a> (<a class="idref" href="hydras.Epsilon0.E0.html#E0fin"><span class="id" title="instance">E0fin</span></a> 3) <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#R1"><span class="id" title="lemma">R1</span></a> <span class="id" title="var">_</span> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#R2"><span class="id" title="lemma">R2</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> <a class="idref" href="hydras.Epsilon0.E0.html#E0_omega"><span class="id" title="instance">E0_omega</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 2.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R4" class="idref" href="#R4"><span class="id" title="lemma">R4</span></a> : <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 3 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 (<a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 2)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R6" class="idref" href="#R6"><span class="id" title="lemma">R6</span></a> : <span class="id" title="keyword">∀</span> <a id="i:5" class="idref" href="#i:5"><span class="id" title="binder">i</span></a>, <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#i:5"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Epsilon0.F_alpha.html#F_"><span class="id" title="definition">F_</span></a> 2 <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#i:5"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Fake_thm" class="idref" href="#Fake_thm"><span class="id" title="lemma">Fake_thm</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.is_F_monotonous.html#S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.isqrt</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.isqrt</h1>
+
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span>  <a class="idref" href="hydras.Ackermann.primRec.html#"><span class="id" title="library">primRec</span></a> <a class="idref" href="hydras.Ackermann.extEqualNat.html#"><span class="id" title="library">extEqualNat</span></a> <a class="idref" href="hydras.Prelude.ssrnat_extracts.html#"><span class="id" title="library">ssrnat_extracts</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Min.html#"><span class="id" title="library">Min</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.setoid_ring.ArithRing.html#"><span class="id" title="library">ArithRing</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Compare_dec.html#"><span class="id" title="library">Compare_dec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Returns smallest value of x less or equal than b such that (P b x). 
+    Otherwise returns b  
+</div>
+<div class="code">
+<span class="id" title="keyword">Check</span> <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a>.<br/>
+<span class="id" title="keyword">Search</span> <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundedSearch3" class="idref" href="#boundedSearch3"><span class="id" title="lemma">boundedSearch3</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:1" class="idref" href="#P:1"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2) (<a id="b:2" class="idref" href="#b:2"><span class="id" title="binder">b</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:2"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:2"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="boundedSearch4" class="idref" href="#boundedSearch4"><span class="id" title="lemma">boundedSearch4</span></a> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:3" class="idref" href="#P:3"><span class="id" title="binder">P</span></a> : <a class="idref" href="hydras.Ackermann.extEqualNat.html#naryRel"><span class="id" title="definition">naryRel</span></a> 2) (<a id="b:4" class="idref" href="#b:4"><span class="id" title="binder">b</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.isqrt.html#P:3"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:4"><span class="id" title="variable">b</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:4"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.isqrt.html#P:3"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:4"><span class="id" title="variable">b</span></a> (<a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#P:3"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#b:4"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="isqrt_spec" class="idref" href="#isqrt_spec"><span class="id" title="definition">isqrt_spec</span></a> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a> <a id="r:6" class="idref" href="#r:6"><span class="id" title="binder">r</span></a> := <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:6"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:6"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea090076ab3dcdee0cfd3882c88b993f"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea090076ab3dcdee0cfd3882c88b993f"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:6"><span class="id" title="variable">r</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:6"><span class="id" title="variable">r</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="sqrtIsPR" class="idref" href="#sqrtIsPR"><span class="id" title="section">sqrtIsPR</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="sqrtIsPR.P" class="idref" href="#sqrtIsPR.P"><span class="id" title="variable">P</span></a> (<a id="n:7" class="idref" href="#n:7"><span class="id" title="binder">n</span></a> <a id="r:8" class="idref" href="#r:8"><span class="id" title="binder">r</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :=  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Arith.PeanoNat.html#Nat.ltb"><span class="id" title="definition">Nat.ltb</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#n:7"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:8"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#r:8"><span class="id" title="variable">r</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="isqrt" class="idref" href="#isqrt"><span class="id" title="definition">isqrt</span></a> := <a class="idref" href="hydras.Ackermann.primRec.html#boundedSearch"><span class="id" title="definition">boundedSearch</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.P"><span class="id" title="variable">P</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="sqrtIsPR.Proof_isqrt" class="idref" href="#sqrtIsPR.Proof_isqrt"><span class="id" title="section">Proof_isqrt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="sqrtIsPR.Proof_isqrt.n" class="idref" href="#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R00" class="idref" href="#R00"><span class="id" title="lemma">R00</span></a> : <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.P"><span class="id" title="variable">P</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R01" class="idref" href="#R01"><span class="id" title="lemma">R01</span></a> :  <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#::nat_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a><a class="idref" href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="R02" class="idref" href="#R02"><span class="id" title="lemma">R02</span></a> : <a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt"><span class="id" title="section">Proof_isqrt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="sqrt_correct" class="idref" href="#sqrt_correct"><span class="id" title="lemma">sqrt_correct</span></a> (<a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt_spec"><span class="id" title="definition">isqrt_spec</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#n:10"><span class="id" title="variable">n</span></a> (<a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a> <a class="idref" href="hydras.solutions_exercises.isqrt.html#n:10"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Instance</span> <a id="issqrtIsPR" class="idref" href="#issqrtIsPR"><span class="id" title="instance">issqrtIsPR</span></a> : <a class="idref" href="hydras.Ackermann.primRec.html#isPR"><span class="id" title="class">isPR</span></a> 1 <a class="idref" href="hydras.solutions_exercises.isqrt.html#isqrt"><span class="id" title="definition">isqrt</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.isqrt.html#sqrtIsPR"><span class="id" title="section">sqrtIsPR</span></a>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Extra work :
+   Define a faster implementation of <span class="inlinecode"><span class="id" title="var">sqrt_spec</span></span>, and prove your function is 
+   extensionnaly equal to <span class="inlinecode"><span class="id" title="var">isqrt</span></span> (hence PR!)
+ 
+</div>
+<div class="code">
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/theories/html/hydras.solutions_exercises.lt_succ_le.html b/theories/html/hydras.solutions_exercises.lt_succ_le.html
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+++ b/theories/html/hydras.solutions_exercises.lt_succ_le.html
@@ -0,0 +1,77 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<title>hydras.solutions_exercises.lt_succ_le</title>
+</head>
+
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+
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+
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+
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+
+<h1 class="libtitle">Library hydras.solutions_exercises.lt_succ_le</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">OrdinalNotations.ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proofs_of_lt_succ_le" class="idref" href="#Proofs_of_lt_succ_le"><span class="id" title="section">Proofs_of_lt_succ_le</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:2" class="idref" href="#lt:2"><span class="id" title="binder">lt</span></a>  : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmp:3" class="idref" href="#cmp:3"><span class="id" title="binder">cmp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="On:4" class="idref" href="#On:4"><span class="id" title="binder">On</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#lt:2"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#cmp:3"><span class="id" title="variable">cmp</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proofs_of_lt_succ_le.Proofs" class="idref" href="#Proofs_of_lt_succ_le.Proofs"><span class="id" title="section">Proofs</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Proofs_of_lt_succ_le.Proofs.alpha" class="idref" href="#Proofs_of_lt_succ_le.Proofs.alpha"><span class="id" title="variable">alpha</span></a> <a id="Proofs_of_lt_succ_le.Proofs.beta" class="idref" href="#Proofs_of_lt_succ_le.Proofs.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+beta is a successor of alpha 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proofs_of_lt_succ_le.Proofs.Halphabeta" class="idref" href="#Proofs_of_lt_succ_le.Proofs.Halphabeta"><span class="id" title="variable">Halphabeta</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proofs_of_lt_succ_le.Proofs.S1" class="idref" href="#Proofs_of_lt_succ_le.Proofs.S1"><span class="id" title="section">S1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proofs_of_lt_succ_le.Proofs.S1.gamma" class="idref" href="#Proofs_of_lt_succ_le.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta" class="idref" href="#Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta"><span class="id" title="variable">HGammaBeta</span></a> :   <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a>: <a class="idref" href="hydras.Prelude.MoreOrders.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs"><span class="id" title="section">Proofs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le"><span class="id" title="section">Proofs_of_lt_succ_le</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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diff --git a/theories/html/hydras.solutions_exercises.predSuccUnicity.html b/theories/html/hydras.solutions_exercises.predSuccUnicity.html
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@@ -0,0 +1,109 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>hydras.solutions_exercises.predSuccUnicity</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library hydras.solutions_exercises.predSuccUnicity</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#"><span class="id" title="library">OrdinalNotations.ON_Generic</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relations.html#"><span class="id" title="library">Relations</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proofs_of_unicity" class="idref" href="#Proofs_of_unicity"><span class="id" title="section">Proofs_of_unicity</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<a id="A:1" class="idref" href="#A:1"><span class="id" title="binder">A</span></a>:<span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:2" class="idref" href="#lt:2"><span class="id" title="binder">lt</span></a> <a id="le:3" class="idref" href="#le:3"><span class="id" title="binder">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Relations.Relation_Definitions.html#relation"><span class="id" title="definition">relation</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cmp:4" class="idref" href="#cmp:4"><span class="id" title="binder">cmp</span></a> : <a class="idref" href="hydras.Prelude.Comparable.html#Compare"><span class="id" title="class">Compare</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#A:1"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="On:5" class="idref" href="#On:5"><span class="id" title="binder">On</span></a> : <a class="idref" href="hydras.OrdinalNotations.ON_Generic.html#ON"><span class="id" title="class">ON</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#lt:2"><span class="id" title="variable">lt</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#cmp:4"><span class="id" title="variable">cmp</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Proofs_of_unicity.Proofs" class="idref" href="#Proofs_of_unicity.Proofs"><span class="id" title="section">Proofs</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variables</span> <a id="Proofs_of_unicity.Proofs.alpha" class="idref" href="#Proofs_of_unicity.Proofs.alpha"><span class="id" title="variable">alpha</span></a> <a id="Proofs_of_unicity.Proofs.beta" class="idref" href="#Proofs_of_unicity.Proofs.beta"><span class="id" title="variable">beta</span></a> : <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+beta is a successor of alpha 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proofs_of_unicity.Proofs.Halphabeta" class="idref" href="#Proofs_of_unicity.Proofs.Halphabeta"><span class="id" title="variable">Halphabeta</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proofs_of_unicity.Proofs.S1" class="idref" href="#Proofs_of_unicity.Proofs.S1"><span class="id" title="section">S1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proofs_of_unicity.Proofs.S1.gamma" class="idref" href="#Proofs_of_unicity.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proofs_of_unicity.Proofs.S1.Halphagamma" class="idref" href="#Proofs_of_unicity.Proofs.S1.Halphagamma"><span class="id" title="variable">Halphagamma</span></a> :   <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L1" class="idref" href="#L1"><span class="id" title="lemma">L1</span></a> :  <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.beta"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1"><span class="id" title="section">S1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Proofs_of_unicity.Proofs.S2" class="idref" href="#Proofs_of_unicity.Proofs.S2"><span class="id" title="section">S2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="Proofs_of_unicity.Proofs.S2.gamma" class="idref" href="#Proofs_of_unicity.Proofs.S2.gamma"><span class="id" title="variable">gamma</span></a>: <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.A"><span class="id" title="variable">A</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span> <a id="Proofs_of_unicity.Proofs.S2.Hgammaalpha" class="idref" href="#Proofs_of_unicity.Proofs.S2.Hgammaalpha"><span class="id" title="variable">Hgammaalpha</span></a> : <a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.gamma"><span class="id" title="variable">gamma</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="L2" class="idref" href="#L2"><span class="id" title="lemma">L2</span></a> : <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.gamma"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2"><span class="id" title="section">S2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs"><span class="id" title="section">Proofs</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Please remind that <span class="inlinecode"><span class="id" title="var">Successor</span></span> <span class="inlinecode"><span class="id" title="keyword">beta</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> must be read as
+    "<span class="inlinecode"><span class="id" title="keyword">beta</span></span> is a successor of <span class="inlinecode"><span class="id" title="var">alpha</span></span>" 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Successor_unicity" class="idref" href="#Successor_unicity"><span class="id" title="lemma">Successor_unicity</span></a> (<a id="alpha:13" class="idref" href="#alpha:13"><span class="id" title="binder">alpha</span></a> <a id="beta:14" class="idref" href="#beta:14"><span class="id" title="binder">beta</span></a> <a id="gamma:15" class="idref" href="#gamma:15"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#beta:14"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#gamma:15"><span class="id" title="variable">gamma</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#alpha:13"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#gamma:15"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#beta:14"><span class="id" title="variable">beta</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Predecessor_unicity" class="idref" href="#Predecessor_unicity"><span class="id" title="lemma">Predecessor_unicity</span></a> (<a id="alpha:16" class="idref" href="#alpha:16"><span class="id" title="binder">alpha</span></a> <a id="beta:17" class="idref" href="#beta:17"><span class="id" title="binder">beta</span></a> <a id="gamma:18" class="idref" href="#gamma:18"><span class="id" title="binder">gamma</span></a> : <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.A"><span class="id" title="variable">A</span></a>):<br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#alpha:16"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="hydras.Prelude.MoreOrders.html#Successor"><span class="id" title="definition">Successor</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#beta:17"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#gamma:18"><span class="id" title="variable">gamma</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
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+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity"><span class="id" title="section">Proofs_of_unicity</span></a>.<br/>
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diff --git a/theories/html/hydras.solutions_exercises.schutte_cnf_counter_example.html b/theories/html/hydras.solutions_exercises.schutte_cnf_counter_example.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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+<title>hydras.solutions_exercises.schutte_cnf_counter_example</title>
+</head>
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+
+<h1 class="libtitle">Library hydras.solutions_exercises.schutte_cnf_counter_example</h1>
+
+<div class="code">
+<span class="id" title="keyword">From</span> <span class="id" title="var">hydras</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="hydras.Schutte.Schutte.html#"><span class="id" title="library">Schutte</span></a> <a class="idref" href="hydras.Schutte.AP.html#"><span class="id" title="library">AP</span></a> <a class="idref" href="hydras.Schutte.CNF.html#"><span class="id" title="library">CNF</span></a>.<br/>
+<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">schutte_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a id="Cter_example" class="idref" href="#Cter_example"><span class="id" title="section">Cter_example</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Hypothesis</span>  <a id="Cter_example.cnf_lt_epsilon0_iff" class="idref" href="#Cter_example.cnf_lt_epsilon0_iff"><span class="id" title="variable">cnf_lt_epsilon0_iff</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="l:1" class="idref" href="#l:1"><span class="id" title="binder">l</span></a> <a id="alpha:2" class="idref" href="#alpha:2"><span class="id" title="binder">alpha</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#l:1"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#alpha:2"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a>  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:3" class="idref" href="#beta:3"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#beta:3"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#alpha:2"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#l:1"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Cter_example.alpha" class="idref" href="#Cter_example.alpha"><span class="id" title="variable">alpha</span></a> := <a class="idref" href="hydras.Schutte.AP.html#phi0"><span class="id" title="abbreviation">phi0</span></a> (<a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Let</span> <a id="Cter_example.l" class="idref" href="#Cter_example.l"><span class="id" title="variable">l</span></a> := <a class="idref" href="hydras.Schutte.Schutte_basics.html#succ"><span class="id" title="definition">succ</span></a> <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R1" class="idref" href="#R1"><span class="id" title="lemma">R1</span></a> : <a class="idref" href="hydras.Schutte.AP.html#epsilon0"><span class="id" title="definition">epsilon0</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.alpha"><span class="id" title="variable">alpha</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R2" class="idref" href="#R2"><span class="id" title="lemma">R2</span></a> : <a class="idref" href="hydras.Schutte.CNF.html#is_cnf_of"><span class="id" title="definition">is_cnf_of</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.l"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Remark</span> <a id="R3" class="idref" href="#R3"><span class="id" title="lemma">R3</span></a> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<span class="id" title="keyword">fun</span> <a id="beta:5" class="idref" href="#beta:5"><span class="id" title="binder">beta</span></a> ⇒  <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#beta:5"><span class="id" title="variable">beta</span></a> <a class="idref" href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.alpha"><span class="id" title="variable">alpha</span></a>) <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.l"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="counter_ex" class="idref" href="#counter_ex"><span class="id" title="lemma">counter_ex</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.15.0/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example"><span class="id" title="section">Cter_example</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
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+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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diff --git a/theories/html/index.html b/theories/html/index.html
new file mode 100644
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+<td>(191 entries)</td>
+</tr>
+<tr>
+<td>Lemma Index</td>
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+<td>(3916 entries)</td>
+</tr>
+<tr>
+<td>Constructor Index</td>
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+<td>(281 entries)</td>
+</tr>
+<tr>
+<td>Axiom Index</td>
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+<td>W</td>
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+<td>_</td>
+<td>other</td>
+<td>(52 entries)</td>
+</tr>
+<tr>
+<td>Inductive Index</td>
+<td><a href="index.html#inductive_A">A</a></td>
+<td><a href="index.html#inductive_B">B</a></td>
+<td><a href="index.html#inductive_C">C</a></td>
+<td><a href="index.html#inductive_D">D</a></td>
+<td><a href="index.html#inductive_E">E</a></td>
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+<td><a href="index.html#inductive_G">G</a></td>
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+<td><a href="index.html#inductive_N">N</a></td>
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+<td>U</td>
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+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(119 entries)</td>
+</tr>
+<tr>
+<td>Projection Index</td>
+<td><a href="index.html#projection_A">A</a></td>
+<td><a href="index.html#projection_B">B</a></td>
+<td><a href="index.html#projection_C">C</a></td>
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+<td>H</td>
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+<td><a href="index.html#projection_T">T</a></td>
+<td><a href="index.html#projection_U">U</a></td>
+<td><a href="index.html#projection_V">V</a></td>
+<td><a href="index.html#projection_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
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+<td>(110 entries)</td>
+</tr>
+<tr>
+<td>Section Index</td>
+<td><a href="index.html#section_A">A</a></td>
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+<td>Q</td>
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+<td>U</td>
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+<td>X</td>
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+<td>Z</td>
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+<td>other</td>
+<td>(284 entries)</td>
+</tr>
+<tr>
+<td>Instance Index</td>
+<td><a href="index.html#instance_A">A</a></td>
+<td><a href="index.html#instance_B">B</a></td>
+<td><a href="index.html#instance_C">C</a></td>
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+<td><a href="index.html#instance_I">I</a></td>
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+<td>U</td>
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+<td>X</td>
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+<td>_</td>
+<td>other</td>
+<td>(366 entries)</td>
+</tr>
+<tr>
+<td>Abbreviation Index</td>
+<td><a href="index.html#abbreviation_A">A</a></td>
+<td><a href="index.html#abbreviation_B">B</a></td>
+<td><a href="index.html#abbreviation_C">C</a></td>
+<td>D</td>
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+<td><a href="index.html#abbreviation_O">O</a></td>
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+<td>Q</td>
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+<td><a href="index.html#abbreviation_U">U</a></td>
+<td><a href="index.html#abbreviation_V">V</a></td>
+<td>W</td>
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+<td>Y</td>
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+<td><a href="index.html#abbreviation__">_</a></td>
+<td><a href="index.html#abbreviation_*">other</a></td>
+<td>(182 entries)</td>
+</tr>
+<tr>
+<td>Definition Index</td>
+<td><a href="index.html#definition_A">A</a></td>
+<td><a href="index.html#definition_B">B</a></td>
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+<td><a href="index.html#definition_E">E</a></td>
+<td><a href="index.html#definition_F">F</a></td>
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+<td>Q</td>
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+<td><a href="index.html#definition_X">X</a></td>
+<td>Y</td>
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+<td><a href="index.html#definition__">_</a></td>
+<td><a href="index.html#definition_*">other</a></td>
+<td>(1487 entries)</td>
+</tr>
+<tr>
+<td>Record Index</td>
+<td><a href="index.html#record_A">A</a></td>
+<td><a href="index.html#record_B">B</a></td>
+<td><a href="index.html#record_C">C</a></td>
+<td><a href="index.html#record_D">D</a></td>
+<td><a href="index.html#record_E">E</a></td>
+<td><a href="index.html#record_F">F</a></td>
+<td><a href="index.html#record_G">G</a></td>
+<td><a href="index.html#record_H">H</a></td>
+<td><a href="index.html#record_I">I</a></td>
+<td>J</td>
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+<td><a href="index.html#record_L">L</a></td>
+<td><a href="index.html#record_M">M</a></td>
+<td>N</td>
+<td><a href="index.html#record_O">O</a></td>
+<td><a href="index.html#record_P">P</a></td>
+<td><a href="index.html#record_Q">Q</a></td>
+<td><a href="index.html#record_R">R</a></td>
+<td><a href="index.html#record_S">S</a></td>
+<td><a href="index.html#record_T">T</a></td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#record_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(52 entries)</td>
+</tr>
+</table>
+<hr/>
+<h1>Global Index</h1>
+<a id="global_A"></a><h2>A </h2>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A">A</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="additions.Monoid_def.html#Abelian_Monoid">Abelian_Monoid</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Abelian_EMonoid">Abelian_EMonoid</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha">Abstract_Properties.alpha</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Abstract_Properties">Abstract_Properties</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#absurd1">absurd1</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#absurd1">absurd1</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#absurd1">absurd1</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun.Rf">AB_given.rel_to_fun.Rf</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun">AB_given.rel_to_fun</a> [section, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf">AB_given.f_given.Rf</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given.f">AB_given.f_given.f</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given">AB_given.f_given</a> [section, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.DB">AB_given.DB</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.DA">AB_given.DA</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.Hb">AB_given.Hb</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.Ha">AB_given.Ha</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.B">AB_given.B</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.A">AB_given.A</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given">AB_given</a> [section, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#ab1:18">ab1:18</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#ab2:19">ab2:19</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#ab:34">ab:34</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Accs_all">Accs_all</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.O2H.html#acc_from_to_round_plus">acc_from_to_round_plus</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GPaths.html#acc_from">acc_from</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Acc_nil">Acc_nil</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="additions.Wf_transparent.html#Acc_inverse_image">Acc_inverse_image</a> [lemma, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Acc_lemma">Acc_lemma</a> [lemma, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from_LT">acc_from_LT</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from_trans">acc_from_trans</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from">acc_from</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.rpo.closure.html#acc_trans">acc_trans</a> [lemma, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#acc_rec">acc_rec</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#acc_imp">acc_imp</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="additions.fib.html#acc1:10">acc1:10</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#acc1:5">acc1:5</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#acc2:11">acc2:11</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#acc2:6">acc2:6</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#acc:108">acc:108</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:113">acc:113</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:118">acc:118</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:126">acc:126</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:133">acc:133</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:140">acc:140</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:144">acc:144</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:148">acc:148</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="additions.Compatibility.html#acc:16">acc:16</a> [binder, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.fib.html#acc:18">acc:18</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Compatibility.html#acc:20">acc:20</a> [binder, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:232">acc:232</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:235">acc:235</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:238">acc:238</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:242">acc:242</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="additions.fib.html#acc:71">acc:71</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#acc:78">acc:78</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.list_set.html#acc:87">acc:87</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack">Ack</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack">Ack</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html">Ack</a> [library]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann">Ackermann</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.addC_CE">Ackermann.addC_CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_normal">Ackermann.add_normal</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le4">Ackermann.add_le4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le3">Ackermann.add_le3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_inj">Ackermann.add_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le2">Ackermann.add_le2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le1">Ackermann.add_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_to_cons">Ackermann.add_to_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_fin_omega">Ackermann.add_fin_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_int">Ackermann.add_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add1Nfin">Ackermann.add1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.all_zeroE">Ackermann.all_zeroE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero">Ackermann.all_zero</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr2CE">Ackermann.ap_pr2CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr4">Ackermann.ap_pr4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr3">Ackermann.ap_pr3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr2">Ackermann.ap_pr2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr1">Ackermann.ap_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_limit">Ackermann.ap_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr0">Ackermann.ap_pr0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Eq">Ackermann.CompareT3Eq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Gt">Ackermann.CompareT3Gt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Lt">Ackermann.CompareT3Lt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.compare_T3">Ackermann.compare_T3</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.conc1">Ackermann.conc1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.cons">Ackermann.cons</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0">Ackermann.epsilon0</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.fincP">Ackermann.fincP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.finite_ltP">Ackermann.finite_ltP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.fooCE">Ackermann.fooCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.le_succ_succE">Ackermann.le_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.le_succ_succ">Ackermann.le_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of">Ackermann.limit_of</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_pr">Ackermann.limit_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_pr1">Ackermann.limit_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit1">Ackermann.limit1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit10">Ackermann.limit10</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit11">Ackermann.limit11</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12a">Ackermann.limit12a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b1">Ackermann.limit12b1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b2">Ackermann.limit12b2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b3">Ackermann.limit12b3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b4">Ackermann.limit12b4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b5">Ackermann.limit12b5</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp">Ackermann.limit12_hyp</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit2">Ackermann.limit2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit3">Ackermann.limit3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit4">Ackermann.limit4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit5">Ackermann.limit5</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit6">Ackermann.limit6</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit7">Ackermann.limit7</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit8">Ackermann.limit8</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit9">Ackermann.limit9</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_4">Ackermann.lt_succ_le_4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_3">Ackermann.lt_succ_le_3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_2">Ackermann.lt_succ_le_2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_1">Ackermann.lt_succ_le_1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_succE">Ackermann.lt_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_succ">Ackermann.lt_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_omega2">Ackermann.lt_omega2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_omega1">Ackermann.lt_omega1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_epsilon0">Ackermann.lt_epsilon0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_not_wf">Ackermann.lt_not_wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi">Ackermann.lt_psi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_rec">Ackermann.lt_rec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi_rec">Ackermann.lt_psi_rec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.minus_le">Ackermann.minus_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.minus_lt">Ackermann.minus_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_add">Ackermann.nf_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_sub">Ackermann.nf_sub</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_pred">Ackermann.nf_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_succ">Ackermann.nf_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_finite">Ackermann.nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_omega">Ackermann.nf_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_split">Ackermann.nf_split</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_Wf">Ackermann.nf_Wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_consE">Ackermann.nf_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_cons_cons">Ackermann.nf_cons_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_int">Ackermann.nf_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_psi">Ackermann.nf_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_0">Ackermann.nf_0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal">Ackermann.normal</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_compose">Ackermann.normal_compose</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_limit">Ackermann.normal_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_id">Ackermann.normal_id</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega">Ackermann.omega</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_minus_one">Ackermann.omega_minus_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_least_inf2">Ackermann.omega_least_inf2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_least_inf1">Ackermann.omega_least_inf1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.one">Ackermann.one</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi">Ackermann.phi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phia">Ackermann.phia</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE">Ackermann.phiE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_3">Ackermann.phiE_3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_2">Ackermann.phiE_2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_1">Ackermann.phiE_1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiL">Ackermann.phiL</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec">Ackermann.phi_rec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec_psi">Ackermann.phi_rec_psi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi0">Ackermann.phi0</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi1">Ackermann.phi1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi10">Ackermann.phi10</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi11">Ackermann.phi11</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12a">Ackermann.phi12a</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b2">Ackermann.phi12b2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b3">Ackermann.phi12b3</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b4">Ackermann.phi12b4</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b5">Ackermann.phi12b5</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi3">Ackermann.phi3</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi3v">Ackermann.phi3v</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi4">Ackermann.phi4</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi5">Ackermann.phi5</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi6">Ackermann.phi6</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi7">Ackermann.phi7</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi8">Ackermann.phi8</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi9">Ackermann.phi9</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_succ">Ackermann.pred_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_lt">Ackermann.pred_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_le">Ackermann.pred_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ_pr1">Ackermann.psi_succ_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ">Ackermann.psi_succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_lt2">Ackermann.psi_lt2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_lt1">Ackermann.psi_lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size">Ackermann.size</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_prop">Ackermann.size_prop</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_prop1">Ackermann.size_prop1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_psi">Ackermann.size_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_d">Ackermann.size_d</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_c">Ackermann.size_c</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_b">Ackermann.size_b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_a">Ackermann.size_a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_add">Ackermann.split_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_le">Ackermann.split_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_pred">Ackermann.split_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_succ">Ackermann.split_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_is_succ">Ackermann.split_is_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_limit">Ackermann.split_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_finite">Ackermann.split_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1rCE">Ackermann.sub_pr1rCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1r">Ackermann.sub_pr1r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_nz">Ackermann.sub_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1">Ackermann.sub_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr">Ackermann.sub_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_le1">Ackermann.sub_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_1bCE">Ackermann.sub_1bCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_1aCE">Ackermann.sub_1aCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_int">Ackermann.sub_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1a">Ackermann.sub1a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1b">Ackermann.sub1b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1Nfin">Ackermann.sub1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psiCE">Ackermann.succ_psiCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_prCE">Ackermann.succ_prCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi2">Ackermann.succ_psi2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi1">Ackermann.succ_psi1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_is_add_one">Ackermann.succ_is_add_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi_lt2">Ackermann.succ_psi_lt2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi_lt">Ackermann.succ_psi_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi">Ackermann.succ_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_nz">Ackermann.succ_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_inj">Ackermann.succ_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_p1">Ackermann.succ_p1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_pred">Ackermann.succ_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_lt">Ackermann.succ_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_Oalpha_succ">Ackermann.sup_Oalpha_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_Oalpha_limit">Ackermann.sup_Oalpha_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_Oalpha_zero">Ackermann.sup_Oalpha_zero</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_unique">Ackermann.sup_unique</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of">Ackermann.sup_of</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.Tf">Ackermann.Tf</a> [abbreviation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.TT1T3_inj">Ackermann.TT1T3_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1succ_nat">Ackermann.T1succ_nat</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1T3_surj">Ackermann.T1T3_surj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1T3_inc">Ackermann.T1T3_inc</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1T3_lt_epsilon0">Ackermann.T1T3_lt_epsilon0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3">Ackermann.T1_T3</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3">Ackermann.T3</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3add">Ackermann.T3add</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3addA">Ackermann.T3addA</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3addn0">Ackermann.T3addn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3addS">Ackermann.T3addS</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3add_eq0">Ackermann.T3add_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3add0n">Ackermann.T3add0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap">Ackermann.T3ap</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad">Ackermann.T3bad</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eq">Ackermann.T3eq</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eqE">Ackermann.T3eqE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eqP">Ackermann.T3eqP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eq_add2l">Ackermann.T3eq_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eq_le">Ackermann.T3eq_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite">Ackermann.T3finite</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite_succ">Ackermann.T3finite_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite1">Ackermann.T3finite1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite2">Ackermann.T3finite2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GeqNotGtn">Ackermann.T3GeqNotGtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GeqNotLtn">Ackermann.T3GeqNotLtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ge1">Ackermann.T3ge1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GtnNotLeq">Ackermann.T3GtnNotLeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3gt1">Ackermann.T3gt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ">Ackermann.T3is_succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lcp0_pr">Ackermann.T3lcp0_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le">Ackermann.T3le</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3leNgt">Ackermann.T3leNgt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lenn">Ackermann.T3lenn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3len0">Ackermann.T3len0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3leP">Ackermann.T3leP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3leq_xor_gtn">Ackermann.T3leq_xor_gtn</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_add2r">Ackermann.T3le_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_add2l">Ackermann.T3le_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_psi">Ackermann.T3le_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_total">Ackermann.T3le_total</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_anti">Ackermann.T3le_anti</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_trans">Ackermann.T3le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_lt_trans">Ackermann.T3le_lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_consE">Ackermann.T3le_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le_eqVlt">Ackermann.T3le_eqVlt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le0n">Ackermann.T3le0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit">Ackermann.T3limit</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt">Ackermann.T3lt</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lta">Ackermann.T3lta</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltE">Ackermann.T3ltE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltgtP">Ackermann.T3ltgtP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltNge">Ackermann.T3ltNge</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltnn">Ackermann.T3ltnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3LtnNotGeq">Ackermann.T3LtnNotGeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltn_xor_geq">Ackermann.T3ltn_xor_geq</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltn0">Ackermann.T3ltn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltP">Ackermann.T3ltP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltW">Ackermann.T3ltW</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_add2r">Ackermann.T3lt_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_add2l">Ackermann.T3lt_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi_a">Ackermann.T3lt_psi_a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi_c">Ackermann.T3lt_psi_c</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi_b">Ackermann.T3lt_psi_b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi_bc">Ackermann.T3lt_psi_bc</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_le_trans">Ackermann.T3lt_le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_trans">Ackermann.T3lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi'">Ackermann.T3lt_psi'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_trichotomy">Ackermann.T3lt_trichotomy</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_anti">Ackermann.T3lt_anti</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_neAle">Ackermann.T3lt_neAle</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_ne'">Ackermann.T3lt_ne'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_ne">Ackermann.T3lt_ne</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_consE">Ackermann.T3lt_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt_psi">Ackermann.T3lt_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt0n">Ackermann.T3lt0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt1">Ackermann.T3lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3nat">Ackermann.T3nat</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf">Ackermann.T3nf</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf_finite">Ackermann.T3nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred">Ackermann.T3pred</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3split">Ackermann.T3split</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3sub">Ackermann.T3sub</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3subnn">Ackermann.T3subnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3subn0">Ackermann.T3subn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3sub0n">Ackermann.T3sub0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ">Ackermann.T3succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3_eqType">Ackermann.T3_eqType</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3_eqMixin">Ackermann.T3_eqMixin</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.zero">Ackermann.zero</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x">_ - _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd">_ + _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&gt;'_x">_ &gt; _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#752c7e761a34d95cac9b695b31aec25e">_ &gt;= _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7">_ &lt;= _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x">_ &lt; _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3">\F _ (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613">[ _ , _ , _ ] (ak_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::cantor_scope:x_'&lt;'_x">_ &lt; _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ackm:13">Ackm:13</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html">AckNotPR</a> [library]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Ackn_IsPR">Ackn_IsPR</a> [instance, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#AckSn_as_PRiterate">AckSn_as_PRiterate</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#AckSn_as_iterate">AckSn_as_iterate</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Ack_not_PR">Ack_not_PR</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#Ack_iterate_rw">Ack_iterate_rw</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_strict_mono_l">Ack_strict_mono_l</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Sn_plus">Ack_Sn_plus</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_mono_r">Ack_mono_r</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_mono_l">Ack_mono_l</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_strict_mono">Ack_strict_mono</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_properties">Ack_properties</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_m_mono_weak">Ack_m_mono_weak</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.Hm">Ack_Properties.Induc_step.Hm</a> [variable, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m">Ack_Properties.Induc_step.m</a> [variable, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step">Ack_Properties.Induc_step</a> [section, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.P">Ack_Properties.P</a> [variable, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties">Ack_Properties</a> [section, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_4_n">Ack_4_n</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_3_n">Ack_3_n</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_2_n">Ack_2_n</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_1_n">Ack_1_n</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_S_S">Ack_S_S</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_S_0">Ack_S_0</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_0">Ack_0</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Compatibility.html#Adaptation">Adaptation</a> [section, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Trace_exercise.html#Add">Add</a> [constructor, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="Goedel.PRrepresentable.html#addExists">addExists</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#addForalls">addForalls</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Schutte.Addition.html">Addition</a> [library]<br/>
+<a href="additions.Addition_Chains.html">Addition_Chains</a> [library]<br/>
+<a href="hydras.Ackermann.folLogic3.html#addPairwiseEquals">addPairwiseEquals</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#add_r_0">add_r_0</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_r">add_r</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus">add_head_r_plus</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_head_r">add_head_r</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.Monoid_def.html#Aeq:64">Aeq:64</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alive">Alive</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Alive_standard">Alive_standard</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Alive_free">Alive_free</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.O2H.html#all_cases">all_cases</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#all_ord_acc">all_ord_acc</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#Almost_done">Almost_done</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha_0_eq">alpha_0_eq</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha_0">alpha_0</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.O2H.html#alpha_pos">alpha_pos</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Schutte.AP.html#alpha_lt_beta">alpha_lt_beta</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#alpha_plus_zero">alpha_plus_zero</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#alpha_sup">alpha_sup</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#alpha_A">alpha_A</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Addition.html#alpha_plus_sup">alpha_plus_sup</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#alpha_plus_zero">alpha_plus_zero</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':137">alpha':137</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha':193">alpha':193</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha':322">alpha':322</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':462">alpha':462</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':505">alpha':505</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':509">alpha':509</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':548">alpha':548</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha':551">alpha':551</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha':92">alpha':92</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#alpha1">alpha1</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:28">alpha1:28</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:36">alpha1:36</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:43">alpha1:43</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:51">alpha1:51</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:58">alpha1:58</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha1:64">alpha1:64</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha1:665">alpha1:665</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha1:674">alpha1:674</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha2:29">alpha2:29</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha2:44">alpha2:44</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#alpha2:52">alpha2:52</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha2:667">alpha2:667</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#alpha2:676">alpha2:676</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#alpha:1">alpha:1</a> [binder, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GHprime.html#alpha:1">alpha:1</a> [binder, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.AP.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
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+<a href="hydras.Epsilon0.Hprime.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#alpha:1">alpha:1</a> [binder, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#alpha:1">alpha:1</a> [binder, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Schutte.Addition.html#alpha:1">alpha:1</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#alpha:1">alpha:1</a> [binder, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#alpha:10">alpha:10</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#alpha:10">alpha:10</a> [binder, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GHprime.html#alpha:10">alpha:10</a> [binder, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#alpha:10">alpha:10</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Schutte.Critical.html#alpha:10">alpha:10</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Addition.html#alpha:10">alpha:10</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
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+<a href="additions.Demo.html#Alt.double">Alt.double</a> [definition, in <a href="additions.Demo.html">additions.Demo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_rpo">Alt.Eps0_rpo</a> [module, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_alg">Alt.Eps0_alg</a> [module, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec_transitive">Alt.Eps0_prec.prec_transitive</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec_antisym">Alt.Eps0_prec.prec_antisym</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec_dec">Alt.Eps0_prec.prec_dec</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.status">Alt.Eps0_prec.status</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.Mul">Alt.Eps0_prec.Mul</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.Lex">Alt.Eps0_prec.Lex</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.status_type">Alt.Eps0_prec.status_type</a> [inductive, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec">Alt.Eps0_prec.prec</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A">Alt.Eps0_prec.A</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec">Alt.Eps0_prec</a> [module, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.arity">Alt.Eps0_sig.arity</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free">Alt.Eps0_sig.Free</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.C">Alt.Eps0_sig.C</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.AC">Alt.Eps0_sig.AC</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.arity_type">Alt.Eps0_sig.arity_type</a> [inductive, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.eq_symbol_dec">Alt.Eps0_sig.eq_symbol_dec</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb">Alt.Eps0_sig.symb</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons">Alt.Eps0_sig.ord_cons</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero">Alt.Eps0_sig.ord_zero</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S">Alt.Eps0_sig.nat_S</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0">Alt.Eps0_sig.nat_0</a> [constructor, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb0">Alt.Eps0_sig.symb0</a> [inductive, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig">Alt.Eps0_sig</a> [module, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alt.hnode">Alt.hnode</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alt.Hydra">Alt.Hydra</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_inc_rpo_0">Alt.lt_inc_rpo_0</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_subterm2">Alt.lt_subterm2</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_subterm1">Alt.lt_subterm1</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term_mono">Alt.nat_2_term_mono</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_lt_cons">Alt.nat_lt_cons</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term">Alt.nat_2_term</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nf_Acc">Alt.nf_Acc</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nf_Wf">Alt.nf_Wf</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.pi2_5IsPR">Alt.pi2_5IsPR</a> [instance, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.predIsPR">Alt.predIsPR</a> [instance, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo_trans">Alt.rpo_trans</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R_inc_rpo">Alt.R_inc_rpo</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R1">Alt.R1</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R2">Alt.R2</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R3">Alt.R3</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R4">Alt.R4</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.succIsPR">Alt.succIsPR</a> [instance, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor">Alt.transfinite_recursor</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor_lt">Alt.transfinite_recursor_lt</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size3">Alt.T1_size3</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size2">Alt.T1_size2</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size1">Alt.T1_size1</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size">Alt.T1_size</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term">Alt.T1_2_term</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars">Alt.Vars</a> [module, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.empty_set">Alt.Vars.empty_set</a> [inductive, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.eq_variable_dec">Alt.Vars.eq_variable_dec</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.var">Alt.Vars.var</a> [definition, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded.R">Alt.well_founded.R</a> [variable, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded">Alt.well_founded</a> [section, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded_rpo">Alt.well_founded_rpo</a> [lemma, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.xpred">Alt.xpred</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.zeroIsPR">Alt.zeroIsPR</a> [instance, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2">Alt2</a> [module, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.Ack">Alt2.Ack</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.test1">Alt2.test1</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.test2">Alt2.test2</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt3">Alt3</a> [module, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt3.ack">Alt3.ack</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.AM.html">AM</a> [library]<br/>
+<a href="additions.AM.html#AM_power_Ok">AM_power_Ok</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#AM_power">AM_power</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Pow.html#AM:105">AM:105</a> [binder, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow_variant.html#AM:96">AM:96</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andE1">andE1</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andE1">andE1</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andE1">andE1</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andE2">andE2</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andE2">andE2</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andE2">andE2</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#andH">andH</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andI">andI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andI">andI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andI">andI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#andRel">andRel</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#andRelPR">andRelPR</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andSym">andSym</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andSym">andSym</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andSym">andSym</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous0:360">anonymous0:360</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous0:366">anonymous0:366</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous0:372">anonymous0:372</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous0:379">anonymous0:379</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:359">anonymous:359</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:365">anonymous:365</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:371">anonymous:371</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:378">anonymous:378</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:386">anonymous:386</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:392">anonymous:392</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#anonymous:399">anonymous:399</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#answer">answer</a> [inductive, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="additions.Addition_Chains.html#any_chain_parametric">any_chain_parametric</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap">ap</a> [inductive, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#ap">ap</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#ap">ap</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP">AP</a> [definition, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html">AP</a> [library]<br/>
+<a href="hydras.Ackermann.fol.html#apply">apply</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#approx">approx</a> [definition, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#approx_ok">approx_ok</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_plusR">ap_plusR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_plus">ap_plus</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_intro">ap_intro</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#ap_cases">ap_cases</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#ap_intro">ap_intro</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_plus_AP">AP_plus_AP</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_to_phi0">AP_to_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_phi0">AP_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_o_segment">AP_o_segment</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed">AP_closed</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_sup">AP_sup</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.denM">AP_closed.denM</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.inhM">AP_closed.inhM</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.OM">AP_closed.OM</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.M">AP_closed.M</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed">AP_closed</a> [section, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_unbounded">AP_unbounded</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_unbounded_0">AP_unbounded_0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.n">AP_Unbounded.ksi_fixed.n</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.lt_ksi">AP_Unbounded.ksi_fixed.lt_ksi</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.ksi">AP_Unbounded.ksi_fixed.ksi</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed">AP_Unbounded.ksi_fixed</a> [section, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.beta">AP_Unbounded.beta</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.seq">AP_Unbounded.seq</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.alpha">AP_Unbounded.alpha</a> [variable, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded">AP_Unbounded</a> [section, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_mult_fin_r_closed">AP_mult_fin_r_closed</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_mult_Sn_closed">AP_mult_Sn_closed</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_plus_closed">AP_plus_closed</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_finite_eq_one">AP_finite_eq_one</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_omega">AP_omega</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_one">AP_one</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#ap_ref">ap_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#Arith_lemmas">Arith_lemmas</a> [section, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Ackermann.fol.html#arityF">arityF</a> [projection, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#arityR">arityR</a> [projection, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#assoc">assoc</a> [projection, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Assoc">Assoc</a> [record, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#assoc">assoc</a> [constructor, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Assoc">Assoc</a> [inductive, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="additions.fib.html#assoc:105">assoc:105</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#assoc:112">assoc:112</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#assoc:123">assoc:123</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.fol.html#atomic">atomic</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Trace_exercise.html#aux:10">aux:10</a> [binder, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AXM">AXM</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Axm">Axm</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Axm">Axm</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Axm">Axm</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AxmEq4">AxmEq4</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AxmEq5">AxmEq5</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Axm:120">Axm:120</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#axm:126">axm:126</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Axm:128">Axm:128</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Axm:141">Axm:141</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Axm:145">Axm:145</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="Goedel.goedel2.html#Axm:15">Axm:15</a> [binder, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Axm:19">Axm:19</a> [binder, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.rosserPA.html#Axm:44">Axm:44</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Axm:47">Axm:47</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_scheme_length1">axp_scheme_length1</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_correct">axp_correct</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_scheme">axp_scheme</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#AX1">AX1</a> [axiom, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#AX2">AX2</a> [axiom, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#AX3">AX3</a> [axiom, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.Addition_Chains.html#ax:252">ax:252</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A_def.m">A_def.m</a> [variable, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A_def">A_def</a> [section, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_full">A_full</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_Cr">A_Cr</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_">A_</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given.LtA">A_given.LtA</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given.A">A_given.A</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given">A_given</a> [section, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="additions.Addition_Chains.html#A_node">A_node</a> [constructor, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A_closed">A_closed</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A_denum">A_denum</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#a'':44">a'':44</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1022">a':1022</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#a':11">a':11</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1150">a':1150</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#a':116">a':116</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1297">a':1297</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':130">a':130</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1301">a':1301</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1305">a':1305</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1309">a':1309</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1335">a':1335</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1381">a':1381</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1391">a':1391</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1419">a':1419</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1425">a':1425</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1433">a':1433</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1497">a':1497</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1505">a':1505</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.more_list.html#a':153">a':153</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1548">a':1548</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1582">a':1582</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1606">a':1606</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1615">a':1615</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1624">a':1624</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1632">a':1632</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#a':164">a':164</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1688">a':1688</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.rpo.html#a':169">a':169</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':1694">a':1694</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':173">a':173</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#a':175">a':175</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#a':18">a':18</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.rpo.rpo.html#a':189">a':189</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#a':191">a':191</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':2063">a':2063</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':2077">a':2077</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':2082">a':2082</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#a':22">a':22</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#a':222">a':222</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a':224">a':224</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#a':228">a':228</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#a':23">a':23</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
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+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#big_to_small">big_to_small</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h">big_h</a> [definition, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#big_to_small">big_to_small</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#big_to_small">big_to_small</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#big0">big0</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.BinaryStrat.html">BinaryStrat</a> [library]<br/>
+<a href="additions.FirstSteps.html#binary_power_mult">binary_power_mult</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#binary_power_mult_ok">binary_power_mult_ok</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#binary_power_mult">binary_power_mult</a> [definition, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#binary_power_mult_ok">binary_power_mult_ok</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#binary_power_mult">binary_power_mult</a> [definition, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.fib.html#binary_power_mult">binary_power_mult</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_chain_length">binary_chain_length</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_generator_not_optimal">binary_generator_not_optimal</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_generator_correct">binary_generator_correct</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_correct">binary_correct</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_eq">binary_power_proof.E_eq</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_one">binary_power_proof.E_one</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_op">binary_power_proof.E_op</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.A">binary_power_proof.A</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof">binary_power_proof</a> [section, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_chain">binary_chain</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.BinaryStrat.html#Binary_strat">Binary_strat</a> [instance, in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="additions.Addition_Chains.html#bin_pow_scheme_length1">bin_pow_scheme_length1</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#bin_pow_scheme">bin_pow_scheme</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.fib.html#bits">bits</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#bitsP">bitsP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#bits_cat">bits_cat</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#bodd">bodd</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boddIsPR">boddIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#bool_eq_iff">bool_eq_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="additions.Monoid_def.html#bool_and_binop">bool_and_binop</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide_eq_false">bool_decide_eq_false</a> [lemma, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide_eq_true">bool_decide_eq_true</a> [lemma, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide">bool_decide</a> [definition, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputation">boundComputation</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputationIsPR">boundComputationIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputationMonotone">boundComputationMonotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded">Bounded</a> [section, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#boundedCheck">boundedCheck</a> [lemma, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#boundedLT">boundedLT</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch">boundedSearch</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearchHelp">boundedSearchHelp</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch1">boundedSearch1</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch2">boundedSearch2</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#boundedSearch3">boundedSearch3</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#boundedSearch4">boundedSearch4</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant">BoundedVariant</a> [record, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#bounded_by">bounded_by</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#bounded_transitionS">bounded_transitionS</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#bounded_transitionS">bounded_transitionS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m_decrease">Bounded.m_decrease</a> [variable, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundMakeTrace">boundMakeTrace</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundSearchIsPR">boundSearchIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp">boundSubFormulaHelp</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp1">boundSubFormulaHelp1</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp2">boundSubFormulaHelp2</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubTerm">boundSubTerm</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubTerms">boundSubTerms</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="Goedel.goedel2.html#box">box</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B">building_ordering_function_by_induction.B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction">building_ordering_function_by_induction</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.g_1">building_ordering_function_1.g_1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta_B">building_ordering_function_1.beta_fixed.beta_B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta">building_ordering_function_1.beta_fixed.beta</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed">building_ordering_function_1.beta_fixed</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.H_B">building_ordering_function_1.H_B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B">building_ordering_function_1.B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1">building_ordering_function_1</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Build_OF">Build_OF</a> [constructor, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_iota_from'">but_last_iota_from'</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_app">but_last_app</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_shift'">but_last_shift'</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_interval">but_last_interval</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_iota_from">but_last_iota_from</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last">but_last</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GHydra.html#bVar">bVar</a> [instance, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#BVar:28">BVar:28</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#B_termination">B_termination</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#b'':45">b'':45</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1023">b':1023</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1151">b':1151</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#b':118">b':118</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1280">b':1280</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1285">b':1285</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1288">b':1288</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1291">b':1291</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1294">b':1294</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1298">b':1298</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#b':13">b':13</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1302">b':1302</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1306">b':1306</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1310">b':1310</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1316">b':1316</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':132">b':132</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1336">b':1336</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1382">b':1382</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1392">b':1392</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1420">b':1420</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1426">b':1426</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1434">b':1434</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1498">b':1498</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1506">b':1506</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1549">b':1549</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1583">b':1583</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#b':16">b':16</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1607">b':1607</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1616">b':1616</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1618">b':1618</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1625">b':1625</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1627">b':1627</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1633">b':1633</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1689">b':1689</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':1695">b':1695</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b':175">b':175</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#b:2022">b:2022</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#b:2031">b:2031</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:2038">b:2038</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:2048">b:2048</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:2051">b:2051</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#b:2074">b:2074</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#b:49">b:49</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#b:49">b:49</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#b:49">b:49</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#b:49">b:49</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Ackermann.code.html#b:49">b:49</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#b:49">b:49</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#b:490">b:490</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#b:491">b:491</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#b:493">b:493</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#b:495">b:495</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Ackermann.codeSubTerm.html#b:50">b:50</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#b:50">b:50</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#b:50">b:50</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
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+<a href="additions.More_on_positive.html#b:51">b:51</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#b:51">b:51</a> [binder, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Ackermann.code.html#B:51">B:51</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
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+<a href="hydras.Prelude.More_Arith.html#b:53">b:53</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
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+<a href="hydras.solutions_exercises.MultisetWf.html#b:53">b:53</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#b:53">b:53</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="hydras.Prelude.DecPreOrder_Instances.html#B:9">B:9</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.extEqualNat.html#b:9">b:9</a> [binder, in <a href="hydras.Ackermann.extEqualNat.html">hydras.Ackermann.extEqualNat</a>]<br/>
+<a href="hydras.rpo.dickson.html#b:9">b:9</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#b:9">b:9</a> [binder, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#b:90">b:90</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:903">b:903</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#b:909">b:909</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Gamma0.T2.html#b:91">b:91</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#b:91">b:91</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#b:91">b:91</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:911">b:911</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#b:912">b:912</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#b:92">b:92</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#b:92">b:92</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#b:92">b:92</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:924">b:924</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:926">b:926</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:928">b:928</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.PRrepresentable.html#B:93">B:93</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#b:93">b:93</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#b:93">b:93</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:930">b:930</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:932">b:932</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#b:936">b:936</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#b:939">b:939</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:939">b:939</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#b:94">b:94</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#b:94">b:94</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#b:94">b:94</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:945">b:945</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#b:95">b:95</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#b:95">b:95</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:954">b:954</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:957">b:957</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:959">b:959</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#b:96">b:96</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:961">b:961</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:963">b:963</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:965">b:965</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#b:97">b:97</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#B:97">B:97</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#b:98">b:98</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#b:98">b:98</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:989">b:989</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#b:99">b:99</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:993">b:993</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#b:999">b:999</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="global_C"></a><h2>C </h2>
+<a href="hydras.Ackermann.cPair.html#callIsPR">callIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#call_lt_wf">call_lt_wf</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#call_lt">call_lt</a> [definition, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#canon">canon</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon">canon</a> [definition, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon">Canon</a> [definition, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon">canon</a> [definition, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html">Canon</a> [library]<br/>
+<a href="gaia_hydras.GCanon.html#canonS">canonS</a> [abbreviation, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonSSn">CanonSSn</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonSSn">canonSSn</a> [abbreviation, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_rel_rounds">canonS_rel_rounds</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota">canonS_iota</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_i">canonS_iota_i</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_final">canonS_iota_final</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_6">canonS_iota_6</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_5">canonS_iota_5</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_4">canonS_iota_4</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_ocons_succE">canonS_ocons_succE</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_zero_inv">canonS_zero_inv</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_phi0_succE">canonS_phi0_succE</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_cons_not_zero">canonS_cons_not_zero</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_lt">canonS_lt</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_LE">canonS_LE</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_phi0_Succ">CanonS_phi0_Succ</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_lt">CanonS_lt</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_phi0_lim">CanonS_phi0_lim</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_LE">canonS_LE</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_mono">canonS_limit_mono</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_lub">canonS_limit_lub</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_strong">canonS_limit_strong</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_cons_not_zero">canonS_cons_not_zero</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_LT">canonS_LT</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_zero_inv">canonS_zero_inv</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_cons_succ_eqn2">canonS_cons_succ_eqn2</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_phi0_succ_eqn">canonS_phi0_succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_succ">canonS_succ</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_lim2">canonS_lim2</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_lim1">canonS_lim1</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#CanonS_Phi0_Succ_eqn">CanonS_Phi0_Succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#CanonS_plus_1">CanonS_plus_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#Canon_plus">Canon_plus</a> [definition, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Canon_plus">Canon_plus</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_of">canon_limit_of</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_mono">canon_limit_mono</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_strong">canon_limit_strong</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim3">canon_lim3</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim2">canon_lim2</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_tail">canon_tail</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim1">canon_lim1</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_SSn_zero">canon_SSn_zero</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lt">canon_lt</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_succ">canon_succ</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#canon_not_null">canon_not_null</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_of_limit_not_null">Canon_of_limit_not_null</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_lt">Canon_lt</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_Omega">Canon_Omega</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_Succ">Canon_Succ</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_limit_mono">canon_limit_mono</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_limit_strong">canon_limit_strong</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lt">canon_lt</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_LT">canon_LT</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_SSn_zero">canon_SSn_zero</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_succ">canon_succ</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim3">canon_lim3</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim2">canon_lim2</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim1">canon_lim1</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_tail">canon_tail</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_zero">canon_zero</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_examples">Canon_examples</a> [section, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_first_step_lim">Canon_plus_first_step_lim</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_first_step">Canon_plus_first_step</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_mono1">Canon_mono1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_inv">Canon_plus_inv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus">Canon_plus</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon0_lt">canon0_lt</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon0_phi0_succE">canon0_phi0_succE</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_LT">canon0_LT</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_cons_succ_eqn2">canon0_cons_succ_eqn2</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_phi0_succ_eqn">canon0_phi0_succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_succ">canon0_succ</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_lim2">canon0_lim2</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal">CantorOrdinal</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.addC_CE">CantorOrdinal.addC_CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.AddLocalNotation">CantorOrdinal.AddLocalNotation</a> [section, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl3">CantorOrdinal.add_simpl3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_normal">CantorOrdinal.add_normal</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le4">CantorOrdinal.add_le4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le3">CantorOrdinal.add_le3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_inj">CantorOrdinal.add_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le2">CantorOrdinal.add_le2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le1">CantorOrdinal.add_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl2">CantorOrdinal.add_simpl2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl1">CantorOrdinal.add_simpl1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_to_cons">CantorOrdinal.add_to_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_fin_omega">CantorOrdinal.add_fin_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_int">CantorOrdinal.add_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add1Nfin">CantorOrdinal.add1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr4">CantorOrdinal.ap_pr4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr3">CantorOrdinal.ap_pr3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr2CE">CantorOrdinal.ap_pr2CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr2">CantorOrdinal.ap_pr2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr1">CantorOrdinal.ap_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr0">CantorOrdinal.ap_pr0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantorCE2">CantorOrdinal.cantorCE2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantorCE3">CantorOrdinal.cantorCE3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_CE1">CantorOrdinal.cantor_CE1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_unique">CantorOrdinal.cantor_unique</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_exists">CantorOrdinal.cantor_exists</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Eq">CantorOrdinal.CompareT1Eq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Gt">CantorOrdinal.CompareT1Gt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Lt">CantorOrdinal.CompareT1Lt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.compare_T1">CantorOrdinal.compare_T1</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons">CantorOrdinal.cons</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.div_by_omega_pr">CantorOrdinal.div_by_omega_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_eq1">CantorOrdinal.expF_eq1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_eq0">CantorOrdinal.expF_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_1n">CantorOrdinal.expF_1n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_mul">CantorOrdinal.expF_mul</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_add">CantorOrdinal.expF_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_add">CantorOrdinal.expO_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_eq1">CantorOrdinal.expO_eq1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_eq0">CantorOrdinal.expO_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_1n">CantorOrdinal.expO_1n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_n0">CantorOrdinal.expO_n0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_mul1">CantorOrdinal.expO_mul1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx_nat">CantorOrdinal.expx_nat</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx_pnat">CantorOrdinal.expx_pnat</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx0">CantorOrdinal.expx0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx1">CantorOrdinal.expx1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx1CE">CantorOrdinal.expx1CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_prod">CantorOrdinal.exp_prod</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_sum">CantorOrdinal.exp_sum</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_FO3">CantorOrdinal.exp_FO3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_FO2">CantorOrdinal.exp_FO2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_FO1">CantorOrdinal.exp_FO1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_FO">CantorOrdinal.exp_FO</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_consCE1">CantorOrdinal.exp_consCE1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_int">CantorOrdinal.exp_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_eq1">CantorOrdinal.exp_eq1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_eq0">CantorOrdinal.exp_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_O">CantorOrdinal.exp_O</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp_F">CantorOrdinal.exp_F</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp0nz">CantorOrdinal.exp0nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp00">CantorOrdinal.exp00</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp1x">CantorOrdinal.exp1x</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.exp2omega">CantorOrdinal.exp2omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ex_middle_pick">CantorOrdinal.ex_middle_pick</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ex_middle">CantorOrdinal.ex_middle</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.fincP">CantorOrdinal.fincP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.fooCE">CantorOrdinal.fooCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.head_lt_cons">CantorOrdinal.head_lt_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.le_succ_succE">CantorOrdinal.le_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.le_succ_succ">CantorOrdinal.le_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_prop">CantorOrdinal.limit_prop</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_fct">CantorOrdinal.limit_fct</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_lub">CantorOrdinal.limit_lub</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_unique">CantorOrdinal.limit_unique</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of">CantorOrdinal.limit_of</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE3">CantorOrdinal.limit_CE3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE2">CantorOrdinal.limit_CE2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE1">CantorOrdinal.limit_CE1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_unique2">CantorOrdinal.limit_unique2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_unique1">CantorOrdinal.limit_unique1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v2">CantorOrdinal.limit_v2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v1">CantorOrdinal.limit_v1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_pr">CantorOrdinal.limit_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_pr1">CantorOrdinal.limit_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit1">CantorOrdinal.limit1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit2">CantorOrdinal.limit2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit3">CantorOrdinal.limit3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.LT">CantorOrdinal.LT</a> [abbreviation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_le_4">CantorOrdinal.lt_succ_le_4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_le_3">CantorOrdinal.lt_succ_le_3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_le_2">CantorOrdinal.lt_succ_le_2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_le_1">CantorOrdinal.lt_succ_le_1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_succE">CantorOrdinal.lt_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ2CE">CantorOrdinal.lt_succ2CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_succ_succ">CantorOrdinal.lt_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.lt_not_wf">CantorOrdinal.lt_not_wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.minus_le">CantorOrdinal.minus_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.minus_lt">CantorOrdinal.minus_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.min_exists">CantorOrdinal.min_exists</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mulA">CantorOrdinal.mulA</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mulnf0">CantorOrdinal.mulnf0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_omega_pr3">CantorOrdinal.mul_omega_pr3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_omega2_pr1">CantorOrdinal.mul_omega2_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_omega_pr1">CantorOrdinal.mul_omega_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_sum_omega">CantorOrdinal.mul_sum_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_omega_limit">CantorOrdinal.mul_omega_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_int_limit">CantorOrdinal.mul_int_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_fin_omega">CantorOrdinal.mul_fin_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_succ">CantorOrdinal.mul_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_distr">CantorOrdinal.mul_distr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_phi0">CantorOrdinal.mul_phi0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_int">CantorOrdinal.mul_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_na">CantorOrdinal.mul_na</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_exp">CantorOrdinal.nf_exp</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_expF">CantorOrdinal.nf_expF</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_expO">CantorOrdinal.nf_expO</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_rev_prod">CantorOrdinal.nf_rev_prod</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_rev_sum">CantorOrdinal.nf_rev_sum</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_rev_unique">CantorOrdinal.nf_rev_unique</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_rev">CantorOrdinal.nf_rev</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_revCE">CantorOrdinal.nf_revCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_div_by_omega">CantorOrdinal.nf_div_by_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_mul">CantorOrdinal.nf_mul</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_toNF">CantorOrdinal.nf_toNF</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_omega_plus_n">CantorOrdinal.nf_omega_plus_n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_sub">CantorOrdinal.nf_sub</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_add">CantorOrdinal.nf_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_pred">CantorOrdinal.nf_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_succ">CantorOrdinal.nf_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_log">CantorOrdinal.nf_log</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_phi0">CantorOrdinal.nf_phi0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_finite">CantorOrdinal.nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_omega">CantorOrdinal.nf_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_split">CantorOrdinal.nf_split</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_Wf'">CantorOrdinal.nf_Wf'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.nf_Wf">CantorOrdinal.nf_Wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal">CantorOrdinal.normal</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal_compose">CantorOrdinal.normal_compose</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal_limit">CantorOrdinal.normal_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.normal_id">CantorOrdinal.normal_id</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_plus_n">CantorOrdinal.omega_plus_n</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_tower_unbounded">CantorOrdinal.omega_tower_unbounded</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_tower_nf">CantorOrdinal.omega_tower_nf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_tower">CantorOrdinal.omega_tower</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.omega_minus_one">CantorOrdinal.omega_minus_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.one">CantorOrdinal.one</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0">CantorOrdinal.phi0</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0_log">CantorOrdinal.phi0_log</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0_lt1">CantorOrdinal.phi0_lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0_le">CantorOrdinal.phi0_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi0_lt">CantorOrdinal.phi0_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi1">CantorOrdinal.phi1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi2">CantorOrdinal.phi2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.phi3">CantorOrdinal.phi3</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.plus_int_Ox">CantorOrdinal.plus_int_Ox</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pow_mon3">CantorOrdinal.pow_mon3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pow_mon2">CantorOrdinal.pow_mon2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pow_mon1">CantorOrdinal.pow_mon1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pow_omega">CantorOrdinal.pow_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_succ_CE">CantorOrdinal.pred_succ_CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_succ">CantorOrdinal.pred_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_lt">CantorOrdinal.pred_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_le">CantorOrdinal.pred_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_add">CantorOrdinal.split_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_limit1">CantorOrdinal.split_limit1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_le">CantorOrdinal.split_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_pred">CantorOrdinal.split_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_succ">CantorOrdinal.split_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_finite">CantorOrdinal.split_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_is_succ">CantorOrdinal.split_is_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_limit">CantorOrdinal.split_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1rCE">CantorOrdinal.sub_pr1rCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_nzCE">CantorOrdinal.sub_nzCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_nz">CantorOrdinal.sub_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1r">CantorOrdinal.sub_pr1r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1CE">CantorOrdinal.sub_pr1CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1">CantorOrdinal.sub_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr">CantorOrdinal.sub_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_le1">CantorOrdinal.sub_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_int">CantorOrdinal.sub_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_1bCE">CantorOrdinal.sub_1bCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_1aCE">CantorOrdinal.sub_1aCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1a">CantorOrdinal.sub1a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1b">CantorOrdinal.sub1b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1Nfin">CantorOrdinal.sub1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_is_add_one">CantorOrdinal.succ_is_add_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_injCE">CantorOrdinal.succ_injCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_inj">CantorOrdinal.succ_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_p1">CantorOrdinal.succ_p1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_predCE">CantorOrdinal.succ_predCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_pred">CantorOrdinal.succ_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_lt">CantorOrdinal.succ_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup">CantorOrdinal.sup</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_limit">CantorOrdinal.sup_Oalpha_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_succ">CantorOrdinal.sup_Oalpha_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_zero">CantorOrdinal.sup_Oalpha_zero</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_unique">CantorOrdinal.sup_unique</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.tail_lt_cons">CantorOrdinal.tail_lt_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf">CantorOrdinal.Tf</a> [abbreviation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF">CantorOrdinal.toNF</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_pred">CantorOrdinal.toNF_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_succ">CantorOrdinal.toNF_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_ex2">CantorOrdinal.toNF_ex2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_ex1">CantorOrdinal.toNF_ex1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_mon">CantorOrdinal.toNF_mon</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_nf">CantorOrdinal.toNF_nf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_nz">CantorOrdinal.toNF_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1">CantorOrdinal.T1</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add">CantorOrdinal.T1add</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addA">CantorOrdinal.T1addA</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addn0">CantorOrdinal.T1addn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addS">CantorOrdinal.T1addS</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add_eq0">CantorOrdinal.T1add_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add0n">CantorOrdinal.T1add0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap">CantorOrdinal.T1ap</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap_phi0">CantorOrdinal.T1ap_phi0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1bad">CantorOrdinal.T1bad</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1div_by_omega">CantorOrdinal.T1div_by_omega</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq">CantorOrdinal.T1eq</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eqE">CantorOrdinal.T1eqE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eqP">CantorOrdinal.T1eqP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq_mul2l">CantorOrdinal.T1eq_mul2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq_add2l">CantorOrdinal.T1eq_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq_le">CantorOrdinal.T1eq_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1exp">CantorOrdinal.T1exp</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite">CantorOrdinal.T1finite</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite_succ">CantorOrdinal.T1finite_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite1">CantorOrdinal.T1finite1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite2">CantorOrdinal.T1finite2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1finite2CE">CantorOrdinal.T1finite2CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1F_inj">CantorOrdinal.T1F_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GeqNotGtn">CantorOrdinal.T1GeqNotGtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GeqNotLtn">CantorOrdinal.T1GeqNotLtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ge1">CantorOrdinal.T1ge1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GtnNotLeq">CantorOrdinal.T1GtnNotLeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1is_succ">CantorOrdinal.T1is_succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le">CantorOrdinal.T1le</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1leNgt">CantorOrdinal.T1leNgt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lenn">CantorOrdinal.T1lenn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1len0">CantorOrdinal.T1len0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1leP">CantorOrdinal.T1leP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1leq_xor_gtn">CantorOrdinal.T1leq_xor_gtn</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_pmull">CantorOrdinal.T1le_pmull</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_mul">CantorOrdinal.T1le_mul</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_mulCE">CantorOrdinal.T1le_mulCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_pmulrl">CantorOrdinal.T1le_pmulrl</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_pmulrCE">CantorOrdinal.T1le_pmulrCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_pmulr">CantorOrdinal.T1le_pmulr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_mul2r">CantorOrdinal.T1le_mul2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_mul2l">CantorOrdinal.T1le_mul2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_add2r">CantorOrdinal.T1le_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_add2l">CantorOrdinal.T1le_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_cons_le">CantorOrdinal.T1le_cons_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_trans">CantorOrdinal.T1le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_lt_trans">CantorOrdinal.T1le_lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_total">CantorOrdinal.T1le_total</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_consE">CantorOrdinal.T1le_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le_eqVlt">CantorOrdinal.T1le_eqVlt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1le0n">CantorOrdinal.T1le0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1limit">CantorOrdinal.T1limit</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log">CantorOrdinal.T1log</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log_exp2">CantorOrdinal.T1log_exp2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log_exp1">CantorOrdinal.T1log_exp1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log_exp0">CantorOrdinal.T1log_exp0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log_prod">CantorOrdinal.T1log_prod</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1log_phi0">CantorOrdinal.T1log_phi0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt">CantorOrdinal.T1lt</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltgtP">CantorOrdinal.T1ltgtP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltNge">CantorOrdinal.T1ltNge</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltnn">CantorOrdinal.T1ltnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1LtnNotGeq">CantorOrdinal.T1LtnNotGeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltn_xor_geq">CantorOrdinal.T1ltn_xor_geq</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltn0">CantorOrdinal.T1ltn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltP">CantorOrdinal.T1ltP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltW">CantorOrdinal.T1ltW</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_mul2r">CantorOrdinal.T1lt_mul2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_mul2l">CantorOrdinal.T1lt_mul2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_add2r">CantorOrdinal.T1lt_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_add2l">CantorOrdinal.T1lt_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_cons_le">CantorOrdinal.T1lt_cons_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_le_trans">CantorOrdinal.T1lt_le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_trans">CantorOrdinal.T1lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_anti">CantorOrdinal.T1lt_anti</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_trichotomy">CantorOrdinal.T1lt_trichotomy</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_neAle">CantorOrdinal.T1lt_neAle</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_ne'">CantorOrdinal.T1lt_ne'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_ne">CantorOrdinal.T1lt_ne</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt0n">CantorOrdinal.T1lt0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt1">CantorOrdinal.T1lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul">CantorOrdinal.T1mul</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1muln0">CantorOrdinal.T1muln0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1muln1">CantorOrdinal.T1muln1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1muln1_CE">CantorOrdinal.T1muln1_CE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul_omega">CantorOrdinal.T1mul_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul_eq1">CantorOrdinal.T1mul_eq1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul_eq0">CantorOrdinal.T1mul_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul0n">CantorOrdinal.T1mul0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul1n">CantorOrdinal.T1mul1n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1mul1nCE">CantorOrdinal.T1mul1nCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nat">CantorOrdinal.T1nat</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nat_inc">CantorOrdinal.T1nat_inc</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf">CantorOrdinal.T1nf</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nfCE">CantorOrdinal.T1nfCE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_rect">CantorOrdinal.T1nf_rect</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_finite">CantorOrdinal.T1nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_finite1">CantorOrdinal.T1nf_finite1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_consb">CantorOrdinal.T1nf_consb</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_consa">CantorOrdinal.T1nf_consa</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_cons_cons">CantorOrdinal.T1nf_cons_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1nf_cons0">CantorOrdinal.T1nf_cons0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1omega">CantorOrdinal.T1omega</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1phi0_zero'">CantorOrdinal.T1phi0_zero'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1phi0_zero">CantorOrdinal.T1phi0_zero</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1pred">CantorOrdinal.T1pred</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1split">CantorOrdinal.T1split</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1sub">CantorOrdinal.T1sub</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1subnn">CantorOrdinal.T1subnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1subn0">CantorOrdinal.T1subn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1sub0">CantorOrdinal.T1sub0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ">CantorOrdinal.T1succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1succ_nat">CantorOrdinal.T1succ_nat</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1transfinite_induction_Q">CantorOrdinal.T1transfinite_induction_Q</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1transfinite_induction">CantorOrdinal.T1transfinite_induction</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1_eqType">CantorOrdinal.T1_eqType</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1_eqMixin">CantorOrdinal.T1_eqMixin</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero">CantorOrdinal.zero</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5">_ ^ _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f">_ * _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x">_ - _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d">_ + _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&gt;'_x">_ &gt; _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#81970e39e8d1381cef754a4956da95e7">_ &gt;= _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9">_ &lt;= _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x">_ &lt; _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220">\F _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Cantor_normal_form">Cantor_normal_form</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#cases_for_mult">cases_for_mult</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#case_Eq">case_Eq</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#case_gt">case_gt</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#case_lt">case_lt</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Ackermann.code.html#cf:11">cf:11</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#cf:14">cf:14</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#cf:17">cf:17</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="Goedel.codeSysPrf.html#cf:2">cf:2</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#cf:2">cf:2</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="additions.Addition_Chains.html#chain">chain</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#chain_gen_OK">chain_gen_OK</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#chain_apply">chain_apply</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Trace_exercise.html#chain_trace">chain_trace</a> [definition, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="additions.Euclidean_Chains.html#chain_gen_OK">chain_gen_OK</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#chain_type">chain_type</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_generator">chain_generator</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_correct">chain_correct</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_correct_nat">chain_correct_nat</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_apply">chain_apply</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_execute">chain_execute</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#chain_length">chain_length</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.extEqualNat.html#charFunction">charFunction</a> [definition, in <a href="hydras.Ackermann.extEqualNat.html">hydras.Ackermann.extEqualNat</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrf">checkPrf</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html">checkPrf</a> [library]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfAXM">checkPrfAXM</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfAXMIsPR">checkPrfAXMIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfCorrect1">checkPrfCorrect1</a> [lemma, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfCorrect2">checkPrfCorrect2</a> [lemma, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfCP">checkPrfCP</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfCPIsPR">checkPrfCPIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQnIsPR">checkPrfEQnIsPR</a> [lemma, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ1">checkPrfEQ1</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ1IsPR">checkPrfEQ1IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ2">checkPrfEQ2</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ2IsPR">checkPrfEQ2IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ3">checkPrfEQ3</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ3IsPR">checkPrfEQ3IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ4">checkPrfEQ4</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ4IsPR">checkPrfEQ4IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ5">checkPrfEQ5</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ5IsPR">checkPrfEQ5IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA1">checkPrfFA1</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA1IsPR">checkPrfFA1IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA2">checkPrfFA2</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA2IsPR">checkPrfFA2IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA3">checkPrfFA3</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA3IsPR">checkPrfFA3IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfGEN">checkPrfGEN</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfGENIsPR">checkPrfGENIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfHelp">checkPrfHelp</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfHelpIsPR">checkPrfHelpIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP1">checkPrfIMP1</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP1IsPR">checkPrfIMP1IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP2">checkPrfIMP2</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP2IsPR">checkPrfIMP2IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIsPR">checkPrfIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfMP">checkPrfMP</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfMPIsPR">checkPrfMPIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#checkSubFormulaTrace">checkSubFormulaTrace</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#checkTraceCorrect">checkTraceCorrect</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#checkTraceIsPR">checkTraceIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Prf">Check_Proof.Prf</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedFormula">Check_Proof.wellFormedFormula</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedTerm">Check_Proof.wellFormedTerm</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula">Check_Proof.Formula</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Terms">Check_Proof.Terms</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Term">Check_Proof.Term</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeRInj">Check_Proof.codeRInj</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeFInj">Check_Proof.codeFInj</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityRIsCorrect2">Check_Proof.codeArityRIsCorrect2</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityRIsCorrect1">Check_Proof.codeArityRIsCorrect1</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityFIsCorrect2">Check_Proof.codeArityFIsCorrect2</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityFIsCorrect1">Check_Proof.codeArityFIsCorrect1</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR">Check_Proof.codeArityR</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF">Check_Proof.codeArityF</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.L">Check_Proof.L</a> [variable, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof">Check_Proof</a> [section, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#ch:345">ch:345</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#ch:346">ch:346</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#ch:349">ch:349</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#ch:350">ch:350</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#classic_battle">classic_battle</a> [definition, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#classification">classification</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.Addition_Chains.html#clog2">clog2</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#close">close</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close">close</a> [section, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#closed">closed</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Closed">Closed</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#closedNatToTerm">closedNatToTerm</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#closedNatToTerm">closedNatToTerm</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.NN.html#closedNN">closedNN</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#closedNN1">closedNN1</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PA.html#closedPA">closedPA</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#closedPA1">closedPA1</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#ClosedSystem">ClosedSystem</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#closedT">closedT</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Closed_Cr">Closed_Cr</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#closeFrom">closeFrom</a> [definition, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#closeList">closeList</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.codeFormula">close.codeFormula</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.Formula">close.Formula</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.L">close.L</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.rpo.closure.html">closure</a> [library]<br/>
+<a href="hydras.Ackermann.code.html#cl:19">cl:19</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="Goedel.codeSysPrf.html#cL:4">cL:4</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#cL:4">cL:4</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#cmpA:3">cmpA:3</a> [binder, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#cmpA:3">cmpA:3</a> [binder, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#cmpA:4">cmpA:4</a> [binder, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#cmpB:7">cmpB:7</a> [binder, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#cmpB:7">cmpB:7</a> [binder, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#cmp:13">cmp:13</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#cmp:14">cmp:14</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#cmp:3">cmp:3</a> [binder, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#cmp:3">cmp:3</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#cmp:4">cmp:4</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#cmp:6">cmp:6</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#cmp:9">cmp:9</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="additions.fib.html#cm:106">cm:106</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#cm:113">cm:113</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#cm:124">cm:124</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#cnf">cnf</a> [projection, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf">cnf</a> [projection, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.CNF.html">CNF</a> [library]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_of_epsilon0">cnf_of_epsilon0</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_lt_epsilon0">cnf_lt_epsilon0</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_exists_unique">cnf_exists_unique</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_unicity">cnf_unicity</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_exists">cnf_exists</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_plus">cnf_plus</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_plus2">cnf_plus2</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_plus1">cnf_plus1</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_eq">cnf_eq</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_head_eq">cnf_head_eq</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_of_ap">cnf_of_ap</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#cnf_t">cnf_t</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_Cons">cnf_Cons</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_Omega_term">cnf_Omega_term</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_Succ">cnf_Succ</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_phi0">cnf_phi0</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_rw">cnf_rw</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_ok">cnf_ok</a> [projection, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.AM.html#code">code</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.code.html">code</a> [library]<br/>
+<a href="hydras.Ackermann.code.html#codeAnd">codeAnd</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeAndCorrect">codeAndCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeAndIsPR">codeAndIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeApp">codeApp</a> [definition, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeAppCorrect">codeAppCorrect</a> [lemma, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeAppIsPR">codeAppIsPR</a> [instance, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#codeArityFIsPR:6">codeArityFIsPR:6</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeArityFIsPR:7">codeArityFIsPR:7</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeArityFIsPR:7">codeArityFIsPR:7</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNFIsCorrect1">codeArityLNNFIsCorrect1</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNFIsCorrect2">codeArityLNNFIsCorrect2</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNR">codeArityLNNR</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect1">codeArityLNNRIsCorrect1</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNRIsCorrect2">codeArityLNNRIsCorrect2</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR">codeArityLNNRIsPR</a> [instance, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTF">codeArityLNTF</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect1">codeArityLNTFIsCorrect1</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTFIsCorrect2">codeArityLNTFIsCorrect2</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR">codeArityLNTFIsPR</a> [instance, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTR">codeArityLNTR</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect1">codeArityLNTRIsCorrect1</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTRIsCorrect2">codeArityLNTRIsCorrect2</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR">codeArityLNTRIsPR</a> [instance, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeArityRIsPR:13">codeArityRIsPR:13</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeArityRIsPR:13">codeArityRIsPR:13</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#codeArityRIsPR:181">codeArityRIsPR:181</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeAxmEqHelp">codeAxmEqHelp</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeAxmEqHelpIsPR">codeAxmEqHelpIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeClose">codeClose</a> [definition, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseCorrect">codeCloseCorrect</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseIsPR">codeCloseIsPR</a> [instance, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseList">codeCloseList</a> [definition, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseListCorrect">codeCloseListCorrect</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseListIsPR">codeCloseListIsPR</a> [instance, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeF">codeF</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeFInj">codeFInj</a> [projection, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="Goedel.goedel1.html#codeFNN">codeFNN</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeForall">codeForall</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeForallCorrect">codeForallCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html#codeForallIsPR">codeForallIsPR</a> [lemma, in <a href="hydras.Ackermann.prLogic.html">hydras.Ackermann.prLogic</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeFormula">codeFormula</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="Goedel.rosserPA.html#codeFormula">codeFormula</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#codeFormula">codeFormula</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeFormulaInj">codeFormulaInj</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html">codeFreeVar</a> [library]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormula">codeFreeVarFormula</a> [definition, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormulaCorrect">codeFreeVarFormulaCorrect</a> [lemma, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormulaIsPR">codeFreeVarFormulaIsPR</a> [instance, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormula">codeFreeVarListFormula</a> [definition, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormulaCorrect">codeFreeVarListFormulaCorrect</a> [lemma, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormulaIsPR">codeFreeVarListFormulaIsPR</a> [instance, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerm">codeFreeVarTerm</a> [definition, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermCorrect">codeFreeVarTermCorrect</a> [lemma, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermIsPR">codeFreeVarTermIsPR</a> [lemma, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTerms">codeFreeVarTerms</a> [definition, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermsCorrect">codeFreeVarTermsCorrect</a> [lemma, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermsIsPR">codeFreeVarTermsIsPR</a> [instance, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTerms">codeFreeVarTermTerms</a> [definition, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTermsIsPR">codeFreeVarTermTermsIsPR</a> [instance, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeIff">codeIff</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeIffCorrect">codeIffCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeIffIsPR">codeIffIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeImp">codeImp</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeImpCorrect">codeImpCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html#codeImpIsPR">codeImpIsPR</a> [instance, in <a href="hydras.Ackermann.prLogic.html">hydras.Ackermann.prLogic</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeIn">codeIn</a> [definition, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeInCorrect">codeInCorrect</a> [lemma, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchema">codeInductionSchema</a> [definition, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchemaCorrect1">codeInductionSchemaCorrect1</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchemaCorrect2">codeInductionSchemaCorrect2</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchemaCorrect3">codeInductionSchemaCorrect3</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchemaIsPR">codeInductionSchemaIsPR</a> [instance, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeInIsPR">codeInIsPR</a> [instance, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeLength">codeLength</a> [definition, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeLengthCorrect">codeLengthCorrect</a> [lemma, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeLengthIsPR">codeLengthIsPR</a> [instance, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#codeList">codeList</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.codeList.html">codeList</a> [library]<br/>
+<a href="hydras.Ackermann.cPair.html#codeListInj">codeListInj</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeListRemove">codeListRemove</a> [definition, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeListRemoveCorrect">codeListRemoveCorrect</a> [lemma, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeListRemoveIsPR">codeListRemoveIsPR</a> [instance, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNNRelation">codeLNNRelation</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNNRelationInj">codeLNNRelationInj</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNTFunction">codeLNTFunction</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNTFunctionInj">codeLNTFunctionInj</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNTRelation">codeLNTRelation</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeLNTRelationInj">codeLNTRelationInj</a> [lemma, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#codeNatToTerm">codeNatToTerm</a> [definition, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html">codeNatToTerm</a> [library]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#codeNatToTermCorrectLNN">codeNatToTermCorrectLNN</a> [lemma, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#codeNatToTermCorrectLNT">codeNatToTermCorrectLNT</a> [lemma, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#codeNatToTermIsPR">codeNatToTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeNewVar">codeNewVar</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeNewVarCorrect">codeNewVarCorrect</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeNewVarIsPR">codeNewVarIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeNoDup">codeNoDup</a> [definition, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeNoDupCorrect">codeNoDupCorrect</a> [lemma, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeNoDupIsPR">codeNoDupIsPR</a> [instance, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeNot">codeNot</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeNotCorrect">codeNotCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html#codeNotIsPR">codeNotIsPR</a> [instance, in <a href="hydras.Ackermann.prLogic.html">hydras.Ackermann.prLogic</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#codeNth">codeNth</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#codeNthCorrect">codeNthCorrect</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#codeNthIsPR">codeNthIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVarsCorrect">codeNVarsCorrect</a> [lemma, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars1">codeNVars1</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars1IsPR">codeNVars1IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars2">codeNVars2</a> [definition, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars2IsPR">codeNVars2IsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeOpen">codeOpen</a> [definition, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeOpenCorrect">codeOpenCorrect</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeOpenIsPR">codeOpenIsPR</a> [instance, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeOr">codeOr</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeOrCorrect">codeOrCorrect</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeOrIsPR">codeOrIsPR</a> [instance, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codePA">codePA</a> [definition, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html">codePA</a> [library]<br/>
+<a href="hydras.Ackermann.codePA.html#codePAcorrect1">codePAcorrect1</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codePAcorrect2">codePAcorrect2</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codePAcorrect3">codePAcorrect3</a> [lemma, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codePAIsPR">codePAIsPR</a> [instance, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.code.html#codePrf">codePrf</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="Goedel.rosserPA.html#codePrf">codePrf</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#codePrf">codePrf</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.code.html#codePrfInjAxm">codePrfInjAxm</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeR">codeR</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeRInj">codeRInj</a> [projection, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeSubFormula">codeSubFormula</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html">codeSubFormula</a> [library]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeSubFormulaCorrect">codeSubFormulaCorrect</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeSubFormulaIsPR">codeSubFormulaIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTerm">codeSubTerm</a> [definition, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html">codeSubTerm</a> [library]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermCorrect">codeSubTermCorrect</a> [lemma, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermIsPR">codeSubTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTerms">codeSubTerms</a> [definition, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermsCorrect">codeSubTermsCorrect</a> [lemma, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermsIsPR">codeSubTermsIsPR</a> [lemma, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermTerms">codeSubTermTerms</a> [definition, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermTermsIsPR">codeSubTermTermsIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPf">codeSysPf</a> [definition, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#codeSysPf">codeSysPf</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#codeSysPf">codeSysPf</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPfCorrect">codeSysPfCorrect</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#codeSysPfCorrect">codeSysPfCorrect</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#codeSysPfCorrect">codeSysPfCorrect</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrf">codeSysPrf</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrf">codeSysPrf</a> [definition, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#codeSysPrf">codeSysPrf</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrf">codeSysPrf</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.codeSysPrf.html">codeSysPrf</a> [library]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfCorrect1">codeSysPrfCorrect1</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfCorrect1">codeSysPrfCorrect1</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfCorrect1">codeSysPrfCorrect1</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfCorrect2">codeSysPrfCorrect2</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfCorrect2">codeSysPrfCorrect2</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#codeSysPrfCorrect2">codeSysPrfCorrect2</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfCorrect2">codeSysPrfCorrect2</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfCorrect3">codeSysPrfCorrect3</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfCorrect3">codeSysPrfCorrect3</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#codeSysPrfCorrect3">codeSysPrfCorrect3</a> [definition, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfCorrect3">codeSysPrfCorrect3</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfNCorrect1">codeSysPrfNCorrect1</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfNCorrect1">codeSysPrfNCorrect1</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfNCorrect1">codeSysPrfNCorrect1</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfNCorrect2">codeSysPrfNCorrect2</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfNCorrect2">codeSysPrfNCorrect2</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfNCorrect2">codeSysPrfNCorrect2</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfNCorrect3">codeSysPrfNCorrect3</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfNCorrect3">codeSysPrfNCorrect3</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfNCorrect3">codeSysPrfNCorrect3</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#codeSysPrfNot">codeSysPrfNot</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#codeSysPrfNot">codeSysPrfNot</a> [definition, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#codeSysPrfNot">codeSysPrfNot</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeTerm">codeTerm</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeTermFreeVar">codeTermFreeVar</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeTermInj">codeTermInj</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeTerms">codeTerms</a> [definition, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeTermsInj">codeTermsInj</a> [lemma, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Terms">Code_Free_Vars.Terms</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Term">Code_Free_Vars.Term</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.System">Code_Free_Vars.System</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Formulas">Code_Free_Vars.Formulas</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Formula">Code_Free_Vars.Formula</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.L">Code_Free_Vars.L</a> [variable, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars">Code_Free_Vars</a> [section, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Terms">Code_Substitute_Term.Terms</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Term">Code_Substitute_Term.Term</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.System">Code_Substitute_Term.System</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Formulas">Code_Substitute_Term.Formulas</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.Formula">Code_Substitute_Term.Formula</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term.L">Code_Substitute_Term.L</a> [variable, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term">Code_Substitute_Term</a> [section, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeTerm">Code_Substitute_Formula.codeTerm</a> [variable, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.codeFormula">Code_Substitute_Formula.codeFormula</a> [variable, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L">Code_Substitute_Formula.L</a> [variable, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula">Code_Substitute_Formula</a> [section, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef">Code_Term_Formula_Proof.codeTermFormDef</a> [section, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.SysPrf">Code_Term_Formula_Proof.SysPrf</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Prf">Code_Term_Formula_Proof.Prf</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Terms">Code_Term_Formula_Proof.Terms</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Term">Code_Term_Formula_Proof.Term</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.System">Code_Term_Formula_Proof.System</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formulas">Code_Term_Formula_Proof.Formulas</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula">Code_Term_Formula_Proof.Formula</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cR">Code_Term_Formula_Proof.LcodeDef.cR</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cF">Code_Term_Formula_Proof.LcodeDef.cF</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef">Code_Term_Formula_Proof.LcodeDef</a> [section, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L">Code_Term_Formula_Proof.L</a> [variable, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof">Code_Term_Formula_Proof</a> [section, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#code_nat_list.drop">code_nat_list.drop</a> [variable, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#code_nat_list">code_nat_list</a> [section, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.expressU2">code_SysPrf.LNN.expressU2</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.expressU1">code_SysPrf.LNN.expressU1</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.freeVarfU">code_SysPrf.LNN.freeVarfU</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.v0">code_SysPrf.LNN.v0</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.fU">code_SysPrf.LNN.fU</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.U">code_SysPrf.LNN.U</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.TextendsNN">code_SysPrf.LNN.TextendsNN</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN.T">code_SysPrf.LNN.T</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN">code_SysPrf.LNN</a> [section, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityRIsCorrect2">code_SysPrf.codeArityRIsCorrect2</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityRIsCorrect1">code_SysPrf.codeArityRIsCorrect1</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityFIsCorrect2">code_SysPrf.codeArityFIsCorrect2</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityFIsCorrect1">code_SysPrf.codeArityFIsCorrect1</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityR">code_SysPrf.codeArityR</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.codeArityF">code_SysPrf.codeArityF</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.L">code_SysPrf.L</a> [variable, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf">code_SysPrf</a> [section, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#Code_PA.codeFormulaInj">Code_PA.codeFormulaInj</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#Code_PA.codeFormula">Code_PA.codeFormula</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#Code_PA.codeTerm">Code_PA.codeTerm</a> [variable, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#Code_PA">Code_PA</a> [section, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#coeff_lt">coeff_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#coeff_lt">coeff_lt</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#comp">comp</a> [instance, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Comparable">Comparable</a> [section, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Comparable">Comparable</a> [record, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html">Comparable</a> [library]<br/>
+<a href="hydras.Prelude.Comparable.html#comparable_comp_spec">comparable_comp_spec</a> [projection, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#comparable_sto">comparable_sto</a> [projection, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Comparable0:15">Comparable0:15</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare">compare</a> [projection, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Compare">Compare</a> [record, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare">compare</a> [constructor, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Compare">Compare</a> [inductive, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:108">compareA:108</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:123">compareA:123</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:135">compareA:135</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:148">compareA:148</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:161">compareA:161</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:173">compareA:173</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:186">compareA:186</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:200">compareA:200</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:31">compareA:31</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:50">compareA:50</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:69">compareA:69</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:83">compareA:83</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareA:95">compareA:95</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:112">compareB:112</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:127">compareB:127</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:139">compareB:139</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:152">compareB:152</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:165">compareB:165</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:177">compareB:177</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:190">compareB:190</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:204">compareB:204</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:35">compareB:35</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:54">compareB:54</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:87">compareB:87</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#compareB:99">compareB:99</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_reflectR">compare_reflectR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_of_phi0">compare_of_phi0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_fin_rw">compare_fin_rw</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_lt_iff">compare_lt_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_lt_impl">compare_lt_impl</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_eq_iff">compare_eq_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_refl">compare_refl</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_rev">compare_rev</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_cons">compare_cons</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_T1">compare_T1</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#compare_omega2">compare_omega2</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_trans">compare_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_ge_iff">compare_ge_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_le_iff">compare_le_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_le_iff_refl">compare_le_iff_refl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_gt_not_lt">compare_gt_not_lt</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_lt_not_gt">compare_lt_not_gt</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_gt_trans">compare_gt_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_gt_irrefl">compare_gt_irrefl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_gt_iff">compare_gt_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_eq_trans">compare_eq_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_refl">compare_refl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_eq_iff">compare_eq_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_lt_irrefl">compare_lt_irrefl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_lt_trans">compare_lt_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare_lt_iff">compare_lt_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#compare_O">compare_O</a> [instance, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#compare_nat_nat">compare_nat_nat</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#compare_h2g">compare_h2g</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#compare_g2h">compare_g2h</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#compare_ref">compare_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#compare_mult">compare_mult</a> [instance, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#compare_plus">compare_plus</a> [instance, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#compare_fin">compare_fin</a> [instance, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#compare_E0">compare_E0</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#compare_nat">compare_nat</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_rw_gt">compare_rw_gt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_rw_eq">compare_rw_eq</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_rw_lt">compare_rw_lt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_Gt">compare_Gt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_Eq">compare_Eq</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_Lt">compare_Lt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_correct">compare_correct</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_reflect">compare_reflect</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_T2">compare_T2</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Trace_exercise.html#compare:2">compare:2</a> [binder, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#comparison_eq_dec">comparison_eq_dec</a> [instance, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="additions.Compatibility.html">Compatibility</a> [library]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815">Compat815</a> [module, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html">Compat815</a> [library]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.ind_0_1_SS">Compat815.ind_0_1_SS</a> [definition, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.le_not_lt">Compat815.le_not_lt</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.le_plus_r">Compat815.le_plus_r</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.le_lt_n_Sm">Compat815.le_lt_n_Sm</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.le_lt_or_eq">Compat815.le_lt_or_eq</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.le_n_0_eq">Compat815.le_n_0_eq</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.lt_not_le">Compat815.lt_not_le</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.lt_n_S">Compat815.lt_n_S</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.lt_S_n">Compat815.lt_S_n</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.lt_n_Sm_le">Compat815.lt_n_Sm_le</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.mult_O_le">Compat815.mult_O_le</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.n_SSn">Compat815.n_SSn</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.n_SSSn">Compat815.n_SSSn</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815.plus_Snm_nSm">Compat815.plus_Snm_nSm</a> [lemma, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#composeFunc">composeFunc</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#composeSigmaFormula">composeSigmaFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#composeSigmaRepresentable">composeSigmaRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_NIsPR">compose1_NIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_3IsPR">compose1_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_2IsPR">compose1_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_1IsPR">compose1_1IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2">compose2</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples">compose2Examples</a> [section, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.f">compose2Examples.f</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.f'">compose2Examples.f'</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.g">compose2Examples.g</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.g'">compose2Examples.g'</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples.h">compose2Examples.h</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2IsPR">compose2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_4IsPR">compose2_4IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_3IsPR">compose2_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_2IsPR">compose2_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_1IsPR">compose2_1IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2_3">compose2_3</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2_2">compose2_2</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2_1">compose2_1</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2_0">compose2_0</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_3IsPR">compose3_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_2IsPR">compose3_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_1IsPR">compose3_1IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose4_3IsPR">compose4_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose4_2IsPR">compose4_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition">Composition</a> [section, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.AM.html#CompositionProofs">CompositionProofs</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App">CompositionProofs.App</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.cn">CompositionProofs.App.cn</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.cp">CompositionProofs.App.cp</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.Hn">CompositionProofs.App.Hn</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.Hp">CompositionProofs.App.Hp</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.n">CompositionProofs.App.n</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App.p">CompositionProofs.App.p</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK">CompositionProofs.FFK</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.cp">CompositionProofs.FFK.cp</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.cq">CompositionProofs.FFK.cq</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.Hp">CompositionProofs.FFK.Hp</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.Hq">CompositionProofs.FFK.Hq</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.p">CompositionProofs.FFK.p</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK.q">CompositionProofs.FFK.q</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FK">CompositionProofs.FK</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FK.cn">CompositionProofs.FK.cn</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FK.Hn">CompositionProofs.FK.Hn</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FK.n">CompositionProofs.FK.n</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK">CompositionProofs.KFK</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.Hpr">CompositionProofs.KFK.Hpr</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.Hq">CompositionProofs.KFK.Hq</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.kpr">CompositionProofs.KFK.kpr</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.mq">CompositionProofs.KFK.mq</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.p">CompositionProofs.KFK.p</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.q">CompositionProofs.KFK.q</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK.r">CompositionProofs.KFK.r</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.f">Composition.f</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.g">Composition.g</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.h">Composition.h</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.k">Composition.k</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.t">Composition.t</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.u">Composition.u</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.v">Composition.v</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.x">Composition.x</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.y">Composition.y</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.z">Composition.z</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.Addition_Chains.html#computation">computation</a> [inductive, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#computation_equiv">computation_equiv</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#computation_eval_rw">computation_eval_rw</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#computation_eval">computation_eval</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#computation_execute">computation_execute</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#computation_length">computation_length</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#computeEvalStrongRecHelp">computeEvalStrongRecHelp</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#comp_res:96">comp_res:96</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="additions.Euclidean_Chains.html#comp_equiv_equivalence">comp_equiv_equivalence</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Comp_equiv">Comp_equiv</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#comp128">comp128</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="Goedel.goedel2.html#Con">Con</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="additions.AM.html#config">config</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#config_equiv">config_equiv</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#cons">cons</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Cons">Cons</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Consistent">Consistent</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="gaia_hydras.GPaths.html#constant_to_standard_path">constant_to_standard_path</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#constant_to_standard_path">constant_to_standard_path</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#constant_to_standard">constant_to_standard</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#constant_to_standard_0">constant_to_standard_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.delta">Constant_to_standard_Proof.delta</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.m">Constant_to_standard_Proof.m</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma">Constant_to_standard_Proof.gamma</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l">Constant_to_standard_Proof.l</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l_def">Constant_to_standard_Proof.l_def</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P">Constant_to_standard_Proof.P</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t">Constant_to_standard_Proof.t</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Hpa">Constant_to_standard_Proof.Hpa</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Hpos">Constant_to_standard_Proof.Hpos</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Halpha">Constant_to_standard_Proof.Halpha</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.n">Constant_to_standard_Proof.n</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.beta">Constant_to_standard_Proof.beta</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.alpha">Constant_to_standard_Proof.alpha</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof">Constant_to_standard_Proof</a> [section, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.fol.html#consTerms">consTerms</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="gaia_hydras.GPaths.html#const_path">const_path</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_LT'">const_pathS_LT'</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_eps_zero">const_pathS_eps_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_first_step">const_pathS_first_step</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_eps_LE_2">const_pathS_eps_LE_2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_eps_trans">const_pathS_eps_trans</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_trans">const_pathS_trans</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_inv_strong">const_pathS_inv_strong</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_inv">const_pathS_inv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_LE">const_pathS_LE</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_path_LT">const_path_LT</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_LT">const_pathS_LT</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_zero">const_pathS_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_nf">const_pathS_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_repeatR">const_pathS_repeatR</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_repeat">const_pathS_repeat</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS_eps">const_pathS_eps</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathSE">const_pathSE</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_path">const_path</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#const_pathS">const_pathS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#const0_NIsPR">const0_NIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#const1_NIsPR">const1_NIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#cons_def">cons_def</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#cons_nf">cons_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#cons_succ">cons_succ</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#cons_standard_path">cons_standard_path</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#cons_standard_pathS">cons_standard_pathS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#cons_lt_epsilon0">cons_lt_epsilon0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#cons_rw">cons_rw</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#continuous">continuous</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#contrad">contrad</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#contrad">contrad</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#contradiction">contradiction</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#contradiction">contradiction</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#contradiction">contradiction</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Contradiction">Contradiction</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness">Correctness</a> [section, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.AM.html#correctness_statement">correctness_statement</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#correctness_statement">correctness_statement</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html">Correctness_E0</a> [library]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.L">Correctness.L</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.P">Correctness.P</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.Q">Correctness.Q</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.R">Correctness.R</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.AM.html#correct_app">correct_app</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#correct_generator">correct_generator</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Cor12">Cor12</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12">Cor12</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_E0">Cor12_E0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_3">Cor12_3</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_2">Cor12_2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_1">Cor12_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable">Countable</a> [section, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html">Countable</a> [library]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#countable_members">countable_members</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#countable_not_Unbounded">countable_not_Unbounded</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#countable_segment_proper">countable_segment_proper</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_image">countable_image</a> [lemma, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_bij_funR">countable_bij_funR</a> [lemma, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_bij_fun">countable_bij_fun</a> [lemma, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_surj">countable_surj</a> [lemma, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection.g_bij">Countable.Countable_bijection.g_bij</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection.g">Countable.Countable_bijection.g</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection.B">Countable.Countable_bijection.B</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection.A">Countable.Countable_bijection.A</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection.V">Countable.Countable_bijection.V</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection">Countable.Countable_bijection</a> [section, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_seq_range">Countable.Countable_seq_range</a> [section, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Definitions">Countable.Definitions</a> [section, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Definitions.A">Countable.Definitions.A</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Definitions.U">Countable.Definitions.U</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.U">Countable.U</a> [variable, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.is_in_seq">Counter_Example.is_in_seq</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.Hwf">Counter_Example.Hwf</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.R">Counter_Example.R</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example">Counter_Example</a> [section, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#counter_ex">counter_ex</a> [lemma, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="additions.AM.html#count:9">count:9</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#CP">CP</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPair">cPair</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html">cPair</a> [library]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInjHelp">cPairInjHelp</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInj1">cPairInj1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInj2">cPairInj2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairIsPR">cPairIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLemma1">cPairLemma1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe1">cPairLe1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe1A">cPairLe1A</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe2">cPairLe2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe2A">cPairLe2A</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe3">cPairLe3</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLt1">cPairLt1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLt2">cPairLt2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi1">cPairPi1</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi1IsPR">cPairPi1IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi2">cPairPi2</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi2IsPR">cPairPi2IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections">cPairProjections</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjectionsHelp">cPairProjectionsHelp</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections1">cPairProjections1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections2">cPairProjections2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_Order">CPair_Order</a> [section, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_projections.searchXY">CPair_projections.searchXY</a> [variable, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_projections">CPair_projections</a> [section, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_Injectivity">CPair_Injectivity</a> [section, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="additions.Addition_Chains.html#cpower">cpower</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#cpower_pos">cpower_pos</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#cp1">cp1</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#cp1">cp1</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#cp1">cp1</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#cp2">cp2</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#cp2">cp2</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#cp2">cp2</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr">Cr</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html">Critical</a> [library]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_1_iff">Cr_1_iff</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_incl">Cr_incl</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_lt">Cr_lt</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos_iff">Cr_pos_iff</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos_inv">Cr_pos_inv</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero_AP">Cr_zero_AP</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero_inv">Cr_zero_inv</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos">Cr_pos</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero">Cr_zero</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_inv">Cr_inv</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_equation">Cr_equation</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_extensional">Cr_extensional</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_fun">Cr_fun</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.code.html#cr:12">cr:12</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#cr:15">cr:15</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#cr:18">cr:18</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="Goedel.codeSysPrf.html#cr:3">cr:3</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#cr:3">cr:3</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr:5">Cr:5</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#cst_correct">cst_correct</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#cst0_correct">cst0_correct</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.l">Cter_example.l</a> [variable, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.alpha">Cter_example.alpha</a> [variable, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example.cnf_lt_epsilon0_iff">Cter_example.cnf_lt_epsilon0_iff</a> [variable, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example">Cter_example</a> [section, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriple">cTriple</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleIsPR">cTripleIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi1">cTriplePi1</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi1IsPR">cTriplePi1IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi2">cTriplePi2</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi2IsPR">cTriplePi2IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi3">cTriplePi3</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi3IsPR">cTriplePi3IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj">cTripleProj</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj1">cTripleProj1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj2">cTripleProj2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj3">cTripleProj3</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.rpo.rpo.html#ctx:317">ctx:317</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#ctx:89">ctx:89</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#c'':46">c'':46</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1025">c':1025</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1153">c':1153</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1337">c':1337</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1383">c':1383</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1393">c':1393</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1421">c':1421</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1427">c':1427</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1435">c':1435</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1499">c':1499</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1507">c':1507</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1550">c':1550</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1584">c':1584</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1608">c':1608</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1617">c':1617</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1619">c':1619</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1620">c':1620</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1626">c':1626</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1628">c':1628</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1634">c':1634</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1690">c':1690</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':1696">c':1696</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#c':2">c':2</a> [binder, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="additions.AM.html#c':20">c':20</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':2065">c':2065</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':2079">c':2079</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':2084">c':2084</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#c':25">c':25</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Monoid_instances.html#c':26">c':26</a> [binder, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#c':28">c':28</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#c':303">c':303</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#c':34">c':34</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#c':42">c':42</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Euclidean_Chains.html#c':46">c':46</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c':544">c':544</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.AM.html#c':58">c':58</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#c':58">c':58</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#c':584">c':584</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Addition_Chains.html#c':81">c':81</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Gamma0.T2.html#c':84">c':84</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':872">c':872</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':909">c':909</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':917">c':917</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c':949">c':949</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Monoid_instances.html#c00">c00</a> [projection, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#c01">c01</a> [projection, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Addition_Chains.html#C1">C1</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html#c10">c10</a> [projection, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#c11">c11</a> [projection, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Fib2.html#c153">c153</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c1:1173">c1:1173</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#c1:1364">c1:1364</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c1:1817">c1:1817</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c1:470">c1:470</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c1:598">c1:598</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#c1:73">c1:73</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#c1:85">c1:85</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c2:1174">c2:1174</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="additions.AM.html#C31_7">C31_7</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c3:1175">c3:1175</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="additions.Addition_Chains.html#C87_ok'">C87_ok'</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#C87_ok">C87_ok</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
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+<a href="hydras.Ackermann.primRec.html#c:780">c:780</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:782">c:782</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:784">c:784</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:787">c:787</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#c:79">c:79</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:790">c:790</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:798">c:798</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#c:8">c:8</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#c:8">c:8</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Addition_Chains.html#c:80">c:80</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#c:804">c:804</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.AM.html#c:81">c:81</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#c:850">c:850</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:854">c:854</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:863">c:863</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:868">c:868</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#c:88">c:88</a> [binder, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#c:9">c:9</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#c:90">c:90</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:905">c:905</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.AM.html#c:91">c:91</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:913">c:913</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.AM.html#c:92">c:92</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Gamma0.T2.html#c:93">c:93</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:940">c:940</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:947">c:947</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#c:95">c:95</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c:995">c:995</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="global_D"></a><h2>D </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#daughters:7">daughters:7</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Schutte.Countable.html#DA:36">DA:36</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#DA:37">DA:37</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#DB:38">DB:38</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="Goedel.rosser.html#DecidableSet">DecidableSet</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.rpo.decidable_set.html">decidable_set</a> [library]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide">decide</a> [projection, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide">decide</a> [constructor, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="Goedel.rosserPA.html#decideAxioms">decideAxioms</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#decideAxioms">decideAxioms</a> [lemma, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#decide_hltE">decide_hltE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide_rel">decide_rel</a> [projection, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide_rel">decide_rel</a> [constructor, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Decision">Decision</a> [record, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Decision">Decision</a> [inductive, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp">decomp</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#decompose">decompose</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decompos2">decompos2</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp1">decomp1</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp2">decomp2</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html">DecPreOrder</a> [library]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html">DecPreOrder_Instances</a> [library]<br/>
+<a href="hydras.Prelude.WfVariant.html#decr">decr</a> [projection, in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#decr_seq">decr_seq</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#dec:20">dec:20</a> [binder, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#dec:33">dec:33</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#dec:42">dec:42</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#dec:66">dec:66</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#dec:77">dec:77</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#dec:90">dec:90</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html">Deduction</a> [library]<br/>
+<a href="hydras.Ackermann.Deduction.html#DeductionTheorem">DeductionTheorem</a> [lemma, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.list_incl">Deduction_Theorem.list_incl</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.SysPrf">Deduction_Theorem.SysPrf</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Prf">Deduction_Theorem.Prf</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Terms">Deduction_Theorem.Terms</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Term">Deduction_Theorem.Term</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.System">Deduction_Theorem.System</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formulas">Deduction_Theorem.Formulas</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formula">Deduction_Theorem.Formula</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.L">Deduction_Theorem.L</a> [variable, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem">Deduction_Theorem</a> [section, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions">Definitions</a> [section, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Definitions">Definitions</a> [section, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions">Definitions</a> [section, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.A">Definitions.A</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.A">Definitions.A</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.B">Definitions.B</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.ltA">Definitions.ltA</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.ltB">Definitions.ltB</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.mult">Definitions.mult</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.one">Definitions.one</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.pair2sig">Definitions.pair2sig</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.wfA">Definitions.wfA</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions.wfB">Definitions.wfB</a> [variable, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.FirstSteps.html#072addf1f952576a9d67e98c63e20939">_ ^ _</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3">_ * _</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.:::'1'">1</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs">Defs</a> [section, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#Defs">Defs</a> [section, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#Defs">Defs</a> [section, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs.A">Defs.A</a> [variable, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs.lt">Defs.lt</a> [variable, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#56ea2d5e19744d3f0cba9664d875d7ec">_ &lt;= _</a> [notation, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x">_ &lt; _</a> [notation, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#delta:593">delta:593</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Addition.html#delta:62">delta:62</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.Monoid_def.html#Demo">Demo</a> [module, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Demo.html">Demo</a> [library]<br/>
+<a href="additions.Demo_power.html">Demo_power</a> [library]<br/>
+<a href="additions.Monoid_def.html#Demo.nat_mult_op">Demo.nat_mult_op</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#den_U">den_U</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depth">depth</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthForall">depthForall</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthImp1">depthImp1</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthImp2">depthImp2</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthNot">depthNot</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#depth_rec_demo.a">depth_rec_demo.a</a> [variable, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#depth_rec_demo.P">depth_rec_demo.P</a> [variable, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#depth_rec_demo.L">depth_rec_demo.L</a> [variable, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#depth_rec_demo">depth_rec_demo</a> [section, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#dest:102">dest:102</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.Dichotomy.html#dicho">dicho</a> [definition, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html">Dichotomy</a> [library]<br/>
+<a href="additions.Dichotomy.html#Dicho_strat">Dicho_strat</a> [instance, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_gt">dicho_gt</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_lt">dicho_lt</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_gt">dicho_aux_gt</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_lt">dicho_aux_lt</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xIXI">dicho_aux_le_xIXI</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xIXO">dicho_aux_le_xIXO</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xOXI">dicho_aux_le_xOXI</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xOXO">dicho_aux_le_xOXO</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xIxI">dicho_aux_xIxI</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xIxO">dicho_aux_xIxO</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xOxI">dicho_aux_xOxI</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xOxO">dicho_aux_xOxO</a> [lemma, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="hydras.rpo.dickson.html">dickson</a> [library]<br/>
+<a href="gaia_hydras.GPrelude.html#diffP">diffP</a> [lemma, in <a href="gaia_hydras.GPrelude.html">gaia_hydras.GPrelude</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.nf_Acc">Direct_proof.nf_Acc</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_implies_Acc_strong">Direct_proof.Acc_implies_Acc_strong</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_strong_stronger">Direct_proof.Acc_strong_stronger</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.acc_impl">Direct_proof.acc_impl</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.Acc_strong">Direct_proof.well_foundedness_proof.Acc_strong</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.wf_LT">Direct_proof.wf_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.First_attempt">Direct_proof.well_foundedness_proof.First_attempt</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_zero">Direct_proof.Acc_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof">Direct_proof.well_foundedness_proof</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof">Direct_proof</a> [module, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.onType.html#disp:27">disp:27</a> [binder, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div_eucl_unique">div_eucl_unique</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div_not_qlt">div_not_qlt</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.Strategies.html#div_gamma_pos">div_gamma_pos</a> [lemma, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#div2IsPR">div2IsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_even_plus">div2_even_plus</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_incr">div2_incr</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_of_Even">div2_of_Even</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_double_is_id">div2_double_is_id</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates">dominates</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#dominates">dominates</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_iterate">dominates_iterate</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_from_le">dominates_from_le</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_trans_strong">dominates_trans_strong</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_trans">dominates_trans</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_from_trans">dominates_from_trans</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_strong">dominates_strong</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_from">dominates_from</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#dominates_strong">dominates_strong</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#dominates_from">dominates_from</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR">dom_AckNotPR</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR.Hf">dom_AckNotPR.Hf</a> [variable, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR.f">dom_AckNotPR.f</a> [variable, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR">dom_AckNotPR</a> [section, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.rpo.more_list.html#dom:152">dom:152</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#Done">Done</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#double">double</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#double">double</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#doubleIsPR">doubleIsPR</a> [lemma, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#doubleIsPR">doubleIsPR</a> [instance, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#doubleS">doubleS</a> [definition, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#doubleS_law">doubleS_law</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_is_even">double_is_even</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_inj">double_inj</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_plus">double_plus</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_S">double_S</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.Hn">DS_iota.Proof_case_6.Hn</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.n">DS_iota.Proof_case_6.n</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.gamma_pos">DS_iota.Proof_case_6.gamma_pos</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.Hgamma">DS_iota.Proof_case_6.Hgamma</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.Hbeta">DS_iota.Proof_case_6.Hbeta</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.gamma">DS_iota.Proof_case_6.gamma</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6.beta">DS_iota.Proof_case_6.beta</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6">DS_iota.Proof_case_6</a> [section, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5.Hn">DS_iota.Proof_case_5.Hn</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5.n">DS_iota.Proof_case_5.n</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5.Hgamma">DS_iota.Proof_case_5.Hgamma</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5.gamma">DS_iota.Proof_case_5.gamma</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5">DS_iota.Proof_case_5</a> [section, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4.Hn">DS_iota.Proof_case_4.Hn</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4.n">DS_iota.Proof_case_4.n</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4.Hlim">DS_iota.Proof_case_4.Hlim</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4.Hlambda">DS_iota.Proof_case_4.Hlambda</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4.lambda">DS_iota.Proof_case_4.lambda</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4">DS_iota.Proof_case_4</a> [section, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_3">DS_iota_3</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_2">DS_iota_2</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_1">DS_iota_1</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Hrec">DS_iota.Hrec</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.nonzero">DS_iota.nonzero</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Halpha">DS_iota.Halpha</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.i">DS_iota.i</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.alpha">DS_iota.alpha</a> [variable, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota">DS_iota</a> [section, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1205">d':1205</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1339">d':1339</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1385">d':1385</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1437">d':1437</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1501">d':1501</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1557">d':1557</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1586">d':1586</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1610">d':1610</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':1692">d':1692</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':2044">d':2044</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':2067">d':2067</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Monoid_instances.html#d':27">d':27</a> [binder, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d':849">d':849</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#d:10">d:10</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:116">d:116</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#d:118">d:118</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#d:13">d:13</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#d:1308">d:1308</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1334">d:1334</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1349">d:1349</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1354">d:1354</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1359">d:1359</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1364">d:1364</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1369">d:1369</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1374">d:1374</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1380">d:1380</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1432">d:1432</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.fib.html#d:146">d:146</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#d:149">d:149</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1496">d:1496</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#d:152">d:152</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:152">d:152</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1535">d:1535</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#d:154">d:154</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1546">d:1546</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1555">d:1555</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:156">d:156</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#d:157">d:157</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1591">d:1591</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:160">d:160</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1605">d:1605</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:164">d:164</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#d:165">d:165</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1654">d:1654</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1678">d:1678</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1687">d:1687</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#d:17">d:17</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:1795">d:1795</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#d:19">d:19</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Schutte.Lub.html#D:2">D:2</a> [binder, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:2029">d:2029</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Monoid_instances.html#d:23">d:23</a> [binder, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:267">d:267</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:271">d:271</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:275">d:275</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:279">d:279</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:283">d:283</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:288">d:288</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:292">d:292</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:296">d:296</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:301">d:301</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:305">d:305</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:309">d:309</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:313">d:313</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:317">d:317</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:321">d:321</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:325">d:325</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:330">d:330</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:334">d:334</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Euclidean_Chains.html#d:335">d:335</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:338">d:338</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#d:34">d:34</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:342">d:342</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:346">d:346</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:36">d:36</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="additions.Euclidean_Chains.html#d:385">d:385</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#d:391">d:391</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#d:398">d:398</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#d:4">d:4</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:44">d:44</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:48">d:48</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#d:5">d:5</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:52">d:52</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#d:56">d:56</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d:567">d:567</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#d:595">d:595</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d:609">d:609</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d:625">d:625</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d:631">d:631</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#d:658">d:658</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.Lub.html#D:8">D:8</a> [binder, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d:845">d:845</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#d:851">d:851</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#d:91">d:91</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="global_E"></a><h2>E </h2>
+<a href="hydras.Epsilon0.T1.html#Elements">Elements</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#embedding">embedding</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="additions.Monoid_def.html#EMonoid">EMonoid</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#empty_interval">empty_interval</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#empty_ordering">empty_ordering</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="additions.Monoid_def.html#Eone_right">Eone_right</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eone_left">Eone_left</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Pow_variant.html#Eop_proper">Eop_proper</a> [instance, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#Eop_proper">Eop_proper</a> [instance, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_comm">Eop_comm</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_assoc">Eop_assoc</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_proper">Eop_proper</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Gamma0.T2.html#epsilon">epsilon</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#epsilon_0">epsilon_0</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#epsilon_equiv">epsilon_equiv</a> [lemma, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#epsilon_ind">epsilon_ind</a> [lemma, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#epsilon_fxp">epsilon_fxp</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.T2.html#epsilon0">epsilon0</a> [abbreviation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.AP.html#epsilon0">epsilon0</a> [definition, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Epsilon0">Epsilon0</a> [instance, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Epsilon0">Epsilon0</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0.html">Epsilon0</a> [library]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html">Epsilon0rpo</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#epsilon0_dec">epsilon0_dec</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#epsilon0_pre_order">epsilon0_pre_order</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Epsilon0_correct">Epsilon0_correct</a> [instance, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Critical.html#epsilon0_Cr1">epsilon0_Cr1</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.AP.html#epsilon0_lfp">epsilon0_lfp</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#epsilon0_AP">epsilon0_AP</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#epsilon0_fxp">epsilon0_fxp</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#epsilon0_as_lub">epsilon0_as_lub</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#epsilon0_fxp">epsilon0_fxp</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html">Epsilon0_Needed_Std</a> [library]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html">Epsilon0_Needed_Free</a> [library]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html">Epsilon0_Needed_Generic</a> [library]<br/>
+<a href="hydras.rpo.more_list.html#eqA:107">eqA:107</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:115">eqA:115</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:139">eqA:139</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:148">eqA:148</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:158">eqA:158</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:167">eqA:167</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:174">eqA:174</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:182">eqA:182</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:190">eqA:190</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:205">eqA:205</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:214">eqA:214</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:224">eqA:224</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:242">eqA:242</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:251">eqA:251</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:263">eqA:263</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:81">eqA:81</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:91">eqA:91</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#eqA:99">eqA:99</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#EqDecision">EqDecision</a> [abbreviation, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.fol.html#eqDepth">eqDepth</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#eqIsPR">eqIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqLT">eqLT</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#eqn_simpl2">eqn_simpl2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#eqn_simpl1">eqn_simpl1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqPlus">eqPlus</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqPlus">eqPlus</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqRefl">eqRefl</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqRefl">eqRefl</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#eqRefl">eqRefl</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqSucc">eqSucc</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqSucc">eqSucc</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqSym">eqSym</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqSym">eqSym</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#eqSym">eqSym</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqTimes">eqTimes</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqTimes">eqTimes</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#eqTrans">eqTrans</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#eqTrans">eqTrans</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#eqTrans">eqTrans</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#equal">equal</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#equalFunction">equalFunction</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.termsMap">Equality_Logic_Rules.termsMap</a> [variable, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.SysPrf">Equality_Logic_Rules.SysPrf</a> [variable, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.Prf">Equality_Logic_Rules.Prf</a> [variable, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules.L">Equality_Logic_Rules.L</a> [variable, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules">Equality_Logic_Rules</a> [section, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#equalRelation">equalRelation</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case3.Hac">Equations_for_addition.case3.Hac</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case3">Equations_for_addition.case3</a> [section, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case2.Hac">Equations_for_addition.case2.Hac</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case2">Equations_for_addition.case2</a> [section, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case1.Hac">Equations_for_addition.case1.Hac</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case1">Equations_for_addition.case1</a> [section, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.beta">Equations_for_addition.beta</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.alpha">Equations_for_addition.alpha</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.Hnfc">Equations_for_addition.Hnfc</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.Hnfa">Equations_for_addition.Hnfa</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.p">Equations_for_addition.p</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.n">Equations_for_addition.n</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.d">Equations_for_addition.d</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.c">Equations_for_addition.c</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.b">Equations_for_addition.b</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.a">Equations_for_addition.a</a> [variable, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition">Equations_for_addition</a> [section, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#equiv">equiv</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="additions.Monoid_def.html#equiv">equiv</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv">Equiv</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#equiv">equiv</a> [constructor, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv">Equiv</a> [inductive, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence">Equivalence</a> [section, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.A">Equivalence.A</a> [variable, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.is_power_of">Equivalence.is_power_of</a> [variable, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.one">Equivalence.one</a> [variable, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.op">Equivalence.op</a> [variable, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.pos_iter_M">Equivalence.pos_iter_M</a> [variable, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#equiv_equiv">equiv_equiv</a> [instance, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#equiv_equiv">equiv_equiv</a> [instance, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#equiv_not_lt">equiv_not_lt</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Trans">Equiv_Trans</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Sym">Equiv_Sym</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Refl">Equiv_Refl</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Equiv">Equiv_Equiv</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#equiv:44">equiv:44</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#equiv:49">equiv:49</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#equiv:52">equiv:52</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#equ:101">equ:101</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:106">equ:106</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:111">equ:111</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:121">equ:121</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:126">equ:126</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:38">equ:38</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:44">equ:44</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:55">equ:55</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:62">equ:62</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:75">equ:75</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:85">equ:85</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#equ:96">equ:96</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:104">eqv:104</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:112">eqv:112</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:118">eqv:118</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:125">eqv:125</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:132">eqv:132</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:162">eqv:162</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:169">eqv:169</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#eqv:175">eqv:175</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#eq_succ_lt">eq_succ_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#eq_succ_LT">eq_succ_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#eq_dec">eq_dec</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#eq_dec">eq_dec</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="additions.Monoid_def.html#eq_equiv">eq_equiv</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eq_equiv">Eq_equiv</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#eq_le">eq_le</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.rpo.rpo.html#eq_A_dec:19">eq_A_dec:19</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#eq_A_dec:14">eq_A_dec:14</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#eq_A_dec:5">eq_A_dec:5</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ1">EQ1</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ2">EQ2</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ3">EQ3</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ4">EQ4</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ5">EQ5</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#essai">essai</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#essai.n">essai.n</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Critical.html#eta:80">eta:80</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="additions.Euclidean_Chains.html">Euclidean_Chains</a> [library]<br/>
+<a href="hydras.Prelude.More_Arith.html#Euc1">Euc1</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Schutte.CNF.html#eval">eval</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalComposeFunc">evalComposeFunc</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalConstFunc">evalConstFunc</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalList">evalList</a> [section, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalList">evalList</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalListComp">evalListComp</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalListCompose2">evalListCompose2</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalListPrimrec_S">evalListPrimrec_S</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalListPrimrec_0">evalListPrimrec_0</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalList_Const">evalList_Const</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalOneParamList">evalOneParamList</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalPrimRec">evalPrimRec</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalPrimRecFunc">evalPrimRecFunc</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalPrimRecParam">evalPrimRecParam</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalPrimRecs">evalPrimRecs</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalProjFunc">evalProjFunc</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#evalProjFuncInd">evalProjFuncInd</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRec">evalStrongRec</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecHelp">evalStrongRecHelp</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecHelpParam">evalStrongRecHelpParam</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecHelp1">evalStrongRecHelp1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecHelp2">evalStrongRecHelp2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecIsPR">evalStrongRecIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#eval_pp">eval_pp</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#even_prod">even_prod</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#every_battle_terminates">every_battle_terminates</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="gaia_hydras.GHydra.html#every_battle_terminates">every_battle_terminates</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Ex">Ex</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="additions.More_on_positive.html#exact_log2_spec">exact_log2_spec</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#exact_log2xOx0">exact_log2xOx0</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#exact_log2">exact_log2</a> [definition, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Examples">Examples</a> [section, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples">Examples</a> [section, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples">Examples</a> [module, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Examples">Examples</a> [module, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples">Examples</a> [module, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.big_correct'''">Examples.big_correct'''</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.big_correct'">Examples.big_correct'</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.big_correct">Examples.big_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.big_chain">Examples.big_chain</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.binary_int31_power">Examples.binary_int31_power</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.C87'">Examples.C87'</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Examples.exact_sqrt">Examples.exact_sqrt</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.exp56789">Examples.exp56789</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Exx">Examples.Exx</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.ex_2">Examples.ex_2</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.ex1">Examples.ex1</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.ex4">Examples.ex4</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Ex5">Examples.Ex5</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Examples.f">Examples.f</a> [variable, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.fast_int63_power">Examples.fast_int63_power</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f1">Examples.f1</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f2">Examples.f2</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f2'">Examples.f2'</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f3">Examples.f3</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f4">Examples.f4</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.height_not_strictly_decreasing">Examples.height_not_strictly_decreasing</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.hsize_bigger">Examples.hsize_bigger</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy">Examples.Hy</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy_Hy'''">Examples.Hy_Hy'''</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy_3">Examples.Hy_3</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy_2">Examples.Hy_2</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy_1">Examples.Hy_1</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'">Examples.Hy'</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'H''">Examples.Hy'H''</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''">Examples.Hy''</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy''_Hy'''">Examples.Hy''_Hy'''</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.Hy'''">Examples.Hy'''</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.M">Examples.M</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.RM">Examples.RM</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples.R2_example">Examples.R2_example</a> [definition, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.slow_int31_power">Examples.slow_int31_power</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Examples.sqrtHelp">Examples.sqrtHelp</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.t1">Examples.t1</a> [variable, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples.Unnamed_thm">Examples.Unnamed_thm</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html">Example_3PlusOmega</a> [library]<br/>
+<a href="additions.AM.html#exec">exec</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#exec_Proper">exec_Proper</a> [instance, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#exec_equiv">exec_equiv</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#exec_app">exec_app</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#exH'">exH'</a> [abbreviation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#existE">existE</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#existE">existE</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#existE">existE</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#existH">existH</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#existI">existI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#existI">existI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#existI">existI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#existSimp">existSimp</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#existSimp">existSimp</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#existSimp">existSimp</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#existSys">existSys</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#existSys">existSys</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#existSys">existSys</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.rpo.more_list.html#exists_map12_without_repetition">exists_map12_without_repetition</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#exists_map_without_repetition">exists_map_without_repetition</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#ExNotIncl">ExNotIncl</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp">exp</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#exp">exp</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#expIsPR">expIsPR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="additions.Trace_exercise.html#exponents">exponents</a> [definition, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Schutte.CNF.html#exponents_lt_eval">exponents_lt_eval</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#exponents_le">exponents_le</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#exponents_lt">exponents_lt</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="additions.Addition_Chains.html#exponent_pos_of_nat">exponent_pos_of_nat</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#exponent_pos2nat">exponent_pos2nat</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#exponent_nat_neq_0">exponent_nat_neq_0</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Expressible">Expressible</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html">expressible</a> [library]<br/>
+<a href="hydras.Ackermann.expressible.html#expressibleAlternate">expressibleAlternate</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html">expressibleExamples</a> [library]<br/>
+<a href="hydras.Ackermann.expressible.html#ExpressibleHelp">ExpressibleHelp</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.rosserPA.html#expressT'1">expressT'1</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.goedel2.html#expressT'1">expressT'1</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.rosserPA.html#expressT'2">expressT'2</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.goedel2.html#expressT'2">expressT'2</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp_F_eq">exp_F_eq</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp_fin_omega">exp_fin_omega</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp_F">exp_F</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#exp_alt_PR">exp_alt_PR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#exp_alt_ok">exp_alt_ok</a> [lemma, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#exp_alt">exp_alt</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="additions.Addition_Chains.html#exp2">exp2</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Prelude.Exp2.html#exp2">exp2</a> [definition, in <a href="hydras.Prelude.Exp2.html">hydras.Prelude.Exp2</a>]<br/>
+<a href="hydras.Prelude.Exp2.html">Exp2</a> [library]<br/>
+<a href="hydras.Prelude.Exp2.html#exp2S">exp2S</a> [lemma, in <a href="hydras.Prelude.Exp2.html">hydras.Prelude.Exp2</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#exp2_as_iterate">exp2_as_iterate</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#exp2_mono_weak">exp2_mono_weak</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#exp2_mono">exp2_mono</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#exp2_ge_S">exp2_ge_S</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#exp2_k_mult_pos">exp2_k_mult_pos</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#exp2_mono1">exp2_mono1</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Addition_Chains.html#exp2_Plus">exp2_Plus</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Prelude.Exp2.html#exp2_gt_id">exp2_gt_id</a> [lemma, in <a href="hydras.Prelude.Exp2.html">hydras.Prelude.Exp2</a>]<br/>
+<a href="hydras.Prelude.Exp2.html#exp2_not_zero">exp2_not_zero</a> [lemma, in <a href="hydras.Prelude.Exp2.html">hydras.Prelude.Exp2</a>]<br/>
+<a href="hydras.Prelude.Exp2.html#exp2_positive">exp2_positive</a> [lemma, in <a href="hydras.Prelude.Exp2.html">hydras.Prelude.Exp2</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp:746">exp:746</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#exp:751">exp:751</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs">Exs</a> [section, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs.f">Exs.f</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs.fdiv2">Exs.fdiv2</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
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+<a href="additions.Monoid_def.html#E_op:47">E_op:47</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_eq:44">E_eq:44</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_one:43">E_one:43</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_op:42">E_op:42</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_eq:39">E_eq:39</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_one:38">E_one:38</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_op:37">E_op:37</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_eq:24">E_eq:24</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_one:23">E_one:23</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#E_op:22">E_op:22</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:197">E_equiv:197</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:196">E_one:196</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:185">E_equiv:185</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:184">E_one:184</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:171">E_equiv:171</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:170">E_one:170</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:165">E_equiv:165</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:164">E_one:164</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:113">E_equiv:113</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:112">E_one:112</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:102">E_equiv:102</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:101">E_one:101</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:91">E_equiv:91</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:90">E_one:90</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:71">E_equiv:71</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:70">E_one:70</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:64">E_equiv:64</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:63">E_one:63</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:56">E_equiv:56</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:55">E_one:55</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_equiv:38">E_equiv:38</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#E_one:37">E_one:37</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:234">E_eq:234</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:233">E_one:233</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:232">E_op:232</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:226">E_eq:226</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:225">E_one:225</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:224">E_op:224</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:74">E_eq:74</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:73">E_one:73</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:72">E_op:72</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:66">E_eq:66</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:65">E_one:65</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:64">E_op:64</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:60">E_eq:60</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:59">E_one:59</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:58">E_op:58</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_eq:49">E_eq:49</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_one:48">E_one:48</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#E_op:47">E_op:47</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.rpo.list_set.html#e':33">e':33</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e':36">e':36</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0">E0</a> [record, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0">E0</a> [record, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html">E0</a> [library]<br/>
+<a href="hydras.Epsilon0.E0.html#E0add">E0add</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.GCanon.html#E0Canon">E0Canon</a> [definition, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0compare">E0compare</a> [instance, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0compare_correct">E0compare_correct</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0eqb">E0eqb</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0fin">E0fin</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0fin">E0fin</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0finS">E0finS</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0fin_cnf">E0fin_cnf</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0g2h_omegaE">E0g2h_omegaE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0g2h_plusE">E0g2h_plusE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0g2h_mulE">E0g2h_mulE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0g2h_phi0">E0g2h_phi0</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0g2h_Fin">E0g2h_Fin</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0is_succE">E0is_succE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0is_succ_succ">E0is_succ_succ</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0is_succ">E0is_succ</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0is_succ">E0is_succ</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0le">E0le</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0le">E0le</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0limit">E0limit</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0limit">E0limit</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0lt">E0lt</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0lt">E0lt</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0lt_wf">E0lt_wf</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0mul">E0mul</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0mul">E0mul</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0one">E0one</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0plus">E0plus</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#e0Sub">e0Sub</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0succb">E0succb</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0zero">E0zero</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0zero">E0zero</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#E0_plus_correct">E0_plus_correct</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="gaia_hydras.GCanon.html#E0_canon_lt">E0_canon_lt</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_comp">E0_comp</a> [instance, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_sto">E0_sto</a> [instance, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_pred_succK">E0_pred_succK</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_diffE">E0_diffE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_eqE">E0_eqE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_g2h_h2gK">E0_g2h_h2gK</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_h2g_g2hK">E0_h2g_g2hK</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_h2g_nf">E0_h2g_nf</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_g2h">E0_g2h</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_h2g">E0_h2g</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_phi0">E0_phi0</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_omega">E0_omega</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_pred">E0_pred</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_succ">E0_succ</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_eqtype">E0_eqtype</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_eq_mixin">E0_eq_mixin</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_lt_ge">E0_lt_ge</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_lt_eq_lt">E0_lt_eq_lt</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_not_Lt_zero">E0_not_Lt_zero</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_Lt_Succ_inv">E0_Lt_Succ_inv</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_Lt_irrefl">E0_Lt_irrefl</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_comp">E0_comp</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_pred_Lt">E0_pred_Lt</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_pred_of_Succ">E0_pred_of_Succ</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_sto">E0_sto</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_pred">E0_pred</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_dec">E0_eq_dec</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_iff">E0_eq_iff</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_intro">E0_eq_intro</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_phi0">E0_phi0</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_succ">E0_succ</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_omega">E0_omega</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#E1">E1</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#e1">e1</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="additions.fib.html#e1:103">e1:103</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#e1:110">e1:110</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.list_set.html#e1:179">e1:179</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#E2">E2</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#e2">e2</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="additions.fib.html#e2:104">e2:104</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#e2:111">e2:111</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.list_set.html#e2:180">e2:180</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:10">e:10</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:100">e:100</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:103">e:103</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:106">e:106</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:109">e:109</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#e:11">e:11</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:112">e:112</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="additions.fib.html#e:119">e:119</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:121">e:121</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#E:126">E:126</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:129">e:129</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:132">e:132</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:136">e:136</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:139">e:139</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:143">e:143</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:147">e:147</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:151">e:151</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:154">e:154</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:157">e:157</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:170">e:170</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:199">e:199</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#E:2">E:2</a> [binder, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#E:2">E:2</a> [binder, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.rpo.term.html#e:204">e:204</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#e:21">e:21</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:22">e:22</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:24">e:24</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:241">e:241</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:245">e:245</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:252">e:252</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:26">e:26</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:28">e:28</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#e:29">e:29</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:30">e:30</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:32">e:32</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#e:32">e:32</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#e:33">e:33</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:35">e:35</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:39">e:39</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:46">e:46</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#e:54">e:54</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.rpo.list_permut.html#e:55">e:55</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:57">e:57</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:61">e:61</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:65">e:65</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:69">e:69</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:73">e:73</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#e:743">e:743</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:75">e:75</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#e:754">e:754</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:78">e:78</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:81">e:81</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:84">e:84</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:85">e:85</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#e:92">e:92</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.rpo.list_set.html#e:97">e:97</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<br/><br/><a id="global_F"></a><h2>F </h2>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html#f">f</a> [definition, in <a href="hydras.OrdinalNotations.Example_3PlusOmega.html">hydras.OrdinalNotations.Example_3PlusOmega</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#F">F</a> [abbreviation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.onType.html#F">F</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#f">f</a> [axiom, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="Goedel.goedel2.html#F">F</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact">fact</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#factIsPR">factIsPR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact_altIsPR">fact_altIsPR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact_alt_ok">fact_alt_ok</a> [lemma, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact_alt">fact_alt</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#fact_correct">fact_correct</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#Fake_thm">Fake_thm</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fapply">Fapply</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.fib.html#fastexp">fastexp</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fastexp2">fastexp2</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fastexp3">fastexp3</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fastexp3P">fastexp3P</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fastexp4">fastexp4</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#FA1">FA1</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#FA2">FA2</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#FA3">FA3</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#fA:181">fA:181</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#fbij">fbij</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#fB:182">fB:182</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain">Fchain</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#Fchain_correct">Fchain_correct</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_proper_prf">Fchain_proper_prf</a> [projection, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_proper">Fchain_proper</a> [record, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_proper_prf">Fchain_proper_prf</a> [constructor, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_proper">Fchain_proper</a> [inductive, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_correct">Fchain_correct</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_correct_nat">Fchain_correct_nat</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose">Fcompose</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_proper">Fcompose_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_correct">Fcompose_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_correct_nat">Fcompose_correct_nat</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc1:120">fc1:120</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc1:129">fc1:129</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc1:133">fc1:133</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc2:121">fc2:121</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc2:130">fc2:130</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc2:134">fc2:134</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc:108">fc:108</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc:34">fc:34</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc:79">fc:79</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fc:86">fc:86</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2">Fexp2</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_proper">Fexp2_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_correct">Fexp2_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_nat_correct">Fexp2_nat_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_nat_proper">Fexp2_nat_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_of_nat">Fexp2_of_nat</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#FF">FF</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="additions.AM.html#FFK">FFK</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK">FFK</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#FFK_correct">FFK_correct</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_proper">FFK_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_correct">FFK_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_correct_nat">FFK_correct_nat</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Fib2.html#fib">fib</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib">fib</a> [definition, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="additions.fib.html#fib">fib</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html">fib</a> [library]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fibIsPR">fibIsPR</a> [instance, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="additions.fib.html#fibm">fibm</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibmP">fibmP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.FirstSteps.html#fibonacci">fibonacci</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#fibonacci">fibonacci</a> [definition, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html">FibonacciPR</a> [library]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fibPR">fibPR</a> [definition, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fibPRIsPR">fibPRIsPR</a> [instance, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="additions.fib.html#fibt">fibt</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibtP">fibtP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibt_aux">fibt_aux</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibZ2P">fibZ2P</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibZ3P">fibZ3P</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Fib2.html#fib_with_chain">fib_with_chain</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_eucl">fib_eucl</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_pos">fib_pos</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_mul2_OK">fib_mul2_OK</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_mul2_OK_0">fib_mul2_OK_0</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_mul2">fib_mul2</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_SSn">fib_SSn</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_ind">fib_ind</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_alt_Ok">fib_alt_Ok</a> [lemma, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_OK0">fib_OK0</a> [lemma, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_altIsPR">fib_altIsPR</a> [instance, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_alt">fib_alt</a> [definition, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="additions.Fib2.html">Fib2</a> [library]<br/>
+<a href="hydras.Ackermann.primRec.html#filter001IsPR">filter001IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter0011IsPR">filter0011IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter01IsPR">filter01IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter010IsPR">filter010IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter011IsPR">filter011IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter10IsPR">filter10IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter100IsPR">filter100IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1000IsPR">filter1000IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter101IsPR">filter101IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1011IsPR">filter1011IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter110IsPR">filter110IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1100IsPR">filter1100IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#fin">fin</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#fin">fin</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#fin">fin</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#fin">fin</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Fin">Fin</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.rpo.more_list.html#find">find</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#find_not_mem">find_not_mem</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite">finite</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#finite_ltR">finite_ltR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#finite_lt">finite_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#finite_succ">finite_succ</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Finite_ref">Finite_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_not_limit">finite_not_limit</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_lt_omega">finite_lt_omega</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_inj">finite_inj</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_mono">finite_mono</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_lt_inv">finite_lt_inv</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#finite_is_finite">finite_is_finite</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#finite_plus_infinite">finite_plus_infinite</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#FinOrd">FinOrd</a> [instance, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_ON">finord_ON</a> [instance, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_lt_wf">finord_lt_wf</a> [lemma, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_comp">finord_comp</a> [instance, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_sto">finord_sto</a> [instance, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_compare_correct">finord_compare_correct</a> [lemma, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_compare">finord_compare</a> [instance, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_lt">finord_lt</a> [definition, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#FinOrd_Omega">FinOrd_Omega</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#FinS_Succ_eq">FinS_Succ_eq</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#FinS_eq">FinS_eq</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#fin_lt_omega">fin_lt_omega</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Addition_Chains.html#fin:90">fin:90</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="Goedel.goedel1.html#FirstIncompletenessA">FirstIncompletenessA</a> [lemma, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#FirstIncompletenessA">FirstIncompletenessA</a> [lemma, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel1.html#FirstIncompletenessB">FirstIncompletenessB</a> [lemma, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="additions.FirstSteps.html">FirstSteps</a> [library]<br/>
+<a href="hydras.Prelude.First_toggle.html#first_toggle">first_toggle</a> [definition, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Depth_Induction">First_Order_Logic.Formula_Depth_Induction</a> [section, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.consTermsHelp">First_Order_Logic.Formula_Decidability.consTermsHelp</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.nilTermsHelp">First_Order_Logic.Formula_Decidability.nilTermsHelp</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.language_eqdec">First_Order_Logic.Formula_Decidability.language_eqdec</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability">First_Order_Logic.Formula_Decidability</a> [section, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.L">First_Order_Logic.L</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic">First_Order_Logic</a> [section, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html">First_toggle</a> [library]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix">Fix</a> [section, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="Goedel.fixPoint.html">fixPoint</a> [library]<br/>
+<a href="gaia_hydras.GHessenberg.html#FixpointDef">FixpointDef</a> [module, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#FixpointDef.oplus">FixpointDef.oplus</a> [definition, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="Goedel.fixPoint.html#FixPointLNN">FixPointLNN</a> [lemma, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#FixPointLNT">FixPointLNT</a> [lemma, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#FixR">FixR</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#FixR_F_eq">FixR_F_eq</a> [lemma, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#FixR_F">FixR_F</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.A">Fix.A</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.E">Fix.E</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.F">Fix.F</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.Hwf">Fix.Hwf</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.P">Fix.P</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.R">Fix.R</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.Rwf">Fix.Rwf</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="additions.AM.html#FK">FK</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FK">FK</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont">Fkont</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_equiv">Fkont_equiv</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper_prf">Fkont_proper_prf</a> [projection, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper">Fkont_proper</a> [record, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper_prf">Fkont_proper_prf</a> [constructor, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper">Fkont_proper</a> [inductive, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#FK_correct">FK_correct</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FK_proper">FK_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FK_correct">FK_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten">flatten</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#flatten">flatten</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_valid_2">flatten_valid_2</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_valid">flatten_valid</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_aux_valid">flatten_aux_valid</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_aux">flatten_aux</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.fib.html#fn1:47">fn1:47</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fn2:48">fn2:48</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fn3:49">fn3:49</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.AM.html#fn:33">fn:33</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#fn:446">fn:446</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#fn:459">fn:459</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#fn:480">fn:480</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#FO">FO</a> [constructor, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Ackermann.fol.html">fol</a> [library]<br/>
+<a href="hydras.rpo.more_list.html#fold_left2">fold_left2</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html">FolExamples</a> [library]<br/>
+<a href="hydras.Ackermann.folLogic.html">folLogic</a> [library]<br/>
+<a href="hydras.Ackermann.folLogic2.html">folLogic2</a> [library]<br/>
+<a href="hydras.Ackermann.folLogic3.html">folLogic3</a> [library]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations">FolNotations</a> [module, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.exH'">FolNotations.exH'</a> [abbreviation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'''_x">_ &lt;-&gt;' _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'''''_x">_ &lt;-&gt;'' _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#bb04b0324214688cf30460e944283f4c">_ /\' _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#e4a489994216e94595670248e88226a2">_ \/' _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x">_ &lt;&gt; _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0">_ = _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32">allH _ .. _ , _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5">exH _ .. _ , _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71">v# _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x">_ &lt;-&gt; _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448">~ _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x">_ -&gt; _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5">_ /\ _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c">_ \/ _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0">_ = _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProof.html">folProof</a> [library]<br/>
+<a href="hydras.Ackermann.folProp.html">folProp</a> [library]<br/>
+<a href="hydras.Ackermann.folReplace.html">folReplace</a> [library]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Substitution_Properties">Fol_Properties.Substitution.Substitution_Properties</a> [section, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Extensions">Fol_Properties.Substitution.Extensions</a> [section, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution">Fol_Properties.Substitution</a> [section, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Free_Variables">Fol_Properties.Free_Variables</a> [section, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.lt_depth">Fol_Properties.lt_depth</a> [variable, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.L">Fol_Properties.L</a> [variable, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties">Fol_Properties</a> [section, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#foo">foo</a> [definition, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#foo:167">foo:167</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#foo:42">foo:42</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#foo:45">foo:45</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallE">forallE</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallE">forallE</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallE">forallE</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#forallH">forallH</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallI">forallI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallI">forallI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallI">forallI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallSimp">forallSimp</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallSimp">forallSimp</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallSimp">forallSimp</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#forall_weak">forall_weak</a> [lemma, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_forall">Forall_forall</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_and">Forall_and</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_inv">Forall_inv</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R">Forall2R</a> [inductive, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_iff">Forall2R_iff</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_cons">Forall2R_cons</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_last">Forall2R_last</a> [constructor, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_nil">Forall2R_nil</a> [constructor, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_indR">Forall2_indR</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_RR">Forall2_RR</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_R">Forall2_R</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_right_induction">Forall2_right_induction</a> [section, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall2_inv">Forall2_inv</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#Formula">Formula</a> [abbreviation, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Formula">Formula</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Formula">Formula</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Formula">Formula</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Formula">Formula</a> [abbreviation, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula">Formula</a> [inductive, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Formula">Formula</a> [abbreviation, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#Formulas">Formulas</a> [abbreviation, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Formulas">Formulas</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Formulas">Formulas</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Formulas">Formulas</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Formulas">Formulas</a> [abbreviation, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formulas">Formulas</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Formulas">Formulas</a> [abbreviation, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="Goedel.PRrepresentable.html#FormulasToFormula">FormulasToFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#FormulasToFuncs">FormulasToFuncs</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_ind2">Formula_depth_ind2</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_ind">Formula_depth_ind</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_forall">Formula_depth_rec2_forall</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_not">Formula_depth_rec2_not</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_imp">Formula_depth_rec2_imp</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2rec_nice">Formula_depth_rec2rec_nice</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2">Formula_depth_rec2</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2rec">Formula_depth_rec2rec</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec_indep">Formula_depth_rec_indep</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec">Formula_depth_rec</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec_rec">Formula_depth_rec_rec</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#formula_eqdec">formula_eqdec</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fp:233">fp:233</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fp:302">fp:302</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fp:318">fp:318</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fp:320">fp:320</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fp:323">Fp:323</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fp:328">Fp:328</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:223">fq:223</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:232">fq:232</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:234">fq:234</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:294">fq:294</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:299">fq:299</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:303">fq:303</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:319">fq:319</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#fq:321">fq:321</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#frec:113">frec:113</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#frec:74">frec:74</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#frec:76">frec:76</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#frec:98">frec:98</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#free">free</a> [instance, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddExists1">freeVarAddExists1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddExists2">freeVarAddExists2</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddForalls1">freeVarAddForalls1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddForalls2">freeVarAddForalls2</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosed">freeVarClosed</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosedList1">freeVarClosedList1</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosedList2">freeVarClosedList2</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPf">freeVarCodeSysPf</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPrf">freeVarCodeSysPrf</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosserPA.html#freeVarCodeSysPrfN">freeVarCodeSysPrfN</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPrfN">freeVarCodeSysPrfN</a> [lemma, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#freeVarCodeSysPrfN">freeVarCodeSysPrfN</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarF">freeVarF</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarFormula">freeVarFormula</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="Goedel.goedel1.html#freeVarG">freeVarG</a> [lemma, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpFormula">freeVarInterpFormula</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpRel">freeVarInterpRel</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpTerm">freeVarInterpTerm</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarListFormula">freeVarListFormula</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarListFormulaApp">freeVarListFormulaApp</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarLT">freeVarLT</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarMap">freeVarMap</a> [definition, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarMap_ext">freeVarMap_ext</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarMap1">freeVarMap1</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarNNHelp">freeVarNNHelp</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarPlus">freeVarPlus</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarPlus">freeVarPlus</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarPrimRecPiFormulaHelp1">freeVarPrimRecPiFormulaHelp1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarPrimRecSigmaFormulaHelp1">freeVarPrimRecSigmaFormulaHelp1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.rosserPA.html#freeVarRepT'">freeVarRepT'</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.goedel2.html#freeVarRepT'">freeVarRepT'</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllFormula1">freeVarSubAllFormula1</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllFormula2">freeVarSubAllFormula2</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerms1">freeVarSubAllTerms1</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerms2">freeVarSubAllTerms2</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerm1">freeVarSubAllTerm1</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerm2">freeVarSubAllTerm2</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula1">freeVarSubFormula1</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula2">freeVarSubFormula2</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula3">freeVarSubFormula3</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula4">freeVarSubFormula4</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms1">freeVarSubTerms1</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms2">freeVarSubTerms2</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms3">freeVarSubTerms3</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms4">freeVarSubTerms4</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm1">freeVarSubTerm1</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm2">freeVarSubTerm2</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm3">freeVarSubTerm3</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm4">freeVarSubTerm4</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarSucc">freeVarSucc</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarSucc">freeVarSucc</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarT">freeVarT</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTApply">freeVarTApply</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTerm">freeVarTerm</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTerms">freeVarTerms</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarTimes">freeVarTimes</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarTimes">freeVarTimes</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTs">freeVarTs</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarZero">freeVarZero</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarZero">freeVarZero</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Free_Variables">Free_Variables</a> [section, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Free_Variables">Free_Variables</a> [section, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="gaia_hydras.GPaths.html#from:16">from:16</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#from:34">from:34</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#from:44">from:44</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#from:47">from:47</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#from:64">from:64</a> [binder, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#from:70">from:70</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#FS">FS</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#FS">FS</a> [abbreviation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#FS">FS</a> [constructor, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#fstar_S">fstar_S</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Fstar_S">Fstar_S</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#FS_rw">FS_rw</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#fs:19">fs:19</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="Goedel.PRrepresentable.html#fs:448">fs:448</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#fs:459">fs:459</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#fs:467">fs:467</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#fs:77">fs:77</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#fuel">fuel</a> [inductive, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Prelude.Fuel.html">Fuel</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#fuel:747">fuel:747</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#fuel:87">fuel:87</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#fuel:94">fuel:94</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Ackermann.model.html#func">func</a> [projection, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Functions">Functions</a> [projection, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#fun_le_trans">fun_le_trans</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#fun_le">fun_le</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#fun_le">fun_le</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection_is_ZL_bijection">fun_bijection_is_ZL_bijection</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection_codomain">fun_bijection_codomain</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_restr">fun_restr</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bij_i">fun_bij_i</a> [constructor, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection">fun_bijection</a> [inductive, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_inj_i">fun_inj_i</a> [constructor, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_injection">fun_injection</a> [inductive, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_inj">fun_inj</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_onto">fun_onto</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_codomain">fun_codomain</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#fun_equiv">fun_equiv</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#fun2PR">fun2PR</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Trace_exercise.html#fusion">fusion</a> [definition, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_not_PR">F_alpha_not_PR</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_not_PR_E0">F_alpha_not_PR_E0</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_restricted_mono_l">F_restricted_mono_l</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_limE">F_limE</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_succE">F_succE</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_0_eq">F_alpha_0_eq</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_mono_l">F_mono_l</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_zeroE">F_zeroE</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_positive">F_alpha_positive</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_Succ_le">F_alpha_Succ_le</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_dom">F_alpha_dom</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_mono">F_alpha_mono</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_gt">F_alpha_gt</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_">F_</a> [definition, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_succ_eqn">f_succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_lim_eqn">f_lim_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_zero_eqn">f_zero_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_iterate">f_star_iterate</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_Succ">f_star_Succ</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_eq2">f_eq2</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_zero_eqn">f_star_zero_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_">f_</a> [definition, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star">f_star</a> [definition, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_mono_l">F_mono_l</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_mono_l_0">F_mono_l_0</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.Hd">F_monotony_l.case_lt.Hd</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.Hlt">F_monotony_l.case_lt.Hlt</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.n">F_monotony_l.case_lt.n</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt">F_monotony_l.case_lt</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_eq.Heq">F_monotony_l.case_eq.Heq</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_eq">F_monotony_l.case_eq</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.H'_beta_alpha">F_monotony_l.H'_beta_alpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta">F_monotony_l.beta</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha">F_monotony_l.alpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l">F_monotony_l</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_ge_S">F_alpha_ge_S</a> [abbreviation, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_restricted_mono_l">F_restricted_mono_l</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_dom">F_alpha_dom</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_Succ_le">F_alpha_Succ_le</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_positive">F_alpha_positive</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_gt">F_alpha_gt</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_mono">F_alpha_mono</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_One_Zero_ge">F_One_Zero_ge</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_One_Zero_dom">F_One_Zero_dom</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_0_eq">F_alpha_0_eq</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_succ_eqn">F_succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_lim_eqn">F_lim_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_zero_eqn">F_zero_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_iterate">F_star_iterate</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_Succ">F_star_Succ</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_eq2">F_eq2</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_zero_eqn">F_star_zero_eqn</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_">F_</a> [definition, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star">F_star</a> [definition, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#f_ok_inv">f_ok_inv</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR_inv">F_alpha_notPR_inv</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_PR_inv">F_alpha_PR_inv</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_n_PR">F_n_PR</a> [instance, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_0_isPR">F_0_isPR</a> [instance, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_not_PR">F_alpha_not_PR</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.H">F_alpha_notPR.H</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.case_lt.Halpha">F_alpha_notPR.case_lt.Halpha</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.case_lt">F_alpha_notPR.case_lt</a> [section, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha">F_alpha_notPR.alpha</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR">F_alpha_notPR</a> [section, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_omega_not_PR">F_omega_not_PR</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_omega_PR:13">F_omega_PR:13</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_omega_notPR">F_omega_notPR</a> [section, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_vs_Ack">F_vs_Ack</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_iterate_rw">F_iterate_rw</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Ackermann.PA.html#f_Sv:55">f_Sv:55</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#f_0:54">f_0:54</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#f_Sv:4">f_Sv:4</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#f_0:3">f_0:3</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#F_incl_ij">F_incl_ij</a> [instance, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#F_alpha_Sn">F_alpha_Sn</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#F_3_eqn">F_3_eqn</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#f_minoration">f_minoration</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#f_sup_commutes">f_sup_commutes</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#F_not_lim">F_not_lim</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.rpo.html#f_l:286">f_l:286</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#f_l:283">f_l:283</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#f_l:274">f_l:274</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html">F_3</a> [library]<br/>
+<a href="hydras.Epsilon0.F_alpha.html">F_alpha</a> [library]<br/>
+<a href="hydras.Epsilon0.F_omega.html">F_omega</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#f':19">f':19</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#f':22">f':22</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#f':33">f':33</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#f':351">f':351</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#f':36">f':36</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#f0">f0</a> [definition, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F0">F0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f0:1239">f0:1239</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f0:1242">f0:1242</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f0:1296">f0:1296</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f0:1299">f0:1299</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.PA.html#f0:13">f0:13</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#f0:148">f0:148</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#f0:151">f0:151</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#f0:158">f0:158</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#f0:160">f0:160</a> [binder, in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:208">f0:208</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:231">f0:231</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:249">f0:249</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:299">f0:299</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:345">f0:345</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:389">f0:389</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:41">f0:41</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:42">f0:42</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:43">f0:43</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:44">f0:44</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.fol.html#f0:441">f0:441</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:45">f0:45</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#f0:46">f0:46</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#f0:62">f0:62</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="additions.AM.html#F1">F1</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#F1">F1</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1">F1</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F1">F1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#f1">f1</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#F1">F1</a> [lemma, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f1rec:67">f1rec:67</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.AM.html#F1_correct">F1_correct</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_proper">F1_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_correct">F1_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_neutral_r">F1_neutral_r</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_neutral_l">F1_neutral_l</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F1_simpl">F1_simpl</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F144">F144</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#F197887">F197887</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#f1:109">f1:109</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#f1:110">f1:110</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#f1:112">f1:112</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#f1:12">f1:12</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="additions.Euclidean_Chains.html#f1:140">f1:140</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#f1:144">f1:144</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#f1:152">f1:152</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#f1:17">f1:17</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="additions.Euclidean_Chains.html#f1:19">f1:19</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.rpo.term.html#f1:2">f1:2</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#f1:2">f1:2</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#f1:200">f1:200</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<br/><br/><a id="global_G"></a><h2>G </h2>
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+<a href="additions.AM.html#Gamma.gamma">Gamma.gamma</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Gamma.gamma">Gamma.gamma</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Gamma0">Gamma0</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0">Gamma0</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html">Gamma0</a> [library]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_rpo">Gamma0_rpo</a> [module, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_alg">Gamma0_alg</a> [module, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_transitive">Gamma0_prec.prec_transitive</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_antisym">Gamma0_prec.prec_antisym</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_dec">Gamma0_prec.prec_dec</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.status">Gamma0_prec.status</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.Mul">Gamma0_prec.Mul</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.Lex">Gamma0_prec.Lex</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.status_type">Gamma0_prec.status_type</a> [inductive, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec">Gamma0_prec.prec</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.A">Gamma0_prec.A</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec">Gamma0_prec</a> [module, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.arity">Gamma0_sig.arity</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free">Gamma0_sig.Free</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.C">Gamma0_sig.C</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.AC">Gamma0_sig.AC</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.arity_type">Gamma0_sig.arity_type</a> [inductive, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.eq_symbol_dec">Gamma0_sig.eq_symbol_dec</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb">Gamma0_sig.symb</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons">Gamma0_sig.ord_cons</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi">Gamma0_sig.ord_psi</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero">Gamma0_sig.ord_zero</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S">Gamma0_sig.nat_S</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0">Gamma0_sig.nat_0</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb0">Gamma0_sig.symb0</a> [inductive, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig">Gamma0_sig</a> [module, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le4">Gamma0.add_le4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le3">Gamma0.add_le3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_inj">Gamma0.add_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le2">Gamma0.add_le2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le1">Gamma0.add_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_to_cons">Gamma0.add_to_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_fin_omega">Gamma0.add_fin_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_int">Gamma0.add_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add1Nfin">Gamma0.add1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr4">Gamma0.ap_pr4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr3">Gamma0.ap_pr3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr2">Gamma0.ap_pr2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr1">Gamma0.ap_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_limit">Gamma0.ap_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr0">Gamma0.ap_pr0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Eq">Gamma0.CompareT2Eq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Gt">Gamma0.CompareT2Gt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Lt">Gamma0.CompareT2Lt</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.compare_T2">Gamma0.compare_T2</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.cons">Gamma0.cons</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.epsilon0">Gamma0.epsilon0</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.le_succ_succE">Gamma0.le_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.le_succ_succ">Gamma0.le_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.limit_pr">Gamma0.limit_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.limit_pr1">Gamma0.limit_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_4">Gamma0.lt_succ_le_4</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_3">Gamma0.lt_succ_le_3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_2">Gamma0.lt_succ_le_2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_1">Gamma0.lt_succ_le_1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_succE">Gamma0.lt_succ_succE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_succ">Gamma0.lt_succ_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_tail">Gamma0.lt_tail</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_psi">Gamma0.lt_psi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_rec">Gamma0.lt_rec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.minus_le">Gamma0.minus_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.minus_lt">Gamma0.minus_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_phi">Gamma0.nf_phi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_add">Gamma0.nf_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_sub">Gamma0.nf_sub</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_pred">Gamma0.nf_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_succ">Gamma0.nf_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_split">Gamma0.nf_split</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_finite">Gamma0.nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_one">Gamma0.nf_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_omega">Gamma0.nf_omega</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_psi">Gamma0.nf_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.no_critical">Gamma0.no_critical</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega">Gamma0.omega</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_minus_one">Gamma0.omega_minus_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_least_inf2">Gamma0.omega_least_inf2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_least_inf1">Gamma0.omega_least_inf1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_lt_epsilon0">Gamma0.omega_lt_epsilon0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.one">Gamma0.one</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi">Gamma0.phi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv3">Gamma0.phi_inv3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv2">Gamma0.phi_inv2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv0">Gamma0.phi_inv0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_ltE">Gamma0.phi_ltE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_eqE">Gamma0.phi_eqE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inj1">Gamma0.phi_inj1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inj">Gamma0.phi_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_mono_c">Gamma0.phi_mono_c</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_mono_b">Gamma0.phi_mono_b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_mono_a">Gamma0.phi_mono_a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv1">Gamma0.phi_inv1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_ab_le2">Gamma0.phi_ab_le2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_ab_le1">Gamma0.phi_ab_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec4c">Gamma0.phi_spec4c</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec4b">Gamma0.phi_spec4b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec4a">Gamma0.phi_spec4a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec3">Gamma0.phi_spec3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec2">Gamma0.phi_spec2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_spec1">Gamma0.phi_spec1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_principalR">Gamma0.phi_principalR</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_cases">Gamma0.phi_cases</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_succ">Gamma0.phi_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_fix2">Gamma0.phi_fix2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_fix1">Gamma0.phi_fix1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_le3">Gamma0.phi_le3</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_le2">Gamma0.phi_le2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_le1">Gamma0.phi_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_ap">Gamma0.phi_ap</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.pred_succ">Gamma0.pred_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.pred_lt">Gamma0.pred_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.pred_le">Gamma0.pred_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi">Gamma0.psi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi_phi1">Gamma0.psi_phi1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi_phi">Gamma0.psi_phi</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi_phi_aux">Gamma0.psi_phi_aux</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi_lt2">Gamma0.psi_lt2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.psi_lt1">Gamma0.psi_lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.size">Gamma0.size</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.size_prop">Gamma0.size_prop</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.size_prop1">Gamma0.size_prop1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_add">Gamma0.split_add</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_le">Gamma0.split_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_pred">Gamma0.split_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_succ">Gamma0.split_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_is_succ">Gamma0.split_is_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_limit">Gamma0.split_limit</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.split_finite">Gamma0.split_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_pr1r">Gamma0.sub_pr1r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_nz">Gamma0.sub_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_pr1">Gamma0.sub_pr1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_pr">Gamma0.sub_pr</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_le1">Gamma0.sub_le1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub_int">Gamma0.sub_int</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub1a">Gamma0.sub1a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub1b">Gamma0.sub1b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.sub1Nfin">Gamma0.sub1Nfin</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_is_add_one">Gamma0.succ_is_add_one</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_psi_lt2">Gamma0.succ_psi_lt2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_psi_lt">Gamma0.succ_psi_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_psi">Gamma0.succ_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_nz">Gamma0.succ_nz</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_inj">Gamma0.succ_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_p1">Gamma0.succ_p1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_pred">Gamma0.succ_pred</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.succ_lt">Gamma0.succ_lt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1succ_nat">Gamma0.T1succ_nat</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2">Gamma0.T1T2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2range1">Gamma0.T1T2range1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2range2">Gamma0.T1T2range2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2_inc">Gamma0.T1T2_inc</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T1T2_inj">Gamma0.T1T2_inj</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2">Gamma0.T2</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2add">Gamma0.T2add</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2addA">Gamma0.T2addA</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2addn0">Gamma0.T2addn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2add_eq0">Gamma0.T2add_eq0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2add0n">Gamma0.T2add0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ap">Gamma0.T2ap</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2eq">Gamma0.T2eq</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2eqE">Gamma0.T2eqE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2eqP">Gamma0.T2eqP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2eq_add2l">Gamma0.T2eq_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2eq_le">Gamma0.T2eq_le</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite">Gamma0.T2finite</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite_succ">Gamma0.T2finite_succ</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite1">Gamma0.T2finite1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2finite2">Gamma0.T2finite2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GeqNotGtn">Gamma0.T2GeqNotGtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GeqNotLtn">Gamma0.T2GeqNotLtn</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ge1">Gamma0.T2ge1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GtnNotLeq">Gamma0.T2GtnNotLeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2gt1">Gamma0.T2gt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2is_succ">Gamma0.T2is_succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le">Gamma0.T2le</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2leNgt">Gamma0.T2leNgt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lenn">Gamma0.T2lenn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2len0">Gamma0.T2len0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2leP">Gamma0.T2leP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2leq_xor_gtn">Gamma0.T2leq_xor_gtn</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_add2r">Gamma0.T2le_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_add2l">Gamma0.T2le_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_psi_b">Gamma0.T2le_psi_b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_psi1">Gamma0.T2le_psi1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_trans">Gamma0.T2le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_lt_trans">Gamma0.T2le_lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_total">Gamma0.T2le_total</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_eqVlt">Gamma0.T2le_eqVlt</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_consE">Gamma0.T2le_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le0n">Gamma0.T2le0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2limit">Gamma0.T2limit</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt">Gamma0.T2lt</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lta">Gamma0.T2lta</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltE">Gamma0.T2ltE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltgtP">Gamma0.T2ltgtP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltNge">Gamma0.T2ltNge</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltnn">Gamma0.T2ltnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2LtnNotGeq">Gamma0.T2LtnNotGeq</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltn_xor_geq">Gamma0.T2ltn_xor_geq</a> [inductive, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltn0">Gamma0.T2ltn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltP">Gamma0.T2ltP</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltW">Gamma0.T2ltW</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_add2r">Gamma0.T2lt_add2r</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_add2l">Gamma0.T2lt_add2l</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_psi_a">Gamma0.T2lt_psi_a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_psi_b">Gamma0.T2lt_psi_b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_le_trans">Gamma0.T2lt_le_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_trans">Gamma0.T2lt_trans</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_psi_aux">Gamma0.T2lt_psi_aux</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_trichotomy">Gamma0.T2lt_trichotomy</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_anti">Gamma0.T2lt_anti</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_neAle">Gamma0.T2lt_neAle</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_ne'">Gamma0.T2lt_ne'</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_ne">Gamma0.T2lt_ne</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_consE">Gamma0.T2lt_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt_psi">Gamma0.T2lt_psi</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt0n">Gamma0.T2lt0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2lt1">Gamma0.T2lt1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nat">Gamma0.T2nat</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nat_inc">Gamma0.T2nat_inc</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf">Gamma0.T2nf</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf_finite">Gamma0.T2nf_finite</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf_consE">Gamma0.T2nf_consE</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2nf_cons_cons">Gamma0.T2nf_cons_cons</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2pred">Gamma0.T2pred</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2split">Gamma0.T2split</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2sub">Gamma0.T2sub</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2subnn">Gamma0.T2subnn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2subn0">Gamma0.T2subn0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2sub0n">Gamma0.T2sub0n</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2succ">Gamma0.T2succ</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr2">Gamma0.T2_pr2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_pr1">Gamma0.T2_pr1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_eqType">Gamma0.T2_eqType</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_eqMixin">Gamma0.T2_eqMixin</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.zero">Gamma0.zero</a> [constructor, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x">_ - _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220">_ + _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&gt;'_x">_ &gt; _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a8834fd6e3c1640c213de7020e0510e2">_ &gt;= _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e">_ &lt;= _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x">_ &lt; _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d9e84494be2f615021318f367a1326d0">\F _ (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#d8473b352e1ea62d853482a112bf8ee4">[ _ , _ ] (g0_scope)</a> [notation, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma0:544">gamma0:544</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#gamma1">gamma1</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:34">gamma1:34</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:41">gamma1:41</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:49">gamma1:49</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:56">gamma1:56</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:62">gamma1:62</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma1:67">gamma1:67</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:35">gamma2:35</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:42">gamma2:42</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:50">gamma2:50</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:57">gamma2:57</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:63">gamma2:63</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma2:68">gamma2:68</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="additions.Strategies.html#gamma:1">gamma:1</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Schutte.AP.html#gamma:10">gamma:10</a> [binder, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma:102">gamma:102</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:111">gamma:111</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:112">gamma:112</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:114">gamma:114</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:117">gamma:117</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Critical.html#gamma:12">gamma:12</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#gamma:122">gamma:122</a> [binder, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:122">gamma:122</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:125">gamma:125</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:126">gamma:126</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:129">gamma:129</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:13">gamma:13</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:131">gamma:131</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:132">gamma:132</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:136">gamma:136</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:138">gamma:138</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:144">gamma:144</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#gamma:15">gamma:15</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:150">gamma:150</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:156">gamma:156</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:160">gamma:160</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:162">gamma:162</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:167">gamma:167</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:17">gamma:17</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:172">gamma:172</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:179">gamma:179</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#gamma:18">gamma:18</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:18">gamma:18</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#gamma:19">gamma:19</a> [binder, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:20">gamma:20</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Addition_Chains.html#gamma:203">gamma:203</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:216">gamma:216</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#gamma:22">gamma:22</a> [binder, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:220">gamma:220</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Fib2.html#gamma:23">gamma:23</a> [binder, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="gaia_hydras.GCanon.html#gamma:25">gamma:25</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:250">gamma:250</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:26">gamma:26</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gamma:27">gamma:27</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:298">gamma:298</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:302">gamma:302</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:309">gamma:309</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:31">gamma:31</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:315">gamma:315</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:33">gamma:33</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:339">gamma:339</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:34">gamma:34</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:34">gamma:34</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:34">gamma:34</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:342">gamma:342</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:35">gamma:35</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:36">gamma:36</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:38">gamma:38</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:38">gamma:38</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#gamma:39">gamma:39</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:396">gamma:396</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#gamma:40">gamma:40</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:40">gamma:40</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:44">gamma:44</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:468">gamma:468</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:47">gamma:47</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:48">gamma:48</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:498">gamma:498</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.CNF.html#gamma:50">gamma:50</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:50">gamma:50</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:509">gamma:509</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:51">gamma:51</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:513">gamma:513</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:513">gamma:513</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:516">gamma:516</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:519">gamma:519</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:52">gamma:52</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:54">gamma:54</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:542">gamma:542</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:545">gamma:545</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.O2H.html#gamma:58">gamma:58</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:58">gamma:58</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gamma:581">gamma:581</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:584">gamma:584</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:59">gamma:59</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:590">gamma:590</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.O2H.html#gamma:60">gamma:60</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:60">gamma:60</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Schutte.Addition.html#gamma:61">gamma:61</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:627">gamma:627</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.O2H.html#gamma:63">gamma:63</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:631">gamma:631</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:637">gamma:637</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:643">gamma:643</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:660">gamma:660</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:663">gamma:663</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:67">gamma:67</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#gamma:676">gamma:676</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.AP.html#gamma:68">gamma:68</a> [binder, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#gamma:693">gamma:693</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Strategies.html#gamma:7">gamma:7</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#gamma:72">gamma:72</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.Critical.html#gamma:74">gamma:74</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Hydra.O2H.html#gamma:82">gamma:82</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#gamma:82">gamma:82</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#gamma:82">gamma:82</a> [binder, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#gamma:85">gamma:85</a> [binder, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:91">gamma:91</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#gamma:96">gamma:96</a> [binder, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#gamma:97">gamma:97</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.GCanon.html">GCanon</a> [library]<br/>
+<a href="gaia_hydras.GCanon.html#gcanon_limit_v2">gcanon_limit_v2</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#gcanon_zero">gcanon_zero</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons">gcons</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons_lt_e0">gcons_lt_e0</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons_nf">gcons_nf</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#ge">ge</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#GEN">GEN</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage.R">General_Relations.elagage.to_nat_elagage.R</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage">General_Relations.elagage.to_nat_elagage</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage">General_Relations.elagage</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R_surj">General_Relations.surjection2injection.R_surj</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R">General_Relations.surjection2injection.R</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection">General_Relations.surjection2injection</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R_inj">General_Relations.injection2surjection.R_inj</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R">General_Relations.injection2surjection.R</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection">General_Relations.injection2surjection</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.DB">General_Relations.DB</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.DA">General_Relations.DA</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.B">General_Relations.B</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.A">General_Relations.A</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R">General_Relations.Definitions.R</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DB">General_Relations.Definitions.DB</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA">General_Relations.Definitions.DA</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.B">General_Relations.Definitions.B</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.A">General_Relations.Definitions.A</a> [variable, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions">General_Relations.Definitions</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations">General_Relations</a> [section, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic">Generic</a> [section, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.A">Generic.A</a> [variable, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging">Generic.Merging</a> [section, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness">Generic.Merging.Correctness</a> [section, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split_permutation">Generic.Merging.Correctness.merge_sort.split_permutation</a> [variable, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split_decr">Generic.Merging.Correctness.merge_sort.split_decr</a> [variable, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split">Generic.Merging.Correctness.merge_sort.split</a> [variable, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort">Generic.Merging.Correctness.merge_sort</a> [section, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le">Generic.Merging.le</a> [variable, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#93d511194bf8dfeef7681d9e29124228">_ &lt;= _</a> [notation, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Splitting">Generic.Splitting</a> [section, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="additions.Euclidean_Chains.html#gen_K">gen_K</a> [constructor, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#gen_F">gen_F</a> [constructor, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#gen:219">gen:219</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#get_decomposition">get_decomposition</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#get_predecessor">get_predecessor</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.solutions_exercises.ge_omega_iff.html#ge_omega_iff">ge_omega_iff</a> [lemma, in <a href="hydras.solutions_exercises.ge_omega_iff.html">hydras.solutions_exercises.ge_omega_iff</a>]<br/>
+<a href="hydras.Schutte.Addition.html#ge_zero">ge_zero</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#ge_o_segment">ge_o_segment</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.solutions_exercises.ge_omega_iff.html">ge_omega_iff</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#gE0le_iff">gE0le_iff</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#gE0lt_wf">gE0lt_wf</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#gE0lt_iff">gE0lt_iff</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#gE0_eq_intro">gE0_eq_intro</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html">GF_alpha</a> [library]<br/>
+<a href="gaia_hydras.GHessenberg.html">GHessenberg</a> [library]<br/>
+<a href="gaia_hydras.GHprime.html">GHprime</a> [library]<br/>
+<a href="gaia_hydras.GHydra.html">GHydra</a> [library]<br/>
+<a href="gaia_hydras.GLarge_Sets.html">GLarge_Sets</a> [library]<br/>
+<a href="gaia_hydras.GL_alpha.html">GL_alpha</a> [library]<br/>
+<a href="gaia_hydras.GPaths.html#gnaw">gnaw</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Gnaw">Gnaw</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw">gnaw</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#gnawS">gnawS</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#GnawS">GnawS</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS">gnawS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#GnawS_omega">GnawS_omega</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#GnawS_Gnaw">GnawS_Gnaw</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_path_toS">gnawS_path_toS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_cut2">gnawS_cut2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_cut1">gnawS_cut1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_SSn">gnawS_SSn</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_tail">gnawS_tail</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_succ_eqn2">gnawS_succ_eqn2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_succ_eqn1">gnawS_succ_eqn1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_lim2">gnawS_lim2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_lim1">gnawS_lim1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaws_rw">gnaws_rw</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_app">gnawS_app</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_to_path_toS">gnawS_to_path_toS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_nf">gnawS_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_zero">gnawS_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS_gnaw">gnawS_gnaw</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_omega_1">gnaw_omega_1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_omega_n_SSn">gnaw_omega_n_SSn</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Gnaw_omega">Gnaw_omega</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Gnaw_GnawS">Gnaw_GnawS</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_last_step">gnaw_last_step</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_n_R">gnaw_n_R</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_finite">gnaw_finite</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_finite_1">gnaw_finite_1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#gnaw_finite_1_iota">gnaw_finite_1_iota</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_path_to">gnaw_path_to</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_app">gnaw_app</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_to_path_to">gnaw_to_path_to</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_rw">gnaw_rw</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_succ">gnaw_succ</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_zero">gnaw_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_nf">gnaw_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw_gnawS">gnaw_gnawS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel'sIncompleteness1st">Goedel'sIncompleteness1st</a> [lemma, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT2">Goedel's_1st_Incompleteness.expressT2</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT1">Goedel's_1st_Incompleteness.expressT1</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.freeVarRepT">Goedel's_1st_Incompleteness.freeVarRepT</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0">Goedel's_1st_Incompleteness.v0</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT">Goedel's_1st_Incompleteness.repT</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.extendsNN">Goedel's_1st_Incompleteness.extendsNN</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T">Goedel's_1st_Incompleteness.T</a> [variable, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness">Goedel's_1st_Incompleteness</a> [section, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.HBL3">Goedel's_2nd_Incompleteness.HBL3</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.HBL2">Goedel's_2nd_Incompleteness.HBL2</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.Goedel1PA">Goedel's_2nd_Incompleteness.Goedel1PA</a> [section, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT2">Goedel's_2nd_Incompleteness.expressT2</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT1">Goedel's_2nd_Incompleteness.expressT1</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.freeVarRepT">Goedel's_2nd_Incompleteness.freeVarRepT</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0">Goedel's_2nd_Incompleteness.v0</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT">Goedel's_2nd_Incompleteness.repT</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.extendsPA">Goedel's_2nd_Incompleteness.extendsPA</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T">Goedel's_2nd_Incompleteness.T</a> [variable, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness">Goedel's_2nd_Incompleteness</a> [section, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel1.html">goedel1</a> [library]<br/>
+<a href="Goedel.goedel2.html">goedel2</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#go:159">go:159</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="gaia_hydras.GPaths.html">GPaths</a> [library]<br/>
+<a href="gaia_hydras.GPrelude.html">GPrelude</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#gP:26">gP:26</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#grec:79">grec:79</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#GRelation">GRelation</a> [definition, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html">GRelations</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#gR:29">gR:29</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="Goedel.goedel2.html#GS">GS</a> [lemma, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#gstrict_lub">gstrict_lub</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#gs:37">gs:37</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gtail">gtail</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#gtIsPR">gtIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#g_bij:91">g_bij:91</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#g_bij:85">g_bij:85</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_1_of">g_1_of</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_1_bij">g_1_bij</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_bij">g_bij</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_mono">g_mono</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_lemma">g_lemma</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_def">g_def</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_beta':130">g_beta':130</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_beta:129">g_beta:129</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_unic">g_unic</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_beta:128">g_beta:128</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#g_def1">g_def1</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.rpo.rpo.html#g_l':275">g_l':275</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0">G0</a> [module, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.compare_correct">G0.compare_correct</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.compare_G0">G0.compare_G0</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Ex42">G0.Ex42</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Finite">G0.Finite</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Gamma0">G0.Gamma0</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Gamma0_comp">G0.Gamma0_comp</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.G0">G0.G0</a> [record, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.le">G0.le</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt">G0.lt</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.LT">G0.LT</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt_wf">G0.lt_wf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt_sto">G0.lt_sto</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt_LT_incl">G0.lt_LT_incl</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Lt_wf">G0.Lt_wf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.mkg0">G0.mkg0</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.nfb_proof_unicity">G0.nfb_proof_unicity</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.nfb_equiv">G0.nfb_equiv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.nfb_a">G0.nfb_a</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.omega">G0.omega</a> [abbreviation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Omega">G0.Omega</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.phi">G0.phi</a> [abbreviation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Phi">G0.Phi</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.phi0">G0.phi0</a> [abbreviation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Plus">G0.Plus</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.vnf">G0.vnf</a> [projection, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.vnf_ok">G0.vnf_ok</a> [projection, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.zero">G0.zero</a> [instance, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.zero_nfb">G0.zero_nfb</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#e70a85d390e2078e674210edfffae3cb">omega^ (g0_scope)</a> [notation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#f8274dee76f0f0f5010365f2cfa67705">_ + _ (g0_scope)</a> [notation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#g0:22">g0:22</a> [binder, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#g0:23">g0:23</a> [binder, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#g0:64">g0:64</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#g0:65">g0:65</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#g1:114">g1:114</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#g1:117">g1:117</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h">g2h</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#g2h">g2h</a> [definition, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.GCanon.html#g2h_canon">g2h_canon</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_seq">g2h_seq</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_E0zero">g2h_E0zero</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_E0_succ">g2h_E0_succ</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="hydras.Ackermann.LNN.html#g:75">g:75</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
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+<a href="Goedel.codeSysPrf.html#g:86">g:86</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#g:89">g:89</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="Goedel.codeSysPrf.html#g:89">g:89</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#g:89">g:89</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
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+<a href="hydras.Schutte.PartialFun.html#g:90">g:90</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#g:92">g:92</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#g:92">g:92</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#g:94">g:94</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#g:95">g:95</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#g:96">g:96</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#g:96">g:96</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#g:98">g:98</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#g:98">g:98</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#g:99">g:99</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<br/><br/><a id="global_H"></a><h2>H </h2>
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+<a href="hydras.Epsilon0.Large_Sets.html#HAB:175">HAB:175</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#HAB:22">HAB:22</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#HAck:88">HAck:88</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="additions.BinaryStrat.html#half">half</a> [definition, in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="gaia_hydras.GHprime.html#Halpha:14">Halpha:14</a> [binder, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Halpha:170">Halpha:170</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Halpha:401">Halpha:401</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Halpha:662">Halpha:662</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Halpha:665">Halpha:665</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Halpha:710">Halpha:710</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#Ha:17">Ha:17</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#HA:173">HA:173</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#HA:20">HA:20</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Ha:390">Ha:390</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Ha:50">Ha:50</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Ha:9">Ha:9</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Ha:98">Ha:98</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="Goedel.goedel2.html#HBL1">HBL1</a> [lemma, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hbounded_transitionS">hbounded_transitionS</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Hb:100">Hb:100</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#HB:174">HB:174</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#HB:21">HB:21</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Hb:391">Hb:391</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#hcanon">hcanon</a> [abbreviation, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hCanon_plus">hCanon_plus</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hcnf">hcnf</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hcons">hcons</a> [abbreviation, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hcons">hcons</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hconst_path">hconst_path</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hconst_pathS">hconst_pathS</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_S2">hcons_mult_S2</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_S1">hcons_mult_S1</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_S0">hcons_mult_S0</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_comm">hcons_mult_comm</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_app">hcons_mult_app</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hcons_mult">hcons_mult</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Hc:102">Hc:102</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#head">head</a> [abbreviation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.fib.html#headbits">headbits</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#head_LT_cons">head_LT_cons</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#head_lt_cons">head_lt_cons</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#head_lt">head_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#head_lt">head_lt</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#head_no_round">head_no_round</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#head_no_round_n">head_no_round_n</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#height">height</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#height_lt_size">height_lt_size</a> [lemma, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#height_bad">height_bad</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#height_var">height_var</a> [instance, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#height:17">height:17</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html">Hessenberg</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0">hE0</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0fin">hE0fin</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0is_succ">hE0is_succ</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0le">hE0le</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0limit">hE0limit</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0lt">hE0lt</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0omega">hE0omega</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0phi0">hE0phi0</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0zero">hE0zero</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#HFLim">HFLim</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#HFsucc">HFsucc</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#hF_">hF_</a> [abbreviation, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf':121">Hf':121</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf':155">Hf':155</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf':94">Hf':94</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#HF0">HF0</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:120">Hf:120</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:154">Hf:154</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:47">Hf:47</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:50">Hf:50</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#Hf:74">Hf:74</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Hf:75">Hf:75</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Hf:79">Hf:79</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Hf:84">Hf:84</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:88">Hf:88</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hf:93">Hf:93</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="additions.AM.html#Hgamma:159">Hgamma:159</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Fib2.html#Hgamma:24">Hgamma:24</a> [binder, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Hgamma:354">Hgamma:354</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Strategies.html#Hgamma:8">Hgamma:8</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hgnaw">hgnaw</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hgnawS">hgnawS</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hgt:163">Hgt:163</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Hydra.O2H.html#hh_2">hh_2</a> [constructor, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#hh_1">hh_1</a> [constructor, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hij:64">Hij:64</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Hincl:170">Hincl:170</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#hinit">hinit</a> [definition, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Hk:74">Hk:74</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hlambda:51">Hlambda:51</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hlarge">hlarge</a> [abbreviation, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hlargeS">hlargeS</a> [abbreviation, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hlim:45">Hlim:45</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hlim:52">Hlim:52</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hlim:63">Hlim:63</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hlim:67">Hlim:67</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hlt_iff">hlt_iff</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hL_spec">hL_spec</a> [abbreviation, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hmlarge">hmlarge</a> [abbreviation, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hmlargeS">hmlargeS</a> [abbreviation, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Hmono:200">Hmono:200</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Hmu">Hmu</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="gaia_hydras.GaiaToHydra.html#hmultA">hmultA</a> [lemma, in <a href="gaia_hydras.GaiaToHydra.html">gaia_hydras.GaiaToHydra</a>]<br/>
+<a href="gaia_hydras.GaiaToHydra.html#hmult_dist">hmult_dist</a> [lemma, in <a href="gaia_hydras.GaiaToHydra.html">gaia_hydras.GaiaToHydra</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Hmu:2">Hmu:2</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Hmu:3">Hmu:3</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hm':165">Hm':165</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#Hm:164">Hm:164</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hnf_g2h">hnf_g2h</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#Hnf:3">Hnf:3</a> [binder, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#Hnf:34">Hnf:34</a> [binder, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hnf:40">Hnf:40</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hnf:44">Hnf:44</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hnf:62">Hnf:62</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#Hnf:66">Hnf:66</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hnil">hnil</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#hoplus">hoplus</a> [abbreviation, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpath">hpath</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpathS">hpathS</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpath_to">hpath_to</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html">Hprime</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#hP:25">hP:25</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:722">Hrecb:722</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:726">Hrecb:726</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:728">Hrecb:728</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:731">Hrecb:731</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:734">Hrecb:734</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:737">Hrecb:737</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:740">Hrecb:740</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:743">Hrecb:743</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:746">Hrecb:746</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:749">Hrecb:749</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:752">Hrecb:752</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:755">Hrecb:755</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:758">Hrecb:758</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:760">Hrecb:760</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:786">Hrecb:786</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:789">Hrecb:789</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:793">Hrecb:793</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:796">Hrecb:796</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:800">Hrecb:800</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:806">Hrecb:806</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:811">Hrecb:811</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Hrecb:814">Hrecb:814</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:128">Hrecn:128</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:144">Hrecn:144</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:147">Hrecn:147</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:151">Hrecn:151</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:155">Hrecn:155</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:157">Hrecn:157</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Hrecn:160">Hrecn:160</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.model.html#Hrecn:178">Hrecn:178</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Hrecn:185">Hrecn:185</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Hrecn:204">Hrecn:204</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Hrecn:211">Hrecn:211</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Hrecn:64">Hrecn:64</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:10">Hrecs:10</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:108">Hrecs:108</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:11">Hrecs:11</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:113">Hrecs:113</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:116">Hrecs:116</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:12">Hrecs:12</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:121">Hrecs:121</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:124">Hrecs:124</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:127">Hrecs:127</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:13">Hrecs:13</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:130">Hrecs:130</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:133">Hrecs:133</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:136">Hrecs:136</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:139">Hrecs:139</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:14">Hrecs:14</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:142">Hrecs:142</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:145">Hrecs:145</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:147">Hrecs:147</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:149">Hrecs:149</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:15">Hrecs:15</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:151">Hrecs:151</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:153">Hrecs:153</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:155">Hrecs:155</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:16">Hrecs:16</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:165">Hrecs:165</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:168">Hrecs:168</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:17">Hrecs:17</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:171">Hrecs:171</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:174">Hrecs:174</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:177">Hrecs:177</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:18">Hrecs:18</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:180">Hrecs:180</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Hrecs:19">Hrecs:19</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:19">Hrecs:19</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:197">Hrecs:197</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:199">Hrecs:199</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:2">Hrecs:2</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:20">Hrecs:20</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:201">Hrecs:201</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:203">Hrecs:203</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:205">Hrecs:205</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:207">Hrecs:207</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Hrecs:21">Hrecs:21</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:21">Hrecs:21</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:213">Hrecs:213</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:218">Hrecs:218</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:22">Hrecs:22</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:221">Hrecs:221</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:226">Hrecs:226</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:229">Hrecs:229</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:23">Hrecs:23</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:232">Hrecs:232</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:235">Hrecs:235</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:238">Hrecs:238</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:24">Hrecs:24</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:241">Hrecs:241</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:244">Hrecs:244</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:247">Hrecs:247</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:25">Hrecs:25</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:250">Hrecs:250</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:253">Hrecs:253</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:256">Hrecs:256</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:259">Hrecs:259</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:26">Hrecs:26</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:27">Hrecs:27</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:276">Hrecs:276</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:278">Hrecs:278</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:28">Hrecs:28</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:280">Hrecs:280</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:282">Hrecs:282</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:284">Hrecs:284</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:286">Hrecs:286</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:29">Hrecs:29</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:30">Hrecs:30</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:31">Hrecs:31</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:32">Hrecs:32</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:34">Hrecs:34</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:36">Hrecs:36</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:41">Hrecs:41</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:42">Hrecs:42</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:44">Hrecs:44</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:46">Hrecs:46</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:46">Hrecs:46</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:48">Hrecs:48</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:49">Hrecs:49</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:5">Hrecs:5</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:54">Hrecs:54</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:57">Hrecs:57</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:60">Hrecs:60</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:63">Hrecs:63</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:66">Hrecs:66</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:69">Hrecs:69</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:7">Hrecs:7</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:7">Hrecs:7</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:72">Hrecs:72</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:74">Hrecs:74</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:76">Hrecs:76</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:78">Hrecs:78</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:80">Hrecs:80</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:82">Hrecs:82</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:84">Hrecs:84</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:86">Hrecs:86</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:9">Hrecs:9</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:90">Hrecs:90</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:92">Hrecs:92</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:94">Hrecs:94</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:96">Hrecs:96</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hR:28">hR:28</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hsize">hsize</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hstandard_gnaw">hstandard_gnaw</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hs_eq_dec">hs_eq_dec</a> [lemma, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs':140">hs':140</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs':142">hs':142</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hs2l">hs2l</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hs2lK">hs2lK</a> [lemma, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs:139">hs:139</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs:141">hs:141</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hs:8">hs:8</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hs:9">hs:9</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GPaths.html#htransition">htransition</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:18">Htrans:18</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:21">Htrans:21</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:25">Htrans:25</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hT1">hT1</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hT2">hT2</a> [abbreviation, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hVar">hVar</a> [instance, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#HVariant">HVariant</a> [instance, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hvariant">Hvariant</a> [record, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#HVariant_0">HVariant_0</a> [instance, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Hvariant_Termination">Hvariant_Termination</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Hvar:2">Hvar:2</a> [binder, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Hvar:2">Hvar:2</a> [binder, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Hwf:130">Hwf:130</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Hwf:21">Hwf:21</a> [binder, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#HX:142">HX:142</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#HX:145">HX:145</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:27">Hx:27</a> [binder, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:31">Hx:31</a> [binder, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:38">Hx:38</a> [binder, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#HX:61">HX:61</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#hyd">hyd</a> [abbreviation, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydra">Hydra</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydrae">Hydrae</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydrae_rect2">Hydrae_rect2</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.HydraGaia_Examples.html">HydraGaia_Examples</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydra_rect2">Hydra_rect2</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html">Hydra_Termination</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Extraction.html">Hydra_Extraction</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html">Hydra_Lemmas</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html">Hydra_Definitions</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html">Hydra_Examples</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html">Hydra_Theorems</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd_mult">hyd_mult</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd1">hyd1</a> [abbreviation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd2">hyd2</a> [abbreviation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd3">hyd3</a> [abbreviation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#hyd4">hyd4</a> [abbreviation, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#hyper_exp2_S">hyper_exp2_S</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#hyper_exp2">hyper_exp2</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos">Hypos</a> [section, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hn">Hypos.Hn</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hnp">Hypos.Hnp</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hp">Hypos.Hp</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.R">Hypos.R</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.search_toggle">Hypos.search_toggle</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.spec">Hypos.spec</a> [variable, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Hyp1:20">Hyp1:20</a> [binder, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Hyp2:21">Hyp2:21</a> [binder, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.fol.html#hyp:220">hyp:220</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#hyp:221">hyp:221</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#hyp:222">hyp:222</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Hyp:24">Hyp:24</a> [binder, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Hyp:46">Hyp:46</a> [binder, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Hyp:48">Hyp:48</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Hydra.O2H.html#hy_app_l_nil">hy_app_l_nil</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hy_app_assoc">hy_app_assoc</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hy_app">hy_app</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Hy'':5">Hy'':5</a> [binder, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hy:28">Hy:28</a> [binder, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Hy:4">Hy:4</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Hy:5">Hy:5</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Hy:6">Hy:6</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hzero">hzero</a> [abbreviation, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h_eq_dec">h_eq_dec</a> [lemma, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h_forall">h_forall</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#H'_F">H'_F</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_omega_cube_min">H'_omega_cube_min</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_alpha_gt">H'_alpha_gt</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_alpha_mono">H'_alpha_mono</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_dom">H'_dom</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_omega_double">H'_omega_double</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_omega">H'_omega</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_eq3">H'_eq3</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_eq2">H'_eq2</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_eq1">H'_eq1</a> [lemma, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_">H'_</a> [definition, in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F">H'_F</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.IHalpha">H'_F.IHalpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.alpha">H'_F.alpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.P">H'_F.P</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F">H'_F</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#H'_L_">H'_L_</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_dom">H'_dom</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_mono_l">H'_mono_l</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_mono_l_1">H'_mono_l_1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_mono_l_0">H'_mono_l_0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_mono_weak">H'_alpha_mono_weak</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_ge_id">H'_alpha_ge_id</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_restricted_mono_l">H'_restricted_mono_l</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_dom">H'_alpha_dom</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_Succ_le">H'_alpha_Succ_le</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_gt">H'_alpha_gt</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_alpha_mono">H'_alpha_mono</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_non_mono1">H'_non_mono1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_min">H'_omega_cube_min</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_sqr_min">H'_omega_sqr_min</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_omega_exact_formula">H'_Phi0_omega_exact_formula</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_omega">H'_Phi0_omega</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N">H'_omega_cube_3.N</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f">H'_omega_cube_3.f</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3">H'_omega_cube_3</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube">H'_omega_cube</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_Si">H'_Phi0_Si</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_succ">H'_Phi0_succ</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_succ_0">H'_Phi0_succ_0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Phi0_succ_1">H'_Phi0_succ_1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_succ_fun">H'_succ_fun</a> [definition, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Omega_term">H'_Omega_term</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Fin_iterate">H'_Fin_iterate</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Omega_term_0">H'_Omega_term_0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Omega_term_1">H'_Omega_term_1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons">H'_cons</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons.i">H'_cons.i</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons.alpha">H'_cons.alpha</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons">H'_cons</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_sqr">H'_omega_sqr</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_i">H'_omega_i</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_4">H'_omega_4</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_3">H'_omega_3</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_double">H'_omega_double</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Plus_Fin">H'_Plus_Fin</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega">H'_omega</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_Fin">H'_Fin</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_eq2">H'_eq2</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
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+<a href="hydras.Epsilon0.Hprime.html#H'_">H'_</a> [definition, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#H'_L_">H'_L_</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
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+<a href="hydras.Hydra.Hydra_Definitions.html#h'':135">h'':135</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h'':83">h'':83</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.O2H.html#h':100">h':100</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
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+<a href="hydras.Hydra.Hydra_Lemmas.html#h':124">h':124</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
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+<a href="hydras.Ackermann.LNN2LNT.html#H1:86">H1:86</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#h2g">h2g</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#h2g">h2g</a> [definition, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_seq">h2g_seq</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_zero">h2g_zero</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_cons">h2g_cons</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_diffE">h2g_diffE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_eqE">h2g_eqE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_g2hK">h2g_g2hK</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#h2g_eqE">h2g_eqE</a> [lemma, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#h2g_g2hK">h2g_g2hK</a> [lemma, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h2_h3">h2_h3</a> [lemma, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#H2:10">H2:10</a> [binder, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#h2:120">h2:120</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#H2:33">H2:33</a> [binder, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h2:37">h2:37</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#H2:374">H2:374</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h2:42">h2:42</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#H2:63">H2:63</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#h3">h3</a> [abbreviation, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h3">h3</a> [definition, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#h:1">h:1</a> [binder, in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#h:1">h:1</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="gaia_hydras.GHydra.html#h:1">h:1</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:1">h:1</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#h:10">h:10</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.GHydra.html#h:10">h:10</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:10">h:10</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:105">h:105</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:106">h:106</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#H:107">H:107</a> [binder, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:109">h:109</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#h:11">h:11</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#h:11">h:11</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="gaia_hydras.GHydra.html#h:11">h:11</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#h:11">h:11</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Schutte.Countable.html#H:11">H:11</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#h:110">h:110</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:112">h:112</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:114">h:114</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:114">h:114</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:117">h:117</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#h:12">h:12</a> [binder, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:12">h:12</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#h:120">h:120</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="additions.fib.html#h:120">h:120</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:120">h:120</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:121">h:121</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#h:123">h:123</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:123">h:123</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:123">h:123</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#h:125">h:125</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:129">h:129</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#h:13">h:13</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.GHydra.html#h:13">h:13</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#H:13">H:13</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:132">h:132</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:133">h:133</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#h:135">h:135</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:135">h:135</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:137">h:137</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#h:138">h:138</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:138">h:138</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#H:14">H:14</a> [binder, in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#H:14">H:14</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:142">h:142</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:144">h:144</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:146">h:146</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:149">h:149</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#h:15">h:15</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:151">h:151</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:152">h:152</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:155">h:155</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#H:157">H:157</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#H:16">H:16</a> [binder, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:162">h:162</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#H:166">H:166</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:166">h:166</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#H:172">H:172</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#H:174">H:174</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#h:175">h:175</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#H:179">H:179</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#H:18">H:18</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:18">h:18</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#H:182">H:182</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.folProp.html#H:195">H:195</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#H:201">H:201</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#H:204">H:204</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#H:204">H:204</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.folProp.html#H:211">H:211</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#H:215">H:215</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Prelude.STDPP_compat.html#H:22">H:22</a> [binder, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#H:220">H:220</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
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+<a href="hydras.Hydra.Hydra_Definitions.html#h:94">h:94</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#h:96">h:96</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#h:96">h:96</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:97">h:97</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#h:98">h:98</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h:99">h:99</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="global_I"></a><h2>I </h2>
+<a href="hydras.Ackermann.primRec.html#idIsPR">idIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.fib.html#idn:126">idn:126</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#id_le_f_alpha">id_le_f_alpha</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="additions.Monoid_instances.html#Id2">Id2</a> [definition, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.fib.html#id:118">id:118</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#iffExist">iffExist</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffE1">iffE1</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffE1">iffE1</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffE1">iffE1</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffE2">iffE2</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffE2">iffE2</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffE2">iffE2</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#iffH">iffH</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffI">iffI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffI">iffI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffI">iffI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffRefl">iffRefl</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffRefl">iffRefl</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffRefl">iffRefl</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffSym">iffSym</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffSym">iffSym</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffSym">iffSym</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffTrans">iffTrans</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffTrans">iffTrans</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffTrans">iffTrans</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ifThenElseH">ifThenElseH</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#if_simpl">if_simpl</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#If2IsPR">If2IsPR</a> [instance, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ignoreParams">ignoreParams</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ignoreParamsIsPR">ignoreParamsIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Ignore_Params">Ignore_Params</a> [section, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#IHn:102">IHn:102</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#image">image</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#image_as_Im">image_as_Im</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#image_B_g_seg">image_B_g_seg</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impE">impE</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impE">impE</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impE">impE</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#impForall">impForall</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.fol.html#impH">impH</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impI">impI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impI">impI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impI">impI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof">ImpossibilityProof</a> [section, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.m">ImpossibilityProof.m</a> [variable, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.mh">ImpossibilityProof.mh</a> [variable, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.mu">ImpossibilityProof.mu</a> [variable, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.nfMu">ImpossibilityProof.nfMu</a> [variable, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_free">Impossibility_free</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [variable, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [variable, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof">Impossibility_Proof</a> [section, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Impossibility_Proof">Impossibility_Proof</a> [section, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_std">Impossibility_std</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [variable, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [variable, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof">Impossibility_Proof</a> [section, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="gaia_hydras.GHydra.html#Impossibility_std">Impossibility_std</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#Impossibility_free">Impossibility_free</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [variable, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota">Impossibility_Proof.iota</a> [variable, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [variable, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m">Impossibility_Proof.m</a> [variable, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof">Impossibility_Proof</a> [section, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [variable, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [variable, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.iota">Impossibility_Proof.iota</a> [variable, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m">Impossibility_Proof.m</a> [variable, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof">Impossibility_Proof</a> [section, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1">Impossibility1</a> [section, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m">Impossibility1.m</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.mDecr">Impossibility1.mDecr</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.x">Impossibility1.x</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.y">Impossibility1.y</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossible">Impossible</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#impossible_nat">impossible_nat</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impRefl">impRefl</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impRefl">impRefl</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impRefl">impRefl</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impTrans">impTrans</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impTrans">impTrans</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impTrans">impTrans</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#IMP1">IMP1</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#IMP2">IMP2</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#im_U_f">im_U_f</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Incl">Incl</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Included_s">Included_s</a> [definition, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.b">Inclusion_ij.b</a> [variable, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.Hij">Inclusion_ij.Hij</a> [variable, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j">Inclusion_ij.j</a> [variable, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i">Inclusion_ij.i</a> [variable, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij">Inclusion_ij</a> [section, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#incl_decomp">incl_decomp</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#incl_inv">incl_inv</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="Goedel.rosser.html#Incompleteness">Incompleteness</a> [lemma, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Inconsistent">Inconsistent</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Inc_U_A">Inc_U_A</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#independent">independent</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#indIsPR">indIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.PA.html#induct">induct</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#InductionSchema">InductionSchema</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step.Hn">inductive_step.Hn</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step.n">inductive_step.n</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step">inductive_step</a> [section, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Schutte.Critical.html#induct_step">induct_step</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ind1ParamIsPR">ind1ParamIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ind2ParamIsPR">ind2ParamIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#Infinite">Infinite</a> [definition, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="additions.Trace_exercise.html#info">info</a> [inductive, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InH">InH</a> [record, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InH">InH</a> [inductive, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#inhNat">inhNat</a> [instance, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InHWit">InHWit</a> [projection, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InHWit">InHWit</a> [constructor, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#inhX:45">inhX:45</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_ord">Inh_ord</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_Ord_Ord">Inh_Ord_Ord</a> [instance, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_OSets">Inh_OSets</a> [instance, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#InH_Ord">InH_Ord</a> [instance, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#inh_Ord">inh_Ord</a> [axiom, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#inh_U">inh_U</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#inh:2">inh:2</a> [binder, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#inh:53">inh:53</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#inh:58">inh:58</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="additions.Trace_exercise.html#Init">Init</a> [constructor, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject">inject</a> [definition, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_lt_epsilon0_ex_unique">inject_lt_epsilon0_ex_unique</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_lt_epsilon0_ex">inject_lt_epsilon0_ex</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_lt_epsilon0_ex_cnf">inject_lt_epsilon0_ex_cnf</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_mult_fin_r">inject_mult_fin_r</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_plus">inject_plus</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_rw">inject_rw</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_lt_epsilon0">inject_lt_epsilon0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_monoR">inject_monoR</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_injective">inject_injective</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_mono">inject_mono</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_mono_0">inject_mono_0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_of_phi0">inject_of_phi0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_of_omega">inject_of_omega</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_of_finite">inject_of_finite</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_of_zero">inject_of_zero</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject_of_finite_pos">inject_of_finite_pos</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#input:61">input:61</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.AM.html#instr">instr</a> [inductive, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpFormula">interpFormula</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpRels">interpRels</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTerm">interpTerm</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTerms">interpTerms</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTermsVector">interpTermsVector</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval">interval</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_lt_not_In">interval_lt_not_In</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_sorted_ge">interval_sorted_ge</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_unroll">interval_unroll</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_app">interval_app</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_singleton">interval_singleton</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_not_empty_iff">interval_not_empty_iff</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_not_empty">interval_not_empty</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_length">interval_length</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#interval_def">interval_def</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="additions.Monoid_instances.html#Int63mult">Int63mult</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#int63_mult_op">int63_mult_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#inv">inv</a> [definition, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#invar:18">invar:18</a> [binder, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Inverse_fun">Inverse_fun</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.Rof">Inverse_Image_transp.Rof</a> [variable, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.f">Inverse_Image_transp.f</a> [variable, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.R">Inverse_Image_transp.R</a> [variable, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.B">Inverse_Image_transp.B</a> [variable, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.A">Inverse_Image_transp.A</a> [variable, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp">Inverse_Image_transp</a> [section, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f_b">inversion_of_bijection.f_b</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inv_spec">inversion_of_bijection.inv_spec</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f">inversion_of_bijection.f</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB">inversion_of_bijection.DB</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA">inversion_of_bijection.DA</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inhA">inversion_of_bijection.inhA</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.B">inversion_of_bijection.B</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.A">inversion_of_bijection.A</a> [variable, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection">inversion_of_bijection</a> [section, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#inv_Pi">inv_Pi</a> [lemma, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_fun_bij">inv_fun_bij</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_composeR">inv_composeR</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_compose">inv_compose</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_fun">inv_fun</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.rpo.closure.html#inv_trans">inv_trans</a> [lemma, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_2_1">In_betaFormula_subst_2_1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_2">In_betaFormula_subst_2</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula">In_betaFormula</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1">In_betaFormula_subst_1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1_2">In_betaFormula_subst_1_2</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1_2_0">In_betaFormula_subst_1_2_0</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#In_sorted_ge_inv">In_sorted_ge_inv</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_remove">in_remove</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_map_in">in_map_in</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_in_map">in_in_map</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarSys">In_freeVarSys</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarListFormulaE">In_freeVarListFormulaE</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarListFormula">In_freeVarListFormula</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#In_list_remove2">In_list_remove2</a> [abbreviation, in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#in_remove_neq">in_remove_neq</a> [lemma, in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#In_add2">In_add2</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#In_add1">In_add1</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#In_cases">In_cases</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota">iota</a> [definition, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#iota">iota</a> [definition, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iotas">iotas</a> [definition, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iotas_succ">iotas_succ</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_phi0">iota_phi0</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_of_succ">iota_of_succ</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_tail">iota_tail</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_rw4">iota_rw4</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_rw3">iota_rw3</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_rw2">iota_rw2</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_rw1">iota_rw1</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_succ_round">iota_succ_round</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_succ_round_n">iota_succ_round_n</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_succ_R1">iota_succ_R1</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota_iotas">iota_iotas</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_lt_not_In">iota_from_lt_not_In</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_sorted_ge">iota_from_sorted_ge</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_unroll">iota_from_unroll</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_plus">iota_from_plus</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_app">iota_from_app</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from">iota_from</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#iota_adapt">iota_adapt</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#iota_ind">iota_ind</a> [lemma, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#iota_eq">iota_eq</a> [lemma, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#iota_spec_1">iota_spec_1</a> [lemma, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#iota_fun_ind">iota_fun_ind</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#iota_fun_e">iota_fun_e</a> [lemma, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#iota_fun">iota_fun</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.le_bottom_zero">iota_demo.Bad.le_bottom_zero</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom_eq">iota_demo.Bad.bottom_eq</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.le_zero_bottom">iota_demo.Bad.le_zero_bottom</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom">iota_demo.Bad.bottom</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad">iota_demo.Bad</a> [module, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.zero_le">iota_demo.zero_le</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.zero">iota_demo.zero</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.zero">iota_demo.zero</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.R">iota_demo.R</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo">iota_demo</a> [module, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#iota_mono">iota_mono</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#iota_compare_commute">iota_compare_commute</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#iota_ij">iota_ij</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:101">iota:101</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:114">iota:114</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:129">iota:129</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:141">iota:141</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:154">iota:154</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:38">iota:38</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:71">iota:71</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iota:89">iota:89</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_op">Iso_same_op</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_fun">Iso_same_fun</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_cst">Iso_same_cst</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_inv2">iso_inv2</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_inv1">iso_inv1</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_compare">iso_compare</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPR">isPR</a> [record, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPR">isPR</a> [inductive, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRextEqual">isPRextEqual</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRrel">isPRrel</a> [record, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRrel">isPRrel</a> [inductive, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRTrans">isPRTrans</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#iSPR_Ack_Sn">iSPR_Ack_Sn</a> [instance, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#isqrt">isqrt</a> [definition, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html">isqrt</a> [library]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#isqrt_spec">isqrt_spec</a> [definition, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#issqrtIsPR">issqrtIsPR</a> [instance, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit_cons">is_limit_cons</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit_0">is_limit_0</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit">is_limit</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_successor">is_successor</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_finite">is_finite</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.Lub.html#is_lub">is_lub</a> [definition, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="hydras.Schutte.CNF.html#is_cnf_of">is_cnf_of</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.AP.html#is_limit_phi0">is_limit_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#is_in_seq_not_Acc">is_in_seq_not_Acc</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr_rel">is_pr_rel</a> [projection, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr_rel">is_pr_rel</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr">is_pr</a> [projection, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr">is_pr</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit_sup_members">is_limit_sup_members</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit_omega">is_limit_omega</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit">is_limit</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_succ">is_succ</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_finite">is_finite</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.onType.html#is_successor_of">is_successor_of</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_ab">is_limit_ab</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_intro">is_limit_intro</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_cons_inv">is_limit_cons_inv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_not_succ">is_limit_not_succ</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_finite_finite">is_finite_finite</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html">is_F_monotonous</a> [library]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate">iterate</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#iterate">iterate</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#iterateIsPR">iterateIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Iterates.html">Iterates</a> [library]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge_diag'">iterate_ge_diag'</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_gt_diag'">iterate_gt_diag'</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge_from">iterate_ge_from</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ext2">iterate_ext2</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_dom_prop">iterate_dom_prop</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono_1">iterate_mono_1</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge''">iterate_ge''</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge'">iterate_ge'</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_Sge">iterate_Sge</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge">iterate_ge</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono">iterate_mono</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono2">iterate_mono2</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le_np_le">iterate_le_np_le</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le_n_Sn">iterate_le_n_Sn</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_lt_from">iterate_lt_from</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_lt">iterate_lt</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le">iterate_le</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ext">iterate_ext</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_rw">iterate_rw</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_S_eqn2">iterate_S_eqn2</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_S_eqn">iterate_S_eqn</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat3">iterate_compat3</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat2">iterate_compat2</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat">iterate_compat</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_comm">iterate_comm</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#iterate_nat_rec">iterate_nat_rec</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.Iterate_compat.html#iterate_extEqual">iterate_extEqual</a> [lemma, in <a href="hydras.MoreAck.Iterate_compat.html">hydras.MoreAck.Iterate_compat</a>]<br/>
+<a href="hydras.MoreAck.Iterate_compat.html#iterate_compat">iterate_compat</a> [lemma, in <a href="hydras.MoreAck.Iterate_compat.html">hydras.MoreAck.Iterate_compat</a>]<br/>
+<a href="hydras.MoreAck.Iterate_compat.html">Iterate_compat</a> [library]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono2">iterate2_mono2</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono3">iterate2_mono3</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono_weak">iterate2_mono_weak</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono">iterate2_mono</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="additions.fib.html#iter_combine">iter_combine</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#iter_comm">iter_comm</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#iter_mul">iter_mul</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.onType.html#I_i_wf">I_i_wf</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#i':179">i':179</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#i1">i1</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#i2">i2</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#i:1">i:1</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#j:34">j:34</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:36">j:36</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#j:39">j:39</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:404">j:404</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:408">j:408</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Addition.html#j:41">j:41</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:414">j:414</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:416">j:416</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#j:420">j:420</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:424">j:424</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:428">j:428</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.fib.html#j:43">j:43</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#j:438">j:438</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:449">j:449</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:455">j:455</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:458">j:458</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.fib.html#j:46">j:46</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:463">j:463</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:465">j:465</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:47">j:47</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:481">j:481</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#j:49">j:49</a> [binder, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:493">j:493</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:497">j:497</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:5">j:5</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:50">j:50</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:501">j:501</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:506">j:506</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:515">j:515</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.fib.html#j:52">j:52</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:521">j:521</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:524">j:524</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:531">j:531</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:54">j:54</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#j:54">j:54</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#j:55">j:55</a> [binder, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.fib.html#j:55">j:55</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:555">j:555</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:559">j:559</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:568">j:568</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#j:57">j:57</a> [binder, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:575">j:575</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:58">j:58</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#j:6">j:6</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:6">j:6</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#j:6">j:6</a> [binder, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:60">j:60</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.fib.html#j:60">j:60</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#j:61">j:61</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="additions.More_on_positive.html#j:61">j:61</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="gaia_hydras.GCanon.html#j:61">j:61</a> [binder, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GPaths.html#j:63">j:63</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:65">j:65</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#j:66">j:66</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#j:69">j:69</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:7">j:7</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#j:7">j:7</a> [binder, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#j:70">j:70</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="gaia_hydras.GPaths.html#j:74">j:74</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:75">j:75</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:77">j:77</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#j:78">j:78</a> [binder, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#j:79">j:79</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#j:79">j:79</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#j:8">j:8</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#j:82">j:82</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:83">j:83</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:85">j:85</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:87">j:87</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#j:9">j:9</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#j:9">j:9</a> [binder, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#j:90">j:90</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:90">j:90</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#j:90">j:90</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#j:92">j:92</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#j:94">j:94</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#j:95">j:95</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#j:96">j:96</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#j:96">j:96</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#j:97">j:97</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.O2H.html#j:99">j:99</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<br/><br/><a id="global_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#kbr:222">kbr:222</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kbr:231">kbr:231</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kbr:293">kbr:293</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kbr:298">kbr:298</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain">Kchain</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#Kchain_correct">Kchain_correct</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_correct_conv">Kchain_correct_conv</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_proper_prf">Kchain_proper_prf</a> [projection, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_proper">Kchain_proper</a> [record, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_proper_prf">Kchain_proper_prf</a> [constructor, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_proper">Kchain_proper</a> [inductive, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_correct">Kchain_correct</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_correct_nat">Kchain_correct_nat</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:181">kc:181</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:191">kc:191</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:192">kc:192</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:204">kc:204</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:218">kc:218</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kc:221">kc:221</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFF">KFF</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF">KFF</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFF_correct">KFF_correct</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF_correct">KFF_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF_proper">KFF_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF_correct_nat">KFF_correct_nat</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFK">KFK</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK">KFK</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFK_correct">KFK_correct</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_correct">KFK_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proper">KFK_proper</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_correct_nat">KFK_correct_nat</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hq_prop">KFK_proof.Hq_prop</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hbr_prop">KFK_proof.Hbr_prop</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hq">KFK_proof.Hq</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hbr">KFK_proof.Hbr</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.fq">KFK_proof.fq</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.kbr">KFK_proof.kbr</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.r">KFK_proof.r</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.q">KFK_proof.q</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.b">KFK_proof.b</a> [variable, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof">KFK_proof</a> [section, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#khi:170">khi:170</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#khi:171">khi:171</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#khi:172">khi:172</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#khi:173">khi:173</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Addition.html#khi:49">khi:49</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#khi:92">khi:92</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#Kkont">Kkont</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kkont_equiv">Kkont_equiv</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kkont_proper">Kkont_proper</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#knp:207">knp:207</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#knp:211">knp:211</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kont_type">kont_type</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#KP">KP</a> [section, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="additions.AM.html#kpr:27">kpr:27</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#kpr:31">kpr:31</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_5_iii">KP_5_iii</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_arrowS_zero">KP_arrowS_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_arrowS">KP_arrowS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.KP_example.html">KP_example</a> [library]<br/>
+<a href="hydras.Hydra.KP_example.html#KP.h'">KP.h'</a> [variable, in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_beta_eq">ksi_plus_beta_eq</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_beta">ksi_plus_beta</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_seq_n'">ksi_plus_seq_n'</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_seq_n">ksi_plus_seq_n</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi:58">ksi:58</a> [binder, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.GPaths.html#KS_thm_2_4">KS_thm_2_4</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_E0">KS_thm_2_4_E0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma5">KS_thm_2_4_lemma5</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma3_0">KS_thm_2_4_lemma3_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma1'">KS_thm_2_4_lemma1'</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#k':105">k':105</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k':116">k':116</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k':174">k':174</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k':200">k':200</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#k17_7">k17_7</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F">K2F</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F_proper">K2F_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#k7_3">k7_3</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#K99_24">K99_24</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#k:10">k:10</a> [binder, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#k:102">k:102</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#k:107">k:107</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#k:107">k:107</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
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+<a href="gaia_hydras.GL_alpha.html#k:11">k:11</a> [binder, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#k:115">k:115</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#k:117">k:117</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#k:118">k:118</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k:12">k:12</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#k:12">k:12</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#k:12">k:12</a> [binder, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#k:122">k:122</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#k:122">k:122</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#k:133">k:133</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#k:137">k:137</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#k:14">k:14</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#k:1402">k:1402</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#k:143">k:143</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k:143">k:143</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.fib.html#k:15">k:15</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.dickson.html#k:15">k:15</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
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+<a href="hydras.Epsilon0.Canon.html#lambda:75">lambda:75</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Language">Language</a> [record, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#language">language</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.Languages.html">Languages</a> [library]<br/>
+<a href="hydras.Ackermann.fol.html#language_decidable">language_decidable</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Large">Large</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large">large</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#large">large</a> [definition, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Largeb">Largeb</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb">largeb</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_n_R">largeb_n_R</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_n">largeb_n</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_finite">largeb_finite</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Largeb_Sb">Largeb_Sb</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_Sb">largeb_Sb</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeS">LargeS</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeS">largeS</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#largeS">largeS</a> [definition, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeSb">LargeSb</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeSb">largeSb</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeSb_b">LargeSb_b</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeSb_b">largeSb_b</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeS_iff">largeS_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_iff">large_omega_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_2">large_omega_2</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_1">large_omega_1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_n_iff">large_n_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_iff">large_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#large_set_check">large_set_check</a> [definition, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html">Large_Sets</a> [library]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html">Large_Sets_Examples</a> [library]<br/>
+<a href="hydras.rpo.more_list.html#la:116">la:116</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.dickson.html#la:7">la:7</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.Ackermann.code.html#Lcode">Lcode</a> [record, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN">LcodeLNN</a> [instance, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT">LcodeLNT</a> [instance, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.rpo.rpo.html#lc:165">lc:165</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#lc:233">lc:233</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#lc:242">lc:242</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#lc:70">lc:70</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE">LE</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le">le</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#le">le</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le">le</a> [abbreviation, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#le">le</a> [definition, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#LE">LE</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#le">le</a> [definition, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#le">le</a> [definition, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le">le</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le">Le</a> [definition, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least">Least</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Least_zero">Least_zero</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.AP.html#least_AP">least_AP</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Least_is_0">Least_is_0</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least_not_Limit">Least_not_Limit</a> [lemma, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least_not_Succ">Least_not_Succ</a> [lemma, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_of_eq">least_member_of_eq</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_ex_unique">least_member_ex_unique</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_unicity">least_member_unicity</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_glb">least_member_glb</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_lower_bound">least_member_lower_bound</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_fixpoint">least_fixpoint</a> [definition, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member">least_member</a> [definition, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#leA:48">leA:48</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#leA:62">leA:62</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#leBool">leBool</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:107">leb:107</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#leB:11">leB:11</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:177">leb:177</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#leB:18">leB:18</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:183">leb:183</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:185">leb:185</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:188">leb:188</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:193">leb:193</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#leb:195">leb:195</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#leB:4">leB:4</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#leB:49">leB:49</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#leB:63">leB:63</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#leIsPR">leIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lemmas_on_length">lemmas_on_length</a> [section, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_5_2">Lemma_4_5_2</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_5_1">Lemma_4_5_1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_2">Lemma_4_4_2</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.H0">Lemma_4_4_Proof.H0</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.H">Lemma_4_4_Proof.H</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Hs">Lemma_4_4_Proof.Hs</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.s">Lemma_4_4_Proof.s</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.n">Lemma_4_4_Proof.n</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Hbeta">Lemma_4_4_Proof.Hbeta</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Halpha">Lemma_4_4_Proof.Halpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.beta">Lemma_4_4_Proof.beta</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.alpha">Lemma_4_4_Proof.alpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof">Lemma_4_4_Proof</a> [section, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_4_0">Lemma_4_4_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_4_1">Lemma_4_4_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3">Lemma_4_3</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_0">Lemma_4_3_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H4">Lemma_4_3_Proof.H4</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H1">Lemma_4_3_Proof.H1</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H0">Lemma_4_3_Proof.H0</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2">Lemma_4_3_Proof.n2</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1">Lemma_4_3_Proof.n1</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n0">Lemma_4_3_Proof.n0</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.nf2">Lemma_4_3_Proof.nf2</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.nf1">Lemma_4_3_Proof.nf1</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H00">Lemma_4_3_Proof.H00</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta">Lemma_4_3_Proof.beta</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha">Lemma_4_3_Proof.alpha</a> [variable, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof">Lemma_4_3_Proof</a> [section, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Lemma2_6_1">Lemma2_6_1</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma2_6_1_E0">Lemma2_6_1_E0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma2_6_1">Lemma2_6_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma4_4">Lemma4_4</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5">Lemma5</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_7">Lemma5_7</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_2">Lemma5_2</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_1">Lemma5_1</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_04">Lemma5_04</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_03">Lemma5_03</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_02">Lemma5_02</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_01">Lemma5_01</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_0">Lemma5_0</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#lengthTerms">lengthTerms</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_abnc">length_abnc</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_ab">length_ab</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_psi">length_psi</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_n">length_n</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_c">length_c</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_b">length_b</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#length_a">length_a</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.rpo.more_list.html#length_map">length_map</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#leq">leq</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#leq_trans">leq_trans</a> [instance, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.rpo.rpo.html#lex">lex</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples">LExamples</a> [section, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex1">LExamples.ex1</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex2">LExamples.ex2</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex3">LExamples.ex3</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.L">LExamples.L</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.P">LExamples.P</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.Q">LExamples.Q</a> [variable, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#lexico">lexico</a> [inductive, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexlt">lexlt</a> [definition, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexnf">lexnf</a> [definition, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexpower">lexpower</a> [inductive, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod">lexprod</a> [section, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A">lexprod.A</a> [variable, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B">lexprod.B</a> [variable, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA">lexprod.leA</a> [variable, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB">lexprod.leB</a> [variable, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;lex'_x">_ &lt;lex _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#e523b9bcd6ab1830bbf762a9717f65e6">_ &lt;=lex _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#29d4d7c85e1747f3f20f3fb958853966">_ =B= _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#6623eebf3a76064b3501b2e089b67493">_ =A= _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;B'_x">_ &lt;B _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#431a2798f744cb20088a983430e54725">_ &lt;=B _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.:::x_'&lt;A'_x">_ &lt;A _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#776f9baa25bd513014f05975f276fe02">_ &lt;=A _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexwf">lexwf</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexwf">lexwf</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod_dec">lex_prod_dec</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_strict_intro_left">lex_strict_intro_left</a> [lemma, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_strict_intro_right">lex_strict_intro_right</a> [lemma, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_inv_right">lex_inv_right</a> [lemma, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_inv_left">lex_inv_left</a> [lemma, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_of_equiv">lex_of_equiv</a> [lemma, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod">lex_prod</a> [inductive, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Lex_nat_wf">Lex_nat_wf</a> [instance, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#lex_nat_wf">lex_nat_wf</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#lex_nat">lex_nat</a> [definition, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#lex_2">lex_2</a> [constructor, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#lex_1">lex_1</a> [constructor, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.rpo.rpo.html#lex_trans">lex_trans</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex1">lex1</a> [constructor, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex1">lex1</a> [constructor, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex2">lex2</a> [constructor, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex2">lex2</a> [constructor, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex3">lex3</a> [constructor, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex4">lex4</a> [constructor, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_LT_eq_dec">LE_LT_eq_dec</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_r">LE_r</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_lt_LT">le_lt_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_phi0">LE_phi0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_LT_trans">LE_LT_trans</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_antisym">LE_antisym</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_trans">LE_trans</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_refl">LE_refl</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_zero">LE_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_le">LE_le</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_nf_r">LE_nf_r</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE_nf_l">LE_nf_l</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_phi0">le_phi0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_tail">le_tail</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_zero_inv">le_zero_inv</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_inv">le_inv</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#le_eq_lt_dec">le_eq_lt_dec</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#le_0">le_0</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#le_intror">le_intror</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_min_bd">le_min_bd</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_min_a">le_min_a</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_max_b">le_max_b</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_max_a">le_max_a</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_not_gt">le_not_gt</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_lt_trans">le_lt_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_trans">le_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le_refl">le_refl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le_lt_equiv_dec">le_lt_equiv_dec</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le_lt_weak">le_lt_weak</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le_lt_trans">le_lt_trans</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le_not_gt">le_not_gt</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Schutte.AP.html#le_phi0">le_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#le_lt_trans">le_lt_trans</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#le_0">le_0</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#le_intror">le_intror</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#le_introl">le_introl</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#le_S_Ack">le_S_Ack</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#le_dec:118">le_dec:118</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#le_total:117">le_total:117</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#LE_ref">LE_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#le_ref">le_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#le_lt_eq">le_lt_eq</a> [lemma, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_alpha_zero">le_alpha_zero</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_sup_members">le_sup_members</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_disj">le_disj</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_not_gt">le_not_gt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_eq_or_lt">le_eq_or_lt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_antisym">le_antisym</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_trans">le_trans</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_lt_trans">le_lt_trans</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le_refl">le_refl</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Le_refl">Le_refl</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Le_trans">Le_trans</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#le_lt_eq_dec">le_lt_eq_dec</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Le_iff">Le_iff</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le_Lt_trans">Le_Lt_trans</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le_trans">Le_trans</a> [instance, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le_antisym">Le_antisym</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le_refl">Le_refl</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_plus_l">le_plus_l</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_plus_r">le_plus_r</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_b_phi_ab">le_b_phi_ab</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_one_cons">le_one_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_cons_tail">le_cons_tail</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_trans">le_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_lt_trans">le_lt_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_inv_nc">le_inv_nc</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_psi_term_le">le_psi_term_le</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#le_zero_alpha">le_zero_alpha</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#le_a_mult_Sn_a">le_a_mult_Sn_a</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#le_plus_r">le_plus_r</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#le_plus_l">le_plus_l</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:16">le:16</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:21">le:21</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:26">le:26</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#le:3">le:3</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#Le:3">Le:3</a> [binder, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#le:31">le:31</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:36">le:36</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#le:40">le:40</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:40">le:40</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:45">le:45</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:50">le:50</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:55">le:55</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:60">le:60</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:65">le:65</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:71">le:71</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#le:75">le:75</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:77">le:77</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:82">le:82</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:85">le:85</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:88">le:88</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#le:94">le:94</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#LF1">LF1</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LF1">LF1</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#LF2">LF2</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LF2">LF2</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LF2_0">LF2_0</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#LF2'">LF2'</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LF2'">LF2'</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#LF3">LF3</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#LF3_2">LF3_2</a> [lemma, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LF3_2">LF3_2</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.rpo.rpo.html#lg:163">lg:163</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#lg:231">lg:231</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#lg:68">lg:68</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#lheight">lheight</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#lhsize">lhsize</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html">LibHyps_Experiments</a> [library]<br/>
+<a href="hydras.Ackermann.subAll.html#liftCloseFrom">liftCloseFrom</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim">lim</a> [section, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Limit">Limit</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#limitb">limitb</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#limitb">limitb</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#limitb">limitb</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Limitb">Limitb</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#limitb_limit">limitb_limit</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#LimitNotSucc">LimitNotSucc</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#limit_is_lub">limit_is_lub</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#limit_is_lub_0">limit_is_lub_0</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Hydra.O2H.html#limit_no_R1">limit_no_R1</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#limit_no_head">limit_no_head</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#Limit_Infinity">Limit_Infinity</a> [lemma, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="gaia_hydras.GCanon.html#limit_canonS_not_zero">limit_canonS_not_zero</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#limit_iff">limit_iff</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#limit_is_omega">limit_is_omega</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Limit_gt_Zero">Limit_gt_Zero</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Limit_not_Zero">Limit_not_Zero</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.onType.html#limit_of">limit_of</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#limit_v2">limit_v2</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#limit_rounds">limit_rounds</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#limit_rounds_0">limit_rounds_0</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#limit_step">limit_step</a> [constructor, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin_ok">limit_plus_fin_ok</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin_inv0">limit_plus_fin_inv0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin_lim">limit_plus_fin_lim</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin_plus">limit_plus_fin_plus</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_F_cons">limit_plus_F_cons</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_F_0">limit_plus_F_0</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin">limit_plus_fin</a> [inductive, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html">Limit_Infinity</a> [library]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.f">lim.f</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.H">lim.H</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.Hlim">lim.Hlim</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.Hnf">lim.Hnf</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.lambda">lim.lambda</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations">LispAbbreviations</a> [module, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.caar">LispAbbreviations.caar</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.caddr">LispAbbreviations.caddr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cadr">LispAbbreviations.cadr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.car">LispAbbreviations.car</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cddddr">LispAbbreviations.cddddr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cdddr">LispAbbreviations.cdddr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr">LispAbbreviations.cddr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr">LispAbbreviations.cdr</a> [abbreviation, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html">ListExt</a> [library]<br/>
+<a href="hydras.Ackermann.cPair.html#listValues">listValues</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#List_length">List_length</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_rec3">list_rec3</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_rec2">list_rec2</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_size_size_eq">list_size_size_eq</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_size_fold">list_size_fold</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_size_app">list_size_app</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_size_tl_compat">list_size_tl_compat</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_size">list_size</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#list_app_length">list_app_length</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove.Aeq_dec">List_Remove.Aeq_dec</a> [variable, in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove.A">List_Remove.A</a> [variable, in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove">List_Remove</a> [section, in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.rpo.list_permut.html">list_permut</a> [library]<br/>
+<a href="hydras.rpo.list_set.html">list_set</a> [library]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#LL">LL</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNN">LNN</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.LNN.html">LNN</a> [library]<br/>
+<a href="hydras.Ackermann.Languages.html#LNNArityF">LNNArityF</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNNArityR">LNNArityR</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNNRelation">LNNRelation</a> [inductive, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#LNN_eqdec">LNN_eqdec</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="Goedel.fixPoint.html#LNN_FixPoint.codeFormula">LNN_FixPoint.codeFormula</a> [variable, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#LNN_FixPoint">LNN_FixPoint</a> [section, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html">LNN_Examples</a> [library]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html">LNN2LNT</a> [library]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_subFormula">LNN2LNT_subFormula</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_freeVarF2">LNN2LNT_freeVarF2</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_freeVarF1">LNN2LNT_freeVarF1</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_freeVarF">LNN2LNT_freeVarF</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_exist">LNN2LNT_exist</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_iff">LNN2LNT_iff</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_and">LNN2LNT_and</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_or">LNN2LNT_or</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_formula">LNN2LNT_formula</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_subTerms">LNN2LNT_subTerms</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_subTerm">LNN2LNT_subTerm</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_freeVarTs">LNN2LNT_freeVarTs</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_freeVarT">LNN2LNT_freeVarT</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_natToTerm">LNN2LNT_natToTerm</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_terms">LNN2LNT_terms</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNN2LNT_term">LNN2LNT_term</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#LNotPR">LNotPR</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT">LNT</a> [module, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNT">LNT</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.LNT.html">LNT</a> [library]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTFunction">LNTFunction</a> [inductive, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTFunctionArity">LNTFunctionArity</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTRelation">LNTRelation</a> [inductive, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTRelationR">LNTRelationR</a> [definition, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT_eqdec">LNT_eqdec</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="Goedel.fixPoint.html#LNT_FixPoint.codeFormula">LNT_FixPoint.codeFormula</a> [variable, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#LNT_FixPoint">LNT_FixPoint</a> [section, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT.Plus">LNT.Plus</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT.Succ">LNT.Succ</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT.Times">LNT.Times</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT.Zero">LNT.Zero</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_subFormula">LNT2LNN_subFormula</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_freeVarF">LNT2LNN_freeVarF</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_freeVarT">LNT2LNN_freeVarT</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_subTerm">LNT2LNN_subTerm</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_formula">LNT2LNN_formula</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_natToTerm">LNT2LNN_natToTerm</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_terms">LNT2LNN_terms</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNN_term">LNT2LNN_term</a> [definition, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNT_formula">LNT2LNT_formula</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#LNT2LNT_term">LNT2LNT_term</a> [lemma, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO">LO</a> [module, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#LocallySorted_inv_In">LocallySorted_inv_In</a> [lemma, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#LocallySorted_trans">LocallySorted_trans</a> [lemma, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#LocallySorted_cons'">LocallySorted_cons'</a> [lemma, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#LocallySorted_cons">LocallySorted_cons</a> [lemma, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Logic">Logic</a> [section, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Logic">Logic</a> [section, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Other_Rules">Logic_Rules.Other_Rules</a> [section, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Not_Rules">Logic_Rules.Not_Rules</a> [section, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.SysPrf">Logic_Rules.SysPrf</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Prf">Logic_Rules.Prf</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Terms">Logic_Rules.Terms</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Term">Logic_Rules.Term</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.System">Logic_Rules.System</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Formulas">Logic_Rules.Formulas</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Formula">Logic_Rules.Formula</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.L">Logic_Rules.L</a> [variable, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules">Logic_Rules</a> [section, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.ap">LO.ap</a> [inductive, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.ap_intro">LO.ap_intro</a> [constructor, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_correct">LO.compare_correct</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_reflect">LO.compare_reflect</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_rev">LO.compare_rev</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_ref">LO.compare_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_oo">LO.compare_oo</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons">LO.cons</a> [abbreviation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons_nf">LO.cons_nf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.eq_ref">LO.eq_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin">LO.fin</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.fin_nf">LO.fin_nf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.FS">LO.FS</a> [abbreviation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.limitb">LO.limitb</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.limitb_ref">LO.limitb_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lo_comparable">LO.lo_comparable</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lo_strorder">LO.lo_strorder</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt">LO.lt</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt_trans">LO.lt_trans</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt_irrefl">LO.lt_irrefl</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lt_ref">LO.lt_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.mult">LO.mult</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.mult_plus_distr_l">LO.mult_plus_distr_l</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.mult_nf">LO.mult_nf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.mult_ref">LO.mult_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf">LO.nf</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf_ref">LO.nf_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf_inv3">LO.nf_inv3</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf_inv2">LO.nf_inv2</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.nf_b">LO.nf_b</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.omega">LO.omega</a> [abbreviation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0">LO.phi0</a> [abbreviation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0_nf">LO.phi0_nf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0_ref">LO.phi0_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus">LO.plus</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus_assoc">LO.plus_assoc</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus_nf">LO.plus_nf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus_ref">LO.plus_ref</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.refine">LO.refine</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_inv1">lt_a_phi0_b_inv1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_le_dec">lt_le_dec</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_inv">lt_a_phi0_b_inv</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_a">lt_a_phi0_a</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_succ">lt_succ</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_c">lt_a_phi0_b_c</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_z">lt_a_phi0_b_z</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b">lt_a_phi0_b</a> [inductive, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_dec">lt_dec</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv_head">lt_inv_head</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_trans">lt_trans</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_fin_iff">lt_fin_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv_tail">lt_inv_tail</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv_coeff_dec">lt_inv_coeff_dec</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv_coeff">lt_inv_coeff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv">lt_inv</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_irrefl">lt_irrefl</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_inv_strong">lt_inv_strong</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_right">lt_right</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_middle">lt_middle</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_left">lt_left</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_cases">lt_cases</a> [inductive, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#lt_omega">lt_omega</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#lt_succ">lt_succ</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#lt_succ_le">lt_succ_le</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_epsilon0">lt_epsilon0</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_7">lt_7</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_6">lt_6</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_5">lt_5</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_4">lt_4</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_3">lt_3</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_2">lt_2</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_1">lt_1</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#lt_lt_Sn">lt_lt_Sn</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Hydra.O2H.html#LT_to_round_plus">LT_to_round_plus</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Lt_n_F_Zero_n">Lt_n_F_Zero_n</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [definition, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_eq_lt">lt_eq_lt</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_incl_le">lt_incl_le</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_le_trans">lt_le_trans</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_not_ge">lt_not_ge</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_not_gt">lt_not_gt</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_irrefl">lt_irrefl</a> [abbreviation, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_trans">lt_trans</a> [abbreviation, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#lt_wf">lt_wf</a> [lemma, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="gaia_hydras.GPaths.html#LT_to_standard_path">LT_to_standard_path</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#LT_path_to">LT_path_to</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#lt_a_phi0_b_oplus">lt_a_phi0_b_oplus</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_transitive">lt_transitive</a> [instance, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_le_trans">lt_le_trans</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_not_equiv">lt_not_equiv</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_not_ge">lt_not_ge</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_irreflexive">lt_irreflexive</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="additions.More_on_positive.html#lt_S_2i">lt_S_2i</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Schutte.AP.html#lt_phi0">lt_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#lt_beta_exists">lt_beta_exists</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#lt_omega">lt_omega</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#lt_succ">lt_succ</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#lt_succ_le">lt_succ_le</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#lt_le_trans">lt_le_trans</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#LT_inv">LT_inv</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#lt_wf_double_ind">lt_wf_double_ind</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#lt_wf_ind">lt_wf_ind</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#LT_ref">LT_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#lt_ref">lt_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [lemma, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#lt_wf">lt_wf</a> [lemma, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#lt_strorder">lt_strorder</a> [instance, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#LT_to_standard_battle">LT_to_standard_battle</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [lemma, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#lt_wf">lt_wf</a> [lemma, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#lt_strorder">lt_strorder</a> [instance, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Ackermann.fol.html#lt_depth">lt_depth</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_omega_finite">lt_omega_finite</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_omega_limit_lt_exists_lt">lt_omega_limit_lt_exists_lt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_omega_limit">lt_omega_limit</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_sup_exists_lt">lt_sup_exists_lt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_sup_exists_leq">lt_sup_exists_leq</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_succ_lt">lt_succ_lt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_succ_le_2">lt_succ_le_2</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_succ_le">lt_succ_le</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_succ">lt_succ</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_or_ge">lt_or_ge</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_irrefl">lt_irrefl</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_le_trans">lt_le_trans</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_trans">lt_trans</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#lt_le">lt_le</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#lt_wf">lt_wf</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#lt_omega_inv">lt_omega_inv</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Lt_Le_incl">Lt_Le_incl</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Lt_trans">Lt_trans</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#lt_Succ_le_2">lt_Succ_le_2</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#lt_Succ_inv">lt_Succ_inv</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Lt_Succ">Lt_Succ</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Lt_Succ_Le">Lt_Succ_Le</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_not_Gt">Lt_not_Gt</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_Le_trans">Lt_Le_trans</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_connect">Lt_connect</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_irreflexive">Lt_irreflexive</a> [projection, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_trans">Lt_trans</a> [projection, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#LT_to_standard_path">LT_to_standard_path</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#LT_acc_from">LT_acc_from</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#LT_path_to">LT_path_to</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#LT_path_toS">LT_path_toS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_not_gt">lt_not_gt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_epsilon0_succ">lt_epsilon0_succ</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_epsilon0_trans">lt_epsilon0_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_epsilon0_okR">lt_epsilon0_okR</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_epsilon0_ok">lt_epsilon0_ok</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_a_phi_ab">lt_a_phi_ab</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_inc_rpo_0">lt_inc_rpo_0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_rpo_cons_cons">lt_rpo_cons_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.H">lt_incl_rpo.H</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.nf2">lt_incl_rpo.nf2</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.nf1">lt_incl_rpo.nf1</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.Hrec">lt_incl_rpo.Hrec</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.Hsize">lt_incl_rpo.Hsize</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n2">lt_incl_rpo.n2</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.n1">lt_incl_rpo.n1</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c2">lt_incl_rpo.c2</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b2">lt_incl_rpo.b2</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a2">lt_incl_rpo.a2</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.c1">lt_incl_rpo.c1</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.b1">lt_incl_rpo.b1</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.a1">lt_incl_rpo.a1</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo.s">lt_incl_rpo.s</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo">lt_incl_rpo</a> [section, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_subterm1">lt_subterm1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_succ_le">lt_succ_le</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_succ">lt_succ</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_compatR">lt_compatR</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_compat">lt_compat</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_omega_is_fin">lt_omega_is_fin</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_omega_inv">lt_omega_inv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_cons_omega_inv">lt_cons_omega_inv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_one_inv">lt_one_inv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_tail">lt_tail</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_tail0">lt_tail0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_alpha_cons">lt_alpha_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_alpha_psi">lt_alpha_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_beta_cons">lt_beta_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_beta_psi">lt_beta_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_le_trans">lt_le_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_trans">lt_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_trans0">lt_trans0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_ge_dec">lt_ge_dec</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_eq_lt_dec">lt_eq_lt_dec</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_than_psi">lt_than_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_irr">lt_irr</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#lt_wf">lt_wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LT_">LT_</a> [constructor, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#lt_not_wf">lt_not_wf</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html">lt_succ_le</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#LT1">LT1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT2">LT2</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT3">LT3</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT4">LT4</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Lt:109">Lt:109</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#lt:12">lt:12</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#lt:129">lt:129</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt:13">lt:13</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Lt:145">Lt:145</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Lt:155">Lt:155</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#lt:2">lt:2</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#Lt:2">Lt:2</a> [binder, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#lt:2">lt:2</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#lt:2">lt:2</a> [binder, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#lt:2">lt:2</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#lt:23">lt:23</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Lub.html#lt:3">lt:3</a> [binder, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#lt:43">lt:43</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#lt:47">lt:47</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt:5">lt:5</a> [binder, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt:52">Lt:52</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt:57">Lt:57</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#lt:8">lt:8</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Schutte.Lub.html#lt:9">lt:9</a> [binder, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Lt:99">Lt:99</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lub">lub</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Lub.html">Lub</a> [library]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lub_mono">lub_mono</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lub_unicity">lub_unicity</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Lvar">Lvar</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#L_3_plus_omega">L_3_plus_omega</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_cube_3_eq">L_omega_cube_3_eq</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_cube_eqn">L_omega_cube_eqn</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_cube_ok">L_omega_cube_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_cube">L_omega_cube</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times_ok">L_omega_square_times_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_times">L_omega_square_times</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult_Sle">L_phi0_mult_Sle</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult_smono">L_phi0_mult_smono</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult_ok">L_phi0_mult_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_phi0_mult">L_phi0_mult</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_ok">L_omega_square_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_smono">L_omega_square_smono</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_Sle">L_omega_square_Sle</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_eqn">L_omega_square_eqn</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_eqn1">L_omega_square_eqn1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square">L_omega_square</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_eqn">L_omega_mult_eqn</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_ok">L_omega_mult_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_Sle">L_omega_mult_Sle</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult">L_omega_mult</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_ok">L_omega_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_smono">L_omega_smono</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_Sle">L_omega_Sle</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega">L_omega</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_fin_ok">L_fin_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_S_succ_rw">L_S_succ_rw</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_fin_smono">L_fin_smono</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_finS_S_le">L_finS_S_le</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_fin">L_fin</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_lim_ok">L_lim_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_lim">L_lim</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_ok">L_succ_ok</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_Sle">L_succ_Sle</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_mono">L_succ_mono</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ">L_succ</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_unicity">L_spec_unicity</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_pos_inv">L_pos_inv</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_zero_inv">L_zero_inv</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_compat">L_spec_compat</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_inv2">L_spec_inv2</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_test">L_test</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec1">L_spec1</a> [constructor, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec0">L_spec0</a> [constructor, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec">L_spec</a> [inductive, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_correct">L_correct</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok">L_ok</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok_lim">L_ok_lim</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok_succ">L_ok_succ</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok0">L_ok0</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_correct_proof.P">L_correct_proof.P</a> [variable, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_correct_proof">L_correct_proof</a> [section, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_succ_ok">L_succ_ok</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ge_S">L_ge_S</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ge_id">L_ge_id</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_omega">L_omega</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_finite">L_finite</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_lim_eqn">L_lim_eqn</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_succ_eqn">L_succ_eqn</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_eq2">L_eq2</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_zero_eqn">L_zero_eqn</a> [lemma, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_">L_</a> [definition, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html#L_3_plus_omega">L_3_plus_omega</a> [instance, in <a href="hydras.OrdinalNotations.Example_3PlusOmega.html">hydras.OrdinalNotations.Example_3PlusOmega</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#l_std_ok">l_std_ok</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#l_std">l_std</a> [definition, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_0_3">L_0_3</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_2_3">L_2_3</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_0_2">L_0_2</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="gaia_hydras.GHydra.html#l_std_ok">l_std_ok</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#l_stdE">l_stdE</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#l_std">l_std</a> [definition, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#L_fin_ok">L_fin_ok</a> [lemma, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#L_spec">L_spec</a> [definition, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#L_3_plus_omega">L_3_plus_omega</a> [definition, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_correct">L_correct</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_pos_inv">L_pos_inv</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_spec1">L_spec1</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_spec0">L_spec0</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_spec">L_spec</a> [definition, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_ge_S">L_ge_S</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_ge_id">L_ge_id</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_omega">L_omega</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_finite">L_finite</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_limE">L_limE</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_succE">L_succE</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_eq2">L_eq2</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_zeroE">L_zeroE</a> [lemma, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_">L_</a> [definition, in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="hydras.Schutte.Addition.html#L_3_plus_omega">L_3_plus_omega</a> [definition, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html">L_alpha</a> [library]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#l'':55">l'':55</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#l'':56">l'':56</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#l'':58">l'':58</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l'1:118">l'1:118</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l'1:150">l'1:150</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#l':105">l':105</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l':114">l':114</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l':123">l':123</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#l':130">l':130</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l':135">l':135</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#l':139">l':139</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#l':14">l':14</a> [binder, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l':140">l':140</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':143">l':143</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l':146">l':146</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':147">l':147</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#l':15">l':15</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':151">l':151</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':158">l':158</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':161">l':161</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':167">l':167</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':171">l':171</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#l':19">l':19</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.rpo.more_list.html#l':192">l':192</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="additions.Trace_exercise.html#l':20">l':20</a> [binder, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':218">l':218</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':222">l':222</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':224">l':224</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
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+<a href="hydras.rpo.rpo.html#l':235">l':235</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':244">l':244</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':267">l':267</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
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+<a href="hydras.Prelude.DecPreOrder_Instances.html#l':30">l':30</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
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+<a href="hydras.Prelude.Sort_spec.html#l':37">l':37</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#l':41">l':41</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#l':43">l':43</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#l':46">l':46</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Hydra.O2H.html#l':48">l':48</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':48">l':48</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#l':52">l':52</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#l':52">l':52</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':52">l':52</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':56">l':56</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Schutte.CNF.html#l':57">l':57</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#l':614">l':614</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#l':618">l':618</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':63">l':63</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#l':66">l':66</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#l':699">l':699</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Prelude.Sort_spec.html#l':8">l':8</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
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+<a href="hydras.rpo.list_permut.html#l2:71">l2:71</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#l2:73">l2:73</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#l2:75">l2:75</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_permut.html#l2:77">l2:77</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#l2:79">l2:79</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#l2:80">l2:80</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#l2:83">l2:83</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#l2:86">l2:86</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
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+<a href="hydras.rpo.list_set.html#l2:92">l2:92</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#l2:94">l2:94</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#L3">L3</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.solutions_exercises.F_3.html#L3">L3</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#L3a">L3a</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#L3_u">L3_u</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#l3:199">l3:199</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#L4">L4</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Hydra.BigBattle.html#L46">L46</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
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+<a href="hydras.rpo.list_permut.html#l4:40">l4:40</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#L5">L5</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.MoreAck.Ack.html#L5">L5</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#L7">L7</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="additions.Addition_Chains.html#L87''">L87''</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L9">L9</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceImp">makeTraceImp</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceImpNice">makeTraceImpNice</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceNot">makeTraceNot</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceNotNice">makeTraceNotNice</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTrace1">makeTrace1</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTrace2">makeTrace2</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.AM.html#make_chain">make_chain</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#make_chain_correct">make_chain_correct</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#make_chain">make_chain</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.acc_build">Make.acc_build</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.acc_lex_drop_proof">Make.acc_lex_drop_proof</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.ac_syntactic">Make.ac_syntactic</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.ac_syntactic_aux">Make.ac_syntactic_aux</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add">Make.add</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_comm">Make.add_comm</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_without_red">Make.add_without_red</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_12">Make.add_12</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_2">Make.add_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_1">Make.add_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_prf">Make.add_prf</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.apply_subst">Make.apply_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.build_list_of_SN_terms">Make.build_list_of_SN_terms</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal">Make.cardinal</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_eq_set">Make.cardinal_eq_set</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union">Make.cardinal_union</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_inter_12">Make.cardinal_union_inter_12</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_2">Make.cardinal_union_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_1">Make.cardinal_union_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_subset">Make.cardinal_subset</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.cons_permut_in">Make.cons_permut_in</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_multiset_extension_step_app2">Make.context_multiset_extension_step_app2</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_trans_clos_multiset_extension_step_app1">Make.context_trans_clos_multiset_extension_step_app1</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_multiset_extension_step_app1">Make.context_multiset_extension_step_app1</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_app2">Make.context_list_permut_app2</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_app1">Make.context_list_permut_app1</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_cons">Make.context_list_permut_cons</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DecVar">Make.DecVar</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DecVar.A">Make.DecVar.A</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DecVar.eq_A_dec">Make.DecVar.eq_A_dec</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson">Make.dickson</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux3">Make.dickson_aux3</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux2">Make.dickson_aux2</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux1">Make.dickson_aux1</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.term.html#Make.direct_subterm">Make.direct_subterm</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DoubleRecursion">Make.DoubleRecursion</a> [section, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DoubleRecursion.Pl2">Make.DoubleRecursion.Pl2</a> [variable, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DoubleRecursion.P2">Make.DoubleRecursion.P2</a> [variable, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.DS">Make.DS</a> [module, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.DS">Make.DS</a> [module, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.DS">Make.DS</a> [module, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.elt">Make.elt</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.elt">Make.elt</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.empty">Make.empty</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.empty_subst_is_id_list">Make.empty_subst_is_id_list</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.empty_subst_is_id">Make.empty_subst_is_id</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Eq">Make.Eq</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.eq_term_dec">Make.eq_term_dec</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_list_permut_support">Make.eq_set_list_permut_support</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_trans">Make.eq_set_trans</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_sym">Make.eq_set_sym</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_refl">Make.eq_set_refl</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_dec">Make.eq_set_dec</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set">Make.eq_set</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_elt_dec">Make.eq_elt_dec</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.eq_elt_dec">Make.eq_elt_dec</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.F">Make.F</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter">Make.filter</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_union">Make.filter_union</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_2">Make.filter_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_2_list">Make.filter_2_list</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_1">Make.filter_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_1_list">Make.filter_1_list</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_aux">Make.filter_aux</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.included_remove_red">Make.included_remove_red</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.included_filter_aux">Make.included_filter_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter">Make.inter</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_12">Make.inter_12</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_12_aux">Make.inter_12_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_2">Make.inter_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_2_aux">Make.inter_2_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_1">Make.inter_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_1_aux">Make.inter_1_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.in_permut_in">Make.in_permut_in</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.in_mult_S">Make.in_mult_S</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.in_sn_sn">Make.in_sn_sn</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.is_a_pos_exists_subtem">Make.is_a_pos_exists_subtem</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.is_a_pos">Make.is_a_pos</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.is_a_set">Make.is_a_set</a> [projection, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex1">Make.lex1</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex1_bis">Make.lex1_bis</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex2">Make.lex2</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex3">Make.lex3</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_acc">Make.list_permut_acc</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_multiset_extension_step_2">Make.list_permut_multiset_extension_step_2</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_multiset_extension_step_1">Make.list_permut_multiset_extension_step_1</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_dec">Make.list_permut_dec</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length_2">Make.list_permut_length_2</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length_1">Make.list_permut_length_1</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_map">Make.list_permut_map</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_size">Make.list_permut_size</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length">Make.list_permut_length</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_remove_hd">Make.list_permut_remove_hd</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_app_app">Make.list_permut_app_app</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_add_cons_inside">Make.list_permut_add_cons_inside</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_add_inside">Make.list_permut_add_inside</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_nil">Make.list_permut_nil</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_trans">Make.list_permut_trans</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_sym">Make.list_permut_sym</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_refl">Make.list_permut_refl</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut">Make.list_permut</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_to_multiset">Make.list_to_multiset</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.list_permut_map_acc">Make.list_permut_map_acc</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul_rest_step">Make.List_mul_rest_step</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul_rest">Make.List_mul_rest</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_eq_rest">Make.List_eq_rest</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_gt_rest">Make.List_gt_rest</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul">Make.List_mul</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_eq">Make.List_eq</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_gt">Make.List_gt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.LP">Make.LP</a> [module, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.LP">Make.LP</a> [module, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.LP">Make.LP</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Lt">Make.Lt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.make_set">Make.make_set</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.map_subst">Make.map_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.mem">Make.mem</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.mem_dec">Make.mem_dec</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.mk_set">Make.mk_set</a> [constructor, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.mk_sn">Make.mk_sn</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.multiplicity_app">Make.multiplicity_app</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.multiset_closure">Make.multiset_closure</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.multiset_extension_step">Make.multiset_extension_step</a> [inductive, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.out_mult_O">Make.out_mult_O</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.o_size3_trans">Make.o_size3_trans</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.o_size3">Make.o_size3</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.o_size2">Make.o_size2</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.o_size">Make.o_size</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.P">Make.P</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.projection_list_of_SN_terms">Make.projection_list_of_SN_terms</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Recursion">Make.Recursion</a> [section, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Recursion.P">Make.Recursion.P</a> [variable, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Recursion.Pl">Make.Recursion.Pl</a> [variable, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.remove_not_common">Make.remove_not_common</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.remove_red_included">Make.remove_red_included</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.remove_red">Make.remove_red</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.remove_context_list_permut_app2">Make.remove_context_list_permut_app2</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.remove_context_list_permut_cons">Make.remove_context_list_permut_cons</a> [lemma, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_list_replace_at_pos_in_subterm">Make.replace_at_pos_list_replace_at_pos_in_subterm</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_is_replace_at_pos2">Make.replace_at_pos_is_replace_at_pos2</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_is_replace_at_pos1">Make.replace_at_pos_is_replace_at_pos1</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_unfold">Make.replace_at_pos_unfold</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_list">Make.replace_at_pos_list</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos">Make.replace_at_pos</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.rmv_case">Make.rmv_case</a> [constructor, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo">Make.rpo</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_add_context">Make.rpo_add_context</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_subst">Make.rpo_subst</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_term">Make.rpo_term</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_trans_clos">Make.rpo_mul_trans_clos</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step">Make.rpo_mul_rest_step</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_rest">Make.rpo_mul_rest</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_rest_same_length">Make.rpo_lex_rest_same_length</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_rest">Make.rpo_lex_rest</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_rest">Make.rpo_rest</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_trans">Make.rpo_trans</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_closure">Make.rpo_closure</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_subterm">Make.rpo_subterm</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_same_length">Make.rpo_lex_same_length</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul">Make.rpo_mul</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex">Make.rpo_lex</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_eq">Make.rpo_eq</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.singleton">Make.singleton</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size">Make.size</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_subterm_at_pos">Make.size_subterm_at_pos</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_direct_subterm">Make.size_direct_subterm</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_ge_one">Make.size_ge_one</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_unfold">Make.size_unfold</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.size2">Make.size2</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.size3">Make.size3</a> [definition, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.sn">Make.sn</a> [projection, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.SN_term">Make.SN_term</a> [record, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset">Make.subset</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_cardinal_not_eq_not_eq_set">Make.subset_cardinal_not_eq_not_eq_set</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_subset_union">Make.subset_subset_union</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat">Make.subset_compat</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat_2">Make.subset_compat_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat_1">Make.subset_compat_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_filter">Make.subset_filter</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_inter_2">Make.subset_inter_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_inter_1">Make.subset_inter_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_union_2">Make.subset_union_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_union_1">Make.subset_union_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_dec">Make.subset_dec</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.substitution">Make.substitution</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp">Make.subst_comp_is_subst_comp</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp_aux2">Make.subst_comp_is_subst_comp_aux2</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp_aux1">Make.subst_comp_is_subst_comp_aux1</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp">Make.subst_comp</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Subterm">Make.Subterm</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos">Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subterm_at_pos">Make.subterm_at_pos</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.support">Make.support</a> [projection, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.symbol">Make.symbol</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.t">Make.t</a> [record, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.T">Make.T</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term">Make.Term</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term">Make.term</a> [inductive, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term_eq_dec.eq_A_dec">Make.Term_eq_dec.eq_A_dec</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term_eq_dec.A">Make.Term_eq_dec.A</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term_eq_dec">Make.Term_eq_dec</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec8">Make.term_rec8</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec7">Make.term_rec7</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec4">Make.term_rec4</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec3">Make.term_rec3</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec2">Make.term_rec2</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_eq_mul">Make.Top_eq_mul</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_eq_lex">Make.Top_eq_lex</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_gt">Make.Top_gt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.tt">Make.tt</a> [projection, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.two_cases">Make.two_cases</a> [lemma, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.two_cases_rpo">Make.two_cases_rpo</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union">Make.union</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_eq_set">Make.union_compat_eq_set</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_subset_2">Make.union_compat_subset_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_subset_1">Make.union_compat_subset_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_assoc">Make.union_assoc</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_comm">Make.union_comm</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_empty_2">Make.union_empty_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_empty_1">Make.union_empty_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_12">Make.union_12</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_12_aux">Make.union_12_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_2">Make.union_2</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_2_aux">Make.union_2_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_1">Make.union_1</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_1_aux">Make.union_1_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Var">Make.Var</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.variable">Make.variable</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.VSet">Make.VSet</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_apply_subst">Make.well_formed_apply_subst</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_subst">Make.well_formed_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_list">Make.well_formed_list</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_fold">Make.well_formed_fold</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_unfold">Make.well_formed_unfold</a> [lemma, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed">Make.well_formed</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_rpo">Make.wf_rpo</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_rpo_term">Make.wf_rpo_term</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_mul_rest">Make.wf_on_mul_rest</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_lex_rest">Make.wf_on_lex_rest</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_rest">Make.wf_on_rest</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size3">Make.wf_size3</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size2">Make.wf_size2</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size">Make.wf_size</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_permut">Make.without_red_permut</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_not_common">Make.without_red_remove_not_common</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_not_common_aux">Make.without_red_remove_not_common_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_add_without_red">Make.without_red_add_without_red</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_singleton">Make.without_red_singleton</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_nil">Make.without_red_nil</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_add">Make.without_red_add</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove">Make.without_red_remove</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_red">Make.without_red_remove_red</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_filter_aux">Make.without_red_filter_aux</a> [lemma, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red">Make.without_red</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.X">Make.X</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#mapListLemma">mapListLemma</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.rpo.more_list.html#map_without_repetition">map_without_repetition</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#map_app">map_app</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#map_map">map_map</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#map12_without_repetition">map12_without_repetition</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#map:296">map:296</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#map:340">map:340</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#map:345">map:345</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#map:350">map:350</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#map:373">map:373</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#mark">mark</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked">marked</a> [inductive, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked0">marked0</a> [constructor, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked1">marked1</a> [constructor, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#match2">match2</a> [definition, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max">max</a> [definition, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxApp">maxApp</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Schutte.Critical.html#maximal_critical">maximal_critical</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#maxIsPR">maxIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma">maxLemma</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma2">maxLemma2</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma3">maxLemma3</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxSubTerm">maxSubTerm</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxSubTerms">maxSubTerms</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxVarFreeVar">maxVarFreeVar</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#max_nf">max_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_regL">max_le_regL</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_regR">max_le_regR</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_plus">max_le_plus</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_assoc">max_assoc</a> [instance, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_refl">max_refl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_ge_b">max_ge_b</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_ge_a">max_ge_a</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_comm">max_comm</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_dec">max_dec</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_tl">max_v_tl</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_ge">max_v_ge</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_lub">max_v_lub</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_2">max_v_2</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v">max_v</a> [definition, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#measure_lt">measure_lt</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#mem">mem</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html#members">members</a> [definition, in <a href="hydras.solutions_exercises.OnCodeList.html">hydras.solutions_exercises.OnCodeList</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#members">members</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html#membersOk">membersOk</a> [lemma, in <a href="hydras.solutions_exercises.OnCodeList.html">hydras.solutions_exercises.OnCodeList</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html#membersOk'">membersOk'</a> [lemma, in <a href="hydras.solutions_exercises.OnCodeList.html">hydras.solutions_exercises.OnCodeList</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#members_omega">members_omega</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#members_eq">members_eq</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#members_proper">members_proper</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#mem_head_mult_inv">mem_head_mult_inv</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#mem_head">mem_head</a> [inductive, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge">merge</a> [definition, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_sort_correct">merge_sort_correct</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_permutation">merge_permutation</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_LocallySorted">merge_LocallySorted</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_Forall">merge_Forall</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_equation">merge_equation</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_rec">merge_rec</a> [definition, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_ind">merge_ind</a> [definition, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_rect">merge_rect</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_aux:115">merge_aux:115</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html">Merge_Sort</a> [library]<br/>
+<a href="hydras.Prelude.Comparable.html#min">min</a> [definition, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#minimize">minimize</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#minimize1">minimize1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minIsPR">minIsPR</a> [instance, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#minIsPR">minIsPR</a> [instance, in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration">minoration</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_3">minoration_3</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_2">minoration_2</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_1">minoration_1</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_0">minoration_0</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html">MinPR</a> [library]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minPR_PR">minPR_PR</a> [instance, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minPR_correct">minPR_correct</a> [lemma, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#minPR_PR">minPR_PR</a> [instance, in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html">MinPR2</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#minus">minus</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#minusIndIsPR">minusIndIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#minusIsPR">minusIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_le">minus_le</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_a_a">minus_a_a</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_lt">minus_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#minus_semi_assoc">minus_semi_assoc</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Schutte.Addition.html#minus_exists">minus_exists</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_assoc">min_assoc</a> [instance, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_refl">min_refl</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_le_b">min_le_b</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_le_ad">min_le_ad</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_dec">min_dec</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_comm">min_comm</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_max_iff">min_max_iff</a> [lemma, in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#min_alt_correct">min_alt_correct</a> [lemma, in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Ackermann.misc.html">misc</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#mIsPR">mIsPR</a> [instance, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mkE0">mkE0</a> [constructor, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mkord">mkord</a> [constructor, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mkP">mkP</a> [constructor, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#mkP">mkP</a> [constructor, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mkQ">mkQ</a> [constructor, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#mks">mks</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge">mlarge</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#mlarge">mlarge</a> [definition, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlargeS">mlargeS</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#mlargeS">mlargeS</a> [definition, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlargeS_iff">mlargeS_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_omega">mlarge_omega</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_FS">mlarge_FS</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_iff">mlarge_iff</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_unshift">mlarge_unshift</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_unicity">mlarge_unicity</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#mlarge_unicity">mlarge_unicity</a> [lemma, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#ml_1">ml_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#ml_psi">ml_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model">Model</a> [record, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#model">model</a> [constructor, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html">model</a> [library]<br/>
+<a href="hydras.Ackermann.model.html#ModelConsistent">ModelConsistent</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory.T">Model_Theory.Consistent_Theory.T</a> [variable, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory">Model_Theory.Consistent_Theory</a> [section, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory.M">Model_Theory.M</a> [variable, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory.L">Model_Theory.L</a> [variable, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory">Model_Theory</a> [section, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_Equiv">mod_Equiv</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_op">mod_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_equiv">mod_equiv</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_eq">mod_eq</a> [definition, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_def.html#Monoid">Monoid</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Monoid_EMonoid">Monoid_EMonoid</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Addition_Chains.html#Monoid_Exp">Monoid_Exp</a> [inductive, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html">Monoid_instances</a> [library]<br/>
+<a href="additions.Monoid_def.html">Monoid_def</a> [library]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_weak">mono_weak</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_injective">mono_injective</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_le">mono_le</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mono_F_Zero">mono_F_Zero</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq_weak2">mono_seq_weak2</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq2">mono_seq2</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq">mono_seq</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html">MoreDecidable</a> [library]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html">MoreEpsilonIota</a> [library]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples">MoreExamples</a> [module, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.cst">MoreExamples.cst</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.cst0">MoreExamples.cst0</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.fact">MoreExamples.fact</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.mult">MoreExamples.mult</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples.plus">MoreExamples.plus</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Prelude.MoreLibHyps.html">MoreLibHyps</a> [library]<br/>
+<a href="hydras.Prelude.MoreLists.html">MoreLists</a> [library]<br/>
+<a href="hydras.Prelude.MoreOrders.html">MoreOrders</a> [library]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType">MoreOrderType</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.disp">MoreOrderType.disp</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.Hlim">MoreOrderType.Succ_no_limit.Hlim</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.Hsucc">MoreOrderType.Succ_no_limit.Hsucc</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.s">MoreOrderType.Succ_no_limit.s</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.y">MoreOrderType.Succ_no_limit.y</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.x">MoreOrderType.Succ_no_limit.x</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit">MoreOrderType.Succ_no_limit</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.T">MoreOrderType.T</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html">MorePRExamples</a> [library]<br/>
+<a href="hydras.Prelude.MoreVectors.html">MoreVectors</a> [library]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.SysPrf">More_Logic_Rules.SysPrf</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Prf">More_Logic_Rules.Prf</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Terms">More_Logic_Rules.Terms</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term">More_Logic_Rules.Term</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.System">More_Logic_Rules.System</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formulas">More_Logic_Rules.Formulas</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formula">More_Logic_Rules.Formula</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L">More_Logic_Rules.L</a> [variable, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules">More_Logic_Rules</a> [section, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html">More_Arith</a> [library]<br/>
+<a href="hydras.rpo.more_list.html">more_list</a> [library]<br/>
+<a href="additions.More_on_positive.html">More_on_positive</a> [library]<br/>
+<a href="hydras.Gamma0.Gamma0.html#moser_lepper">moser_lepper</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#MP">MP</a> [constructor, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#MPSys">MPSys</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="additions.AM.html#mp:29">mp:29</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#mq:28">mq:28</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#mq:30">mq:30</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#mq:32">mq:32</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms">ms</a> [definition, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ms">ms</a> [definition, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_eqn3">ms_eqn3</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_nf">ms_nf</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_eqn2">ms_eqn2</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="additions.AM.html#MUL">MUL</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult">mult</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Mult">Mult</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.Addition_Chains.html#Mult">Mult</a> [constructor, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#multE4">multE4</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#multE5">multE5</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#multIndIsPR">multIndIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html">MultisetWf</a> [library]<br/>
+<a href="hydras.Ackermann.primRec.html#multIsPR">multIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#multLemma1">multLemma1</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#multLemma2">multLemma2</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="additions.AM.html#mults_squares">mults_squares</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#mults:5">mults:5</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_plus_distr_l">mult_plus_distr_l</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_mono">mult_mono</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_nf">mult_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_Sn_add">mult_Sn_add</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_0_a">mult_0_a</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_nf_fin">mult_nf_fin</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_a_1">mult_a_1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_1_a">mult_1_a</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_a_0">mult_a_0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_compat">mult_compat</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_fin_omega">mult_fin_omega</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#mult_Sn_dist">mult_Sn_dist</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#mult_fin_l">mult_fin_l</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#mult_fin_r">mult_fin_r</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#mult_lt_lt">mult_lt_lt</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mult_ref">mult_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mult_ref0">mult_ref0</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="additions.Monoid_def.html#mult_op">mult_op</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Mult_op">Mult_op</a> [record, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#mult_op">mult_op</a> [constructor, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Mult_op">Mult_op</a> [inductive, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#mult_comp">mult_comp</a> [instance, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="additions.Monoid_instances.html#mult_mod_associative">mult_mod_associative</a> [lemma, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mult_mod_proper">mult_mod_proper</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mult_mod">mult_mod</a> [definition, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mult_plus_distr_l">mult_plus_distr_l</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mult_fin_rw">mult_fin_rw</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Mult_i">Mult_i</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn_mono3">mult_Sn_mono3</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn_mono2">mult_Sn_mono2</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r_mono">mult_fin_r_mono</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r_one">mult_fin_r_one</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r">mult_fin_r</a> [definition, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn">mult_Sn</a> [definition, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#mult_correct">mult_correct</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#mult_eqn1">mult_eqn1</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.FirstSteps.html#mult:35">mult:35</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#mult:41">mult:41</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Addition_Chains.html#Mul_node">Mul_node</a> [constructor, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Fib2.html#Mul2">Mul2</a> [instance, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#mul2">mul2</a> [definition, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.fib.html#mul:75">mul:75</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#mul:82">mul:82</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#mul:88">mul:88</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#mul:94">mul:94</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu_beta_h">mu_beta_h</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu_positive">mu_positive</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#mu:1">mu:1</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#mu:1">mu:1</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#mu:160">mu:160</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu:2">mu:2</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="gaia_hydras.GHydra.html#mu:29">mu:29</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#mu:32">mu:32</a> [binder, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#mVariant">mVariant</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="additions.fib.html#my_pow_m2lmul">my_pow_m2lmul</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#my_pow_m3lmul">my_pow_m3lmul</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#my_powP">my_powP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#my_pow">my_pow</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_lt">m_lt</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_variant_LT">m_variant_LT</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_ge">m_ge</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.Power_of_op">M_given.Power_of_op</a> [section, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.About_power">M_given.About_power</a> [section, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_eq">M_given.E_eq</a> [variable, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_one">M_given.E_one</a> [variable, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_op">M_given.E_op</a> [variable, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.A">M_given.A</a> [variable, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given">M_given</a> [section, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#M_given.Power_of_op">M_given.Power_of_op</a> [section, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#M_given.E_one">M_given.E_one</a> [variable, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#M_given.A">M_given.A</a> [variable, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#M_given">M_given</a> [section, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_dominates_Ack">m_dominates_Ack</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_dominates_Ack_from_3">m_dominates_Ack_from_3</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_ge_Ack">m_ge_Ack</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_ge_F_omega">m_ge_F_omega</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_eqn">m_eqn</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#m_ms">m_ms</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#m_nf">m_nf</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_lt">m_lt</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_decrease">m_decrease</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_ge">m_ge</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="additions.Monoid_instances.html#m_neq_0">m_neq_0</a> [lemma, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="gaia_hydras.GHydra.html#m_nf">m_nf</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#m_ref">m_ref</a> [lemma, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#m_strict_mono">m_strict_mono</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#m_ge_generic">m_ge_generic</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#m_ge_0">m_ge_0</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_ge">m_ge</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_lt">m_lt</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_big_h_not_null">m_big_h_not_null</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#m_def">m_def</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#m_ge">m_ge</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#m_lt">m_lt</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#m_bounded">m_bounded</a> [projection, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.Naive.html#M'':37">M'':37</a> [binder, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Monoid_instances.html#m':19">m':19</a> [binder, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Naive.html#M':29">M':29</a> [binder, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M':32">M':32</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M':36">M':36</a> [binder, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="hydras.rpo.dickson.html#m':47">m':47</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m':91">m':91</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#m0:60">m0:60</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m1:12">m1:12</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:14">m1:14</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="additions.fib.html#m1:152">m1:152</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m1:16">m1:16</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:19">m1:19</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m1:20">m1:20</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:208">m1:208</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:213">m1:213</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:218">m1:218</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:223">m1:223</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:228">m1:228</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:24">m1:24</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:29">m1:29</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m1:51">m1:51</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m1:659">m1:659</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m1:663">m1:663</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.FirstSteps.html#M2">M2</a> [module, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2">M2</a> [module, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Monoid_instances.html#M2">M2</a> [record, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.fib.html#m2lfib">m2lfib</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lfibP">m2lfibP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lmul">m2lmul</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lmulP">m2lmulP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lmx">m2lmx</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lpow">m2lpow</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Monoid_instances.html#M2N">M2N</a> [definition, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_Monoid">M2_Monoid</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_op">M2_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_eq_intros">M2_eq_intros</a> [lemma, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_mult">M2_mult</a> [definition, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.rt">M2_def.rt</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#3d24eb4e8ded7098f168acd26997b293">_ * _</a> [notation, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#b0d4ffdfad44d7e0423365f5b6c1abdc">_ + _</a> [notation, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.:::'1'">1</a> [notation, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.:::'0'">0</a> [notation, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.mult">M2_def.mult</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.plus">M2_def.plus</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.one">M2_def.one</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.zero">M2_def.zero</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.A">M2_def.A</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def">M2_def</a> [section, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c00">M2.c00</a> [projection, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c00">M2.c00</a> [projection, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c01">M2.c01</a> [projection, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c01">M2.c01</a> [projection, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c10">M2.c10</a> [projection, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c10">M2.c10</a> [projection, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c11">M2.c11</a> [projection, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c11">M2.c11</a> [projection, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions">M2.Definitions</a> [section, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.A">M2.Definitions.A</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.mult">M2.Definitions.mult</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.one">M2.Definitions.one</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.plus">M2.Definitions.plus</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.rt">M2.Definitions.rt</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.zero">M2.Definitions.zero</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#67a4b07b9250dfdc7f4120645cd84088">_ ** _</a> [notation, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#4d985c683f0b20e66a34edc4883055ad">_ * _</a> [notation, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#617a2cc97dac8fe5e951278b3bb11609">_ + _</a> [notation, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.:::'0'">0</a> [notation, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.:::'1'">1</a> [notation, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.Id2">M2.Id2</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.Id2">M2.Id2</a> [definition, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Id2_neutral">M2.Id2_neutral</a> [lemma, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.mat">M2.mat</a> [constructor, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.mat">M2.mat</a> [constructor, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_mult">M2.M2_mult</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#e19429236df0008b7680aa787e1bf5a5">_ * _</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#792efe48a5e5591a3ac8904aa574c15c">_ + _</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.:::'1'">1</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.:::'0'">0</a> [notation, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.rt">M2.M2_Definitions.rt</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.mult">M2.M2_Definitions.mult</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.plus">M2.M2_Definitions.plus</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.one">M2.M2_Definitions.one</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.zero">M2.M2_Definitions.zero</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.A">M2.M2_Definitions.A</a> [variable, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions">M2.M2_Definitions</a> [section, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.M2_mult_assoc">M2.M2_mult_assoc</a> [lemma, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
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+<a href="additions.Naive.html#M2.power_of_plus">M2.power_of_plus</a> [lemma, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
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+<a href="additions.Naive.html#M2.t">M2.t</a> [record, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
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+<a href="hydras.Ackermann.subAll.html#m2:15">m2:15</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="additions.fib.html#m2:153">m2:153</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m2:17">m2:17</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:20">m2:20</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:209">m2:209</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m2:21">m2:21</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:214">m2:214</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:219">m2:219</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:224">m2:224</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:229">m2:229</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:25">m2:25</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:30">m2:30</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m2:52">m2:52</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m2:660">m2:660</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m2:664">m2:664</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.fib.html#m3lfib">m3lfib</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m3lfibP">m3lfibP</a> [lemma, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="additions.AM.html#M:102">M:102</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#m:102">m:102</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#m:117">m:117</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#m:1183">m:1183</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.rpo.rpo.html#m:269">m:269</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
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+<a href="hydras.Ackermann.subAll.html#m:290">m:290</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m:3">m:3</a> [binder, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#m:574">m:574</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:575">m:575</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:575">m:575</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:576">m:576</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:577">m:577</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:578">m:578</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:579">m:579</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:580">m:580</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:581">m:581</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:582">m:582</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:583">m:583</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:584">m:584</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:585">m:585</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:586">m:586</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:587">m:587</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:588">m:588</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:59">m:59</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#m:59">m:59</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.rpo.dickson.html#m:59">m:59</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:590">m:590</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:592">m:592</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:594">m:594</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:596">m:596</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:598">m:598</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:6">m:6</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:6">m:6</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:60">m:60</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="additions.Pow_variant.html#M:60">M:60</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="additions.Monoid_def.html#M:60">M:60</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:602">m:602</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:604">m:604</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:606">m:606</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:606">m:606</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:608">m:608</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:609">m:609</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:609">m:609</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:609">m:609</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:61">m:61</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Addition_Chains.html#M:61">M:61</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.rpo.dickson.html#m:61">m:61</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:611">m:611</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:612">m:612</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:612">m:612</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:614">m:614</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:615">m:615</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:615">m:615</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:616">m:616</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:617">m:617</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:619">m:619</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#m:62">m:62</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:62">m:62</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:621">m:621</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:623">m:623</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:623">m:623</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:625">m:625</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:627">m:627</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:627">m:627</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="additions.AM.html#M:63">M:63</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
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+<a href="additions.fib.html#m:63">m:63</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:635">m:635</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:635">m:635</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:638">m:638</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:639">m:639</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:64">m:64</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:642">m:642</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:643">m:643</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:644">m:644</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:645">m:645</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:646">m:646</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:646">m:646</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#m:648">m:648</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Monoid_def.html#M:65">M:65</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#M:65">M:65</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#m:68">m:68</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="hydras.MoreAck.Ack.html#m:69">m:69</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.fib.html#m:69">m:69</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Schutte.Lub.html#M:7">M:7</a> [binder, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#m:704">m:704</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:71">m:71</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:714">m:714</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:72">m:72</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#m:754">m:754</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Ackermann.primRec.html#m:8">m:8</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:807">m:807</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:809">m:809</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:82">m:82</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:83">m:83</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:83">m:83</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="additions.fib.html#m:83">m:83</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:837">m:837</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:840">m:840</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:845">m:845</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#m:85">m:85</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.fib.html#m:85">m:85</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:857">m:857</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.AM.html#M:86">M:86</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#m:862">m:862</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:868">m:868</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:87">m:87</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:87">m:87</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:871">m:871</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:89">m:89</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:892">m:892</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:895">m:895</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:9">m:9</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:908">m:908</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:91">m:91</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:911">m:911</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.Euclidean_Chains.html#M:92">M:92</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:93">m:93</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:935">m:935</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#m:938">m:938</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:94">m:94</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#m:95">m:95</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:966">m:966</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:968">m:968</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#m:97">m:97</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="additions.AM.html#M:97">M:97</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:970">m:970</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:974">m:974</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:978">m:978</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#m:98">m:98</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:984">m:984</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#m:986">m:986</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="global_N"></a><h2>N </h2>
+<a href="hydras.Hydra.BigBattle.html#N">N</a> [definition, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Naive.html">Naive</a> [library]<br/>
+<a href="hydras.Ackermann.LNN.html#nAnd">nAnd</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nAnd">nAnd</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nAnd">nAnd</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.extEqualNat.html#naryFunc">naryFunc</a> [definition, in <a href="hydras.Ackermann.extEqualNat.html">hydras.Ackermann.extEqualNat</a>]<br/>
+<a href="hydras.Ackermann.model.html#naryFunc">naryFunc</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#naryIf">naryIf</a> [definition, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Ackermann.extEqualNat.html#naryRel">naryRel</a> [definition, in <a href="hydras.Ackermann.extEqualNat.html">hydras.Ackermann.extEqualNat</a>]<br/>
+<a href="hydras.Ackermann.model.html#naryRel">naryRel</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natLE">natLE</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natLT">natLT</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.PAconsistent.html#natModel">natModel</a> [definition, in <a href="hydras.Ackermann.PAconsistent.html">hydras.Ackermann.PAconsistent</a>]<br/>
+<a href="additions.Monoid_instances.html#Natmult">Natmult</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natNE">natNE</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="additions.Monoid_instances.html#Natplus">Natplus</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natPlus">natPlus</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natTimes">natTimes</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#natToTerm">natToTerm</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#natToTerm">natToTerm</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#natToTermLNN">natToTermLNN</a> [definition, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#natToTermLNT">natToTermLNT</a> [definition, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Nat_le_TO">Nat_le_TO</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Nat_le_dec">Nat_le_dec</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#nat_double_or_s_double">nat_double_or_s_double</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.Compatibility.html#nat_power_ok">nat_power_ok</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#nat_eqb_false">nat_eqb_false</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Monoid_instances.html#nat_plus_op">nat_plus_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#nat_mult_op">nat_mult_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_2_term_mono">nat_2_term_mono</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_lt_psi">nat_lt_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_lt_cons">nat_lt_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_2_term">nat_2_term</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#NA:4">NA:4</a> [binder, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#NA:4">NA:4</a> [binder, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nbterms">nbterms</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.rpo.more_list.html#nb_occ_app">nb_occ_app</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#nb_occ">nb_occ</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#NB:8">NB:8</a> [binder, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#NB:8">NB:8</a> [binder, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#neqIsPR">neqIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#nested_Ack_bound">nested_Ack_bound</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Fib2.html#neutral_r">neutral_r</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#neutral_l">neutral_l</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.Ackermann.NewNotations.html">NewNotations</a> [library]<br/>
+<a href="hydras.Ackermann.folProp.html#newVar">newVar</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#newVar1">newVar1</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#newVar2">newVar2</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nExist">nExist</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nExist">nExist</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nExist">nExist</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#next">next</a> [definition, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Fib2.html#next_fib">next_fib</a> [lemma, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#next_step">next_step</a> [definition, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#next_round">next_round</a> [definition, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#next_round_dec">next_round_dec</a> [definition, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf">nf</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf">nf</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF">NF</a> [abbreviation, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nfb">nfb</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nForall">nForall</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nForall">nForall</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nForall">nForall</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html">nfwfgaia</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_LT_right">nf_LT_right</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_exp_F">nf_exp_F</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_LT_iff">nf_LT_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega_omega">nf_omega_omega</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_phi0">nf_phi0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega">nf_omega</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_Wf">nf_Wf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_Acc">nf_Acc</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_tail_lt">nf_tail_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_LT">nf_cons_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_rect">nf_rect</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega_tower">nf_omega_tower</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_coeff_irrelevance">nf_coeff_irrelevance</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_intro'">nf_intro'</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_intro">nf_intro</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_tail_lt_nf">nf_tail_lt_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_of_finite">nf_of_finite</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_fin">nf_fin</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_FS">nf_FS</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_b_cons_eq">nf_b_cons_eq</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_iff">nf_cons_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_inv">nf_cons_inv</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv3">nf_inv3</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv2">nf_inv2</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv1">nf_inv1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_b">nf_b</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_epsilon">nf_epsilon</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_epsilon0">nf_epsilon0</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_omega">nf_omega</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_fin">nf_fin</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.CNF.html#nf_bounded">nf_bounded</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_Acc">NF_Acc</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_inv2">NF_inv2</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_inv1">NF_inv1</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#nf_ref">nf_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#nf_canon">nf_canon</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#nf_proof_unicity">nf_proof_unicity</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#nf_m">nf_m</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_intro">nf_intro</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_coeff_irrelevance">nf_coeff_irrelevance</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_Wf">nf_Wf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c2">nf_c2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c1">nf_c1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b2">nf_b2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b1">nf_b1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a2">nf_a2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a1">nf_a1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_subterm">nf_subterm</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_fin_inv">nf_fin_inv</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_inv_tail">nf_inv_tail</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c">nf_c</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b">nf_b</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a">nf_a</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#nh">nh</a> [projection, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Ackermann.fol.html#nilTerms">nilTerms</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nImp">nImp</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nImp">nImp</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nImp">nImp</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmodulo">Nmodulo</a> [section, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmodulo.m">Nmodulo.m</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmodulo.m_gt_1">Nmodulo.m_gt_1</a> [variable, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmod_Monoid">Nmod_Monoid</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#NMult">NMult</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN">NN</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN">NN</a> [section, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html">NN</a> [library]<br/>
+<a href="hydras.Ackermann.LNN.html#nnE">nnE</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nnE">nnE</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nnE">nnE</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.model.html#nnHelp">nnHelp</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nnI">nnI</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nnI">nnI</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nnI">nnI</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations">NNnotations</a> [module, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.LT">NNnotations.LT</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.Plus">NNnotations.Plus</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.Succ">NNnotations.Succ</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.S_">NNnotations.S_</a> [abbreviation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.Times">NNnotations.Times</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.Zero">NNnotations.Zero</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#3eb208196255ec4fdf2292f5d7aaaf16">fol:_ * _</a> [notation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#018d848f24ca8d774a2830f1a5e4a70f">fol:_ + _</a> [notation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.::fol_scope:x_'&lt;'_x">_ &lt; _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#d5e10e8b47737e9c618b2250abdff48c">_ * _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#138f411abfc162998ee0d11413343f95">_ + _ (fol_scope)</a> [notation, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#nnPlusNotNeeded">nnPlusNotNeeded</a> [lemma, in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html">NNtheory</a> [library]<br/>
+<a href="hydras.Ackermann.model.html#nnTranslate">nnTranslate</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn1">nn1</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN1">NN1</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn2">nn2</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN2">NN2</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN2PA.html#NN2PA">NN2PA</a> [lemma, in <a href="hydras.Ackermann.NN2PA.html">hydras.Ackermann.NN2PA</a>]<br/>
+<a href="hydras.Ackermann.NN2PA.html">NN2PA</a> [library]<br/>
+<a href="hydras.Ackermann.NN.html#nn3">nn3</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN3">NN3</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn4">nn4</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN4">NN4</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn5">nn5</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN5">NN5</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn6">nn6</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN6">NN6</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn7">nn7</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN7">NN7</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN72PA">NN72PA</a> [lemma, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn8">nn8</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN8">NN8</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN82PA">NN82PA</a> [lemma, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn9">nn9</a> [lemma, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN9">NN9</a> [definition, in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN92PA">NN92PA</a> [lemma, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#node">node</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#noMiddle">noMiddle</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#noMiddle">noMiddle</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#noMiddle">noMiddle</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.rpo.more_list.html#none_nb_occ_O">none_nb_occ_O</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#non_commutativity_of_plus">non_commutativity_of_plus</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Non_denum">Non_denum</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.Addition_Chains.html#non_optimality_proof.binary_opt">non_optimality_proof.binary_opt</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#non_optimality_proof">non_optimality_proof</a> [section, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nOr">nOr</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nOr">nOr</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nOr">nOr</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#normal">normal</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.AP.html#normal_phi0">normal_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Addition.html#normal_plus_alpha">normal_plus_alpha</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="Goedel.PRrepresentable.html#notBoundedForall">notBoundedForall</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.wConsistent.html#notCon2wNotCon">notCon2wNotCon</a> [lemma, in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="hydras.Ackermann.fol.html#notH">notH</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#NotIncl">NotIncl</a> [section, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#notInFreeVarSys">notInFreeVarSys</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notRel">notRel</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notRelPR">notRelPR</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notZero">notZero</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notZeroIsPR">notZeroIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_LT_zero">not_LT_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_zero_lt">not_zero_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_zero_gt_0">not_zero_gt_0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_lt_zero">not_lt_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#not_double_is_s_double">not_double_is_s_double</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#not_le_gt">not_le_gt</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#not_le_ge">not_le_ge</a> [lemma, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Schutte.CNF.html#not_AP_inv2">not_AP_inv2</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#not_AP_inv_0">not_AP_inv_0</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#not_acc">not_acc</a> [lemma, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_countable_unbounded">not_countable_unbounded</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Not_Unbounded_countable">Not_Unbounded_countable</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Not_Unbounded_bounded">Not_Unbounded_bounded</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_succ_limit">not_is_succ_limit</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_limit_succ">not_is_limit_succ</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_limit_zero">not_is_limit_zero</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_succ_zero">not_is_succ_zero</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_lt_zero">not_lt_zero</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_gt_le">not_gt_le</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#not_lt_zero">not_lt_zero</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#not_decreasing">not_decreasing</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_decreasing">not_decreasing</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_decreasing_aux">not_decreasing_aux</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_acc">not_acc</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="additions.Monoid_instances.html#NPlus">NPlus</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Compatibility.html#Npos_power_compat">Npos_power_compat</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.More_on_positive.html#Npos_gt_0">Npos_gt_0</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#Npos_diff_zero">Npos_diff_zero</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.rpo.more_list.html#nth_error_ok_in">nth_error_ok_in</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#nth_error_map">nth_error_map</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#nVars">nVars</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#nv:118">nv:118</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#nv:13">nv:13</a> [binder, in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#nv:2">nv:2</a> [binder, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#nv:283">nv:283</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="Goedel.codeSysPrf.html#nv:30">nv:30</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#nv:404">nv:404</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#nv:432">nv:432</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#nv:437">nv:437</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#nv:48">nv:48</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#nv:65">nv:65</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.Compatibility.html#N_size_gt">N_size_gt</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#N_pow_compat">N_pow_compat</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#N_pow_power">N_pow_power</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#N_pow">N_pow</a> [definition, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.FirstSteps.html#N_bpow">N_bpow</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_commute">N_bpow_commute</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_ok_R">N_bpow_ok_R</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_ok">N_bpow_ok</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow">N_bpow</a> [definition, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#N_bpow_commute">N_bpow_commute</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#N_bpow_ok_R">N_bpow_ok_R</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#N_bpow_ok">N_bpow_ok</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#N_bpow">N_bpow</a> [definition, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.More_on_positive.html#N_le_mul_pos">N_le_mul_pos</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_q0">N_pos_div_eucl_q0</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_rest">N_pos_div_eucl_rest</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_divides">N_pos_div_eucl_divides</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_N2pos">N_pos_N2pos</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_le_1_pos">N_le_1_pos</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_0_le_n">N_0_le_n</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Naive.html#N_mod.power">N_mod.power</a> [definition, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#N_mod.mult_mod">N_mod.mult_mod</a> [definition, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#N_mod.m_fixed.m">N_mod.m_fixed.m</a> [variable, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#N_mod.m_fixed">N_mod.m_fixed</a> [section, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#N_mod">N_mod</a> [module, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Monoid_instances.html#N_plus_op">N_plus_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#N_mult_op">N_mult_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#N_simpl">N_simpl</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n'':11">n'':11</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n'':8">n'':8</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':10">n':10</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1024">n':1024</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1152">n':1152</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':117">n':117</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1204">n':1204</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':131">n':131</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1338">n':1338</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1384">n':1384</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#n':139">n':139</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.dickson.html#n':14">n':14</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1436">n':1436</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1500">n':1500</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1556">n':1556</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1585">n':1585</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1609">n':1609</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':165">n':165</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':1691">n':1691</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':174">n':174</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':176">n':176</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n':19">n':19</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':194">n':194</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':2043">n':2043</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':2066">n':2066</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':223">n':223</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':225">n':225</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':231">n':231</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#n':24">n':24</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n':25">n':25</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':258">n':258</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.rpo.html#n':259">n':259</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#n':27">n':27</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':285">n':285</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#n':29">n':29</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':294">n':294</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':3">n':3</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n':30">n':30</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#n':301">n':301</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#n':305">n':305</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':324">n':324</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':329">n':329</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':34">n':34</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':347">n':347</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':359">n':359</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n':36">n':36</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':366">n':366</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':380">n':380</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n':42">n':42</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':425">n':425</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':429">n':429</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n':430">n':430</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n':434">n':434</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Prelude.Iterates.html#n:110">n:110</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:110">n:110</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#n:111">n:111</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#n:111">n:111</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#n:112">n:112</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Gamma0.T2.html#n:112">n:112</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:112">n:112</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#n:112">n:112</a> [binder, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:112">n:112</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:112">n:112</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#n:113">n:113</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1133">n:1133</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n:114">n:114</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#n:114">n:114</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="additions.fib.html#n:114">n:114</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:114">n:114</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1140">n:1140</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1142">n:1142</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1146">n:1146</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#n:115">n:115</a> [binder, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:115">n:115</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#n:115">n:115</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1157">n:1157</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#n:116">n:116</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#n:116">n:116</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#n:116">n:116</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:116">n:116</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#n:117">n:117</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1178">n:1178</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.PRrepresentable.html#n:118">n:118</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:118">n:118</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#n:118">n:118</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1181">n:1181</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1184">n:1184</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1187">n:1187</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:119">n:119</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#n:119">n:119</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:119">n:119</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1190">n:1190</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#n:12">n:12</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="Goedel.PRrepresentable.html#n:12">n:12</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#n:12">n:12</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#n:12">n:12</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#n:12">n:12</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#n:12">n:12</a> [binder, in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<a href="Goedel.fixPoint.html#n:12">n:12</a> [binder, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#n:12">n:12</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#n:12">n:12</a> [binder, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#n:12">n:12</a> [binder, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.Ackermann.fol.html#n:12">n:12</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="gaia_hydras.onType.html#n:12">n:12</a> [binder, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:12">n:12</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#n:12">n:12</a> [binder, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#n:12">n:12</a> [binder, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:120">n:120</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#n:120">n:120</a> [binder, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#N:120">N:120</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:120">n:120</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#n:121">n:121</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#n:121">n:121</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:121">n:121</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1215">n:1215</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1219">n:1219</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#n:122">n:122</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#n:122">n:122</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1226">n:1226</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1229">n:1229</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:123">n:123</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#N:123">N:123</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:123">n:123</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:124">n:124</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n:124">n:124</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1245">n:1245</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1247">n:1247</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1248">n:1248</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Gamma0.T2.html#n:125">n:125</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:125">n:125</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#n:1252">n:1252</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1254">n:1254</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:126">n:126</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Epsilon0.Hprime.html#n:126">n:126</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1260">n:1260</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1262">n:1262</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1264">n:1264</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:127">n:127</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Ackermann.subProp.html#n:127">n:127</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:1277">n:1277</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.PRrepresentable.html#n:128">n:128</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="hydras.Prelude.More_Arith.html#n:13">n:13</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#n:13">n:13</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:13">n:13</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Schutte.AP.html#n:13">n:13</a> [binder, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#n:13">n:13</a> [binder, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#n:13">n:13</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#n:13">n:13</a> [binder, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#n:13">n:13</a> [binder, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#n:13">n:13</a> [binder, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.wConsistent.html#n:13">n:13</a> [binder, in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="hydras.rpo.dickson.html#n:13">n:13</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#n:130">n:130</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#n:130">n:130</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#n:130">n:130</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:1302">n:1302</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#n:16">n:16</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="hydras.OrdinalNotations.ON_Finite.html#n:16">n:16</a> [binder, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="additions.fib.html#n:16">n:16</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#n:205">n:205</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#n:205">n:205</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#n:206">n:206</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#n:206">n:206</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:206">n:206</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="hydras.Epsilon0.Large_Sets.html#n:208">n:208</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:208">n:208</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="hydras.Ackermann.codeList.html#n:209">n:209</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="Goedel.fixPoint.html#n:21">n:21</a> [binder, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#N:21">N:21</a> [binder, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#n:21">n:21</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#n:21">n:21</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.code.html#n:21">n:21</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#n:21">n:21</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#n:21">n:21</a> [binder, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#n:21">n:21</a> [binder, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Schutte.Countable.html#n:21">n:21</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:210">n:210</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#n:210">n:210</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#n:210">n:210</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#n:210">n:210</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
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+<a href="additions.Addition_Chains.html#n:211">n:211</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#n:211">n:211</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.rpo.term.html#n:212">n:212</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#n:212">n:212</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#n:212">n:212</a> [binder, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
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+<a href="hydras.Ackermann.subAll.html#n:22">n:22</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#n:22">n:22</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Hydra.O2H.html#n:22">n:22</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
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+<a href="hydras.Ackermann.LNN2LNT.html#n:22">n:22</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
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+<a href="hydras.Epsilon0.Canon.html#n:22">n:22</a> [binder, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#n:22">n:22</a> [binder, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#n:22">n:22</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#n:220">n:220</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#n:220">n:220</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#n:225">n:225</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="additions.Pow.html#n:96">n:96</a> [binder, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="Goedel.rosserPA.html#n:96">n:96</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#n:96">n:96</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
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+<a href="additions.Pow_variant.html#n:97">n:97</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="hydras.OrdinalNotations.ON_Generic.html#OB:153">OB:153</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
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+<a href="hydras.Schutte.Ordering_Functions.html#OF_total">OF_total</a> [projection, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#OF_segment">OF_segment</a> [projection, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Addition.html#of_u">of_u</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#of_g">of_g</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.AM.html#OK">OK</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Ok">Ok</a> [constructor, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#OK">OK</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#OK87">OK87</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Olt">Olt</a> [instance, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#Olt">Olt</a> [instance, in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Olt">Olt</a> [instance, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol'':40">ol'':40</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol'':45">ol'':45</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol':30">ol':30</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol':34">ol':34</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol':37">ol':37</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol':39">ol':39</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol':43">ol':43</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol:29">ol:29</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol:33">ol:33</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol:35">ol:35</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol:38">ol:38</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#ol:41">ol:41</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega">omega</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#omega">omega</a> [abbreviation, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#omega">omega</a> [abbreviation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega">omega</a> [abbreviation, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega">omega</a> [abbreviation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#Omega">Omega</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#omeganf">omeganf</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html">OmegaOmega</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_limit">omega_limit</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_omega">omega_omega</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_term_def">omega_term_def</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_term_plus_rw">omega_term_plus_rw</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_tower">omega_tower</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_term">omega_term</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#omega_ap">omega_ap</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Omega_limit_limitb">Omega_limit_limitb</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#omega_is_limit">omega_is_limit</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_ex2">omega_ex2</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_ex1">omega_ex1</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_square_thrice_eqn">omega_square_thrice_eqn</a> [definition, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_mult_mlarge_0">omega_mult_mlarge_0</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.AP.html#omega_tower_mono">omega_tower_mono</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#omega_eqn">omega_eqn</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#omega_second_AP">omega_second_AP</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#omega_tower">omega_tower</a> [definition, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Omega_as_lub">Omega_as_lub</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega_not_succ">omega_not_succ</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega_is_limit">omega_is_limit</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Omega_plus_Omega">Omega_plus_Omega</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#omega_omega_1_3">omega_omega_1_3</a> [definition, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#omega_omega_1_4">omega_omega_1_4</a> [definition, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#omega_ref">omega_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit_Limit">Omega_limit_Limit</a> [lemma, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit_not_Succ">Omega_limit_not_Succ</a> [lemma, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit_type">Omega_limit_type</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit">Omega_limit</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega_limit_least_gt">omega_limit_least_gt</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega_limit_least">omega_limit_least</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega_limit">omega_limit</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Omega_term_plus">Omega_term_plus</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Omega_2">Omega_2</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Omega_term">Omega_term</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.onType.html#omega_lt_wf">omega_lt_wf</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#omega_ok">omega_ok</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#Omega_comp">Omega_comp</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#omega_lt_epsilon">omega_lt_epsilon</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#omega_lt_epsilon0">omega_lt_epsilon0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html">Omega_Small</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Omega2">Omega2</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html">Omega2_Small</a> [library]<br/>
+<a href="gaia_hydras.onType.html#om12">om12</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#om67">om67</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON">ON</a> [record, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html">OnCodeList</a> [library]<br/>
+<a href="gaia_hydras.onType.html#ONDef">ONDef</a> [module, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports">ONDef.Exports</a> [module, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas">ONDef.Exports.Lemmas</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.disp">ONDef.Exports.Lemmas.disp</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.U">ONDef.Exports.Lemmas.U</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.ON">ONDef.Exports.ON</a> [abbreviation, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.on">ONDef.Exports.on</a> [abbreviation, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.ONMixin">ONDef.Exports.ONMixin</a> [abbreviation, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.tricho">ONDef.Exports.tricho</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.trichoP">ONDef.Exports.trichoP</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.wf">ONDef.Exports.wf</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Mixin">ONDef.Mixin</a> [constructor, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.mixin_of">ONDef.mixin_of</a> [record, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.on_struct">ONDef.on_struct</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Pack">ONDef.Pack</a> [constructor, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing">ONDef.Packing</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing.cT">ONDef.Packing.cT</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing.disp">ONDef.Packing.disp</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.pack_type">ONDef.pack_type</a> [record, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.type">ONDef.type</a> [projection, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#one">one</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#one">one</a> [abbreviation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="gaia_hydras.onType.html#ONEpsilon0Def">ONEpsilon0Def</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#one_ref">one_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="additions.Monoid_def.html#one_right">one_right</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#one_left">one_left</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#one_step">one_step</a> [inductive, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="additions.Monoid_instances.html#one_mod_neutral_r">one_mod_neutral_r</a> [lemma, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#one_mod_neutral_l">one_mod_neutral_l</a> [lemma, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Addition_Chains.html#One_node">One_node</a> [constructor, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Addition.html#one_plus_infinite">one_plus_infinite</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#one_plus_omega">one_plus_omega</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.AM.html#one:100">one:100</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#one:103">one:103</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#one:105">one:105</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#one:110">one:110</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#one:111">one:111</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#one:117">one:117</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#one:120">one:120</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#one:124">one:124</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#one:125">one:125</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#one:131">one:131</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#one:161">one:161</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#one:168">one:168</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#one:174">one:174</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.FirstSteps.html#one:36">one:36</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.AM.html#one:37">one:37</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.FirstSteps.html#one:42">one:42</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.AM.html#one:43">one:43</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#one:54">one:54</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Monoid_def.html#one:59">one:59</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.AM.html#one:61">one:61</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Monoid_def.html#one:63">one:63</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.AM.html#one:74">one:74</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Monoid_def.html#one:8">one:8</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.AM.html#one:84">one:84</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#one:95">one:95</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteDef">onFiniteDef</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteDef.i">onFiniteDef.i</a> [variable, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteMixin">onFiniteMixin</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteType">onFiniteType</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaDef">onOmegaDef</a> [section, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaMixin">onOmegaMixin</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaType">onOmegaType</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html">onType</a> [library]<br/>
+<a href="hydras.Gamma0.T2.html#on_length">on_length</a> [section, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S2.H">On_alpha.S2.H</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S2">On_alpha.S2</a> [section, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.Hbeta">On_alpha.S1.Hbeta</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.beta">On_alpha.S1.beta</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.H">On_alpha.S1.H</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1">On_alpha.S1</a> [section, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.HnonZero">On_alpha.HnonZero</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.Halpha">On_alpha.Halpha</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha">On_alpha.alpha</a> [variable, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha">On_alpha</a> [section, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#ON_O">ON_O</a> [instance, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#ON_O_comp">ON_O_comp</a> [instance, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.unique_P">On_Iota.unique_P</a> [variable, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.inhA">On_Iota.inhA</a> [variable, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P">On_Iota.P</a> [variable, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.A">On_Iota.A</a> [variable, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota">On_Iota</a> [section, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#ON_mult">ON_mult</a> [instance, in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#ON_plus">ON_plus</a> [instance, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_op_ok">ON_op_ok</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_fun_ok">ON_fun_ok</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_cst_ok">ON_cst_ok</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#On_compare_spec">On_compare_spec</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_correct_onto">ON_correct_onto</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_correct_inj">ON_correct_inj</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_correct">ON_correct</a> [record, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_Iso">ON_Iso</a> [record, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_le">ON_le</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_lt">ON_lt</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_compare">ON_compare</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_t">ON_t</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_wf">ON_wf</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_comp">ON_comp</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html">ON_Generic</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html">ON_mult</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html">ON_Finite</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">ON_Omega_plus_omega</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html">ON_Omega</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html">ON_Omega2</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html">ON_plus</a> [library]<br/>
+<a href="gaia_hydras.ON_gfinite.html">ON_gfinite</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html">ON_O</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#on:10">on:10</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#on:14">on:14</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#On:4">On:4</a> [binder, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#On:5">On:5</a> [binder, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO">OO</a> [module, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.compare_OO">OO.compare_OO</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.data">OO.data</a> [projection, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.data_ok">OO.data_ok</a> [projection, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.embed">OO.embed</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Fin">OO.Fin</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.le">OO.le</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt">OO.lt</a> [definition, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt_wf">OO.lt_wf</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.lt_embed">OO.lt_embed</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.mkord">OO.mkord</a> [constructor, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.mult">OO.mult</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.nf_proof_unicity">OO.nf_proof_unicity</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.omega">OO.omega</a> [abbreviation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.ON_OO">OO.ON_OO</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO">OO.OO</a> [record, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO_comp">OO.OO_comp</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO_eq_intro">OO.OO_eq_intro</a> [lemma, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.oo_str">OO.oo_str</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.phi0">OO.phi0</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.plus">OO.plus</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Succ">OO.Succ</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Zero">OO.Zero</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO._Omega">OO._Omega</a> [instance, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#3e75d3b5e7edc50985c52d117590a4bb">_ * _ (OO_scope)</a> [notation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#1a72aa5d15ab5318d1af7e45f6175964">omega^ (OO_scope)</a> [notation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#3e75d3b5e7edc50985c52d117590a4bb">_ * _ (OO_scope)</a> [notation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#f0892e4f827b9668045c34fe437da503">_ + _ (OO_scope)</a> [notation, in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#OpaqueWf">OpaqueWf</a> [module, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#OpaqueWf.lt_wf">OpaqueWf.lt_wf</a> [lemma, in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#opA:194">opA:194</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#opB:195">opB:195</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.PA.html#open">open</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#openClosed">openClosed</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus">oplus</a> [definition, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Oplus">Oplus</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus">oplus</a> [definition, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplusA">oplusA</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplusa0">oplusa0</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplusC">oplusC</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplusE">oplusE</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt_phi0">oplus_lt_phi0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_of_phi0">oplus_of_phi0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_of_phi0_0">oplus_of_phi0_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_bi">oplus_strict_mono_bi</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_LT_r">oplus_strict_mono_LT_r</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_LT_l">oplus_strict_mono_LT_l</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_l">oplus_strict_mono_l</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_r">oplus_strict_mono_r</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_strict_mono_0">oplus_strict_mono_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_le2">oplus_le2</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_le1">oplus_le1</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt2">oplus_lt2</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt1">oplus_lt1</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt_0">oplus_lt_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_assoc">oplus_assoc</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_assoc_0">oplus_assoc_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt_rw2">oplus_lt_rw2</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_comm">oplus_comm</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_comm_0">oplus_comm_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_nf">oplus_nf</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_nf_0">oplus_nf_0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_bounded_phi0">oplus_bounded_phi0</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_cons_cons">oplus_cons_cons</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_eqn">oplus_eqn</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_compare_Gt">oplus_compare_Gt</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_gt_rw">oplus_gt_rw</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_eq_rw">oplus_eq_rw</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_lt_rw">oplus_lt_rw</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_compare_Lt">oplus_compare_Lt</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_0_l">oplus_0_l</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_0_r">oplus_0_r</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus_aux:8">oplus_aux:8</a> [binder, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#oPlus_rw">oPlus_rw</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Oplus_assoc">Oplus_assoc</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_lt_phi0">oplus_lt_phi0</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_strict_mono_r">oplus_strict_mono_r</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_strict_mono_l">oplus_strict_mono_l</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_lt2">oplus_lt2</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_lt1">oplus_lt1</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_nf">oplus_nf</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus_aux:14">oplus_aux:14</a> [binder, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus0b">oplus0b</a> [lemma, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="additions.Addition_Chains.html#optimal">optimal</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#optimal_upper_bound">optimal_upper_bound</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#optimal_generator">optimal_generator</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#op_comm">op_comm</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#op_assoc">op_assoc</a> [projection, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:100">op:100</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.fib.html#op:102">op:102</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Addition_Chains.html#op:102">op:102</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#op:104">op:104</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#op:109">op:109</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.fib.html#op:109">op:109</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Addition_Chains.html#op:110">op:110</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:111">op:111</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#op:116">op:116</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.fib.html#op:117">op:117</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.AM.html#op:119">op:119</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#op:123">op:123</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.AM.html#op:124">op:124</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#op:130">op:130</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#op:160">op:160</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:163">op:163</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#op:167">op:167</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:169">op:169</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#op:173">op:173</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:183">op:183</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:195">op:195</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#op:36">op:36</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:36">op:36</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#op:42">op:42</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#op:42">op:42</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:43">op:43</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:48">op:48</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#op:5">op:5</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:51">op:51</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#op:53">op:53</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:54">op:54</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#op:54">op:54</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#op:58">op:58</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.AM.html#op:60">op:60</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Monoid_def.html#op:62">op:62</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:62">op:62</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:69">op:69</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#op:7">op:7</a> [binder, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.AM.html#op:73">op:73</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#op:83">op:83</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#op:89">op:89</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#op:94">op:94</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#op:99">op:99</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Ord">Ord</a> [axiom, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ord">ord</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_least_least">ordering_function_least_least</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_unbounded_unbounded">ordering_unbounded_unbounded</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_ex">ordering_function_ex</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_segment_ex_unique">ordering_segment_ex_unique</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_segments_of_B">ordering_segments_of_B</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_seg_unicity">ordering_function_seg_unicity</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity">ordering_function_unicity</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.O2">ordering_function_unicity.O2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.O1">ordering_function_unicity.O1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f2">ordering_function_unicity.f2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f1">ordering_function_unicity.f1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A2">ordering_function_unicity.A2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A1">ordering_function_unicity.A1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.B">ordering_function_unicity.B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity">ordering_function_unicity</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.O2">ordering_function_unicity_1.O2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.O1">ordering_function_unicity_1.O1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.f2">ordering_function_unicity_1.f2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.f1">ordering_function_unicity_1.f1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.A2">ordering_function_unicity_1.A2</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.A1">ordering_function_unicity_1.A1</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.B">ordering_function_unicity_1.B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1">ordering_function_unicity_1</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_le">ordering_le</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_seg">ordering_function_seg</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_mono_weakR">ordering_function_mono_weakR</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Ordering_bijection">Ordering_bijection</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_monoR">ordering_function_monoR</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_mono_weak">ordering_function_mono_weak</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_mono">ordering_function_mono</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_In">ordering_function_In</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_segment">ordering_segment</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function">ordering_function</a> [record, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html">Ordering_Functions</a> [library]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#ordinal">ordinal</a> [abbreviation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#ordinal_ok">ordinal_ok</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordinal_segment">ordinal_segment</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Prelude.OrdNotations.html">OrdNotations</a> [library]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ord_eq">ord_eq</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ord_ok">ord_ok</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#orE">orE</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#orE">orE</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#orE">orE</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.fol.html#orH">orH</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#orI1">orI1</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#orI1">orI1</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#orI1">orI1</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#orI2">orI2</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#orI2">orI2</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#orI2">orI2</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#orRel">orRel</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#orRelPR">orRelPR</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#orSym">orSym</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#orSym">orSym</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#orSym">orSym</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#orSys">orSys</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#orSys">orSys</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#orSys">orSys</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#output:62">output:62</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.rpo.more_list.html#o_length">o_length</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_mono">o_finite_mult_mono</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_lt_phi0_1">o_finite_mult_lt_phi0_1</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_rw">o_finite_mult_rw</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_nf">o_finite_mult_nf</a> [lemma, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult">o_finite_mult</a> [definition, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#o_finite_mult_S_rw">o_finite_mult_S_rw</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#o_36_of_42">o_36_of_42</a> [definition, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#o_33_of_42">o_33_of_42</a> [definition, in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#o_finite_mult">o_finite_mult</a> [definition, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o':423">o':423</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o':434">o':434</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o':437">o':437</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#o':61">o':61</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#o':64">o':64</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#o1:15">o1:15</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#o1:20">o1:20</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#o1:6">o1:6</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.O2H.html">O2H</a> [library]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota">o2iota</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota_1">o2iota_1</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota_0">o2iota_0</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.rpo.rpo.html#o2:16">o2:16</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#o2:21">o2:21</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#o2:7">o2:7</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html#O3O">O3O</a> [definition, in <a href="hydras.OrdinalNotations.Example_3PlusOmega.html">hydras.OrdinalNotations.Example_3PlusOmega</a>]<br/>
+<a href="hydras.Hydra.O2H.html#o:10">o:10</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#o:124">o:124</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Hydra.O2H.html#o:13">o:13</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#o:131">o:131</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Addition.html#o:18">o:18</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#o:19">o:19</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:280">o:280</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#o:36">o:36</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:424">o:424</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:435">o:435</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:436">o:436</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#o:537">o:537</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:619">o:619</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#o:62">o:62</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#o:628">o:628</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#o:63">o:63</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Schutte.CNF.html#o:63">o:63</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#o:64">o:64</a> [binder, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#o:75">o:75</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="global_P"></a><h2>P </h2>
+<a href="hydras.Epsilon0.F_alpha.html#P">P</a> [record, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#P">P</a> [axiom, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#P">P</a> [record, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PA">PA</a> [definition, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PA">PA</a> [projection, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA">PA</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA">PA</a> [section, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA">PA</a> [projection, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.PA.html">PA</a> [library]<br/>
+<a href="hydras.Ackermann.NN2PA.html#PAboundedLT">PAboundedLT</a> [lemma, in <a href="hydras.Ackermann.NN2PA.html">hydras.Ackermann.NN2PA</a>]<br/>
+<a href="hydras.Ackermann.PAconsistent.html#PAconsistent">PAconsistent</a> [lemma, in <a href="hydras.Ackermann.PAconsistent.html">hydras.Ackermann.PAconsistent</a>]<br/>
+<a href="hydras.Ackermann.PAconsistent.html">PAconsistent</a> [library]<br/>
+<a href="hydras.Ackermann.PA.html#PAdec">PAdec</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="Goedel.rosserPA.html#PAIncomplete">PAIncomplete</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#PairwiseEqual">PairwiseEqual</a> [definition, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:134">pair:134</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:145">pair:145</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:156">pair:156</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:239">pair:239</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:245">pair:245</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:460">pair:460</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#pair:468">pair:468</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#paPlusSym">paPlusSym</a> [lemma, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="additions.Addition_Chains.html#parametric">parametric</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_for_free">param_correctness_for_free</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness">param_correctness</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_nat">param_correctness_nat</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_aux">param_correctness_aux</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html">PartialFun</a> [library]<br/>
+<a href="gaia_hydras.GPaths.html#path">path</a> [abbreviation, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path">path</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html">PAtheory</a> [library]<br/>
+<a href="gaia_hydras.GPaths.html#pathS">pathS</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#pathS">pathS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html">Paths</a> [library]<br/>
+<a href="hydras.Hydra.O2H.html#path_to_round_plus">path_to_round_plus</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#path_toS_round_plus">path_toS_round_plus</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#path_toS_trace">path_toS_trace</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_LT">path_to_LT</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_iff1">path_to_iff1</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_inv1">path_to_inv1</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_tob">path_tob</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to">path_to</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#path_to_omega_mult">path_to_omega_mult</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_standard_equiv">path_to_standard_equiv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_standardS_equiv">path_toS_standardS_equiv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_S_standard_pathS">path_to_S_standard_pathS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_S_iota_from">path_to_S_iota_from</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_LT">path_to_LT</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_LT">path_toS_LT</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_gnaw">path_to_gnaw</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_mult">path_to_mult</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_mult">path_toS_mult</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_tail">path_to_tail</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_tail">path_toS_tail</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_not_In_zero">path_to_not_In_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_zero">path_to_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero">path_toS_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_inv">path_toS_zero_inv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_one">path_toS_zero_one</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_but_last">path_toS_zero_but_last</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_appR">path_to_appR</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_appR">path_toS_appR</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_decompose">path_to_decompose</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_decompose">path_toS_decompose</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_app">path_to_app</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_app">path_toS_app</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_gnawS">path_toS_gnawS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_acc_from">path_acc_from</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_nf">path_toS_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_path_toS_iff">path_to_path_toS_iff</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_path_toS">path_to_path_toS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_path_to">path_toS_path_to</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_interval_inv_le">path_to_interval_inv_le</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_cons">path_toS_cons</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_1">path_toS_1</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS">path_toS</a> [inductive, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_not_nil">path_to_not_nil</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_cons">path_to_cons</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_1">path_to_1</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to">path_to</a> [inductive, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#paZeroOrSucc">paZeroOrSucc</a> [lemma, in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA_Succ">PA_Succ</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA_Zero">PA_Zero</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa1">pa1</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA1">PA1</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa2">pa2</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA2">PA2</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa3">pa3</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA3">PA3</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa4">pa4</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA4">PA4</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa5">pa5</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA5">PA5</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa6">pa6</a> [lemma, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA6">PA6</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA7">PA7</a> [definition, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#pA:50">pA:50</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#PA:64">PA:64</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PB">PB</a> [definition, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PB">PB</a> [projection, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PB">PB</a> [projection, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PB:5">PB:5</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#pB:51">pB:51</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#PB:65">PB:65</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PC">PC</a> [projection, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PC">PC</a> [projection, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PC_Zero">PC_Zero</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PD">PD</a> [projection, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PD">PD</a> [projection, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Pdec:3">Pdec:3</a> [binder, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PD_Zero">PD_Zero</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PE">PE</a> [projection, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PE">PE</a> [projection, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut">Permut</a> [module, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.DS">Permut.DS</a> [module, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.elt">Permut.elt</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.eq_elt_dec">Permut.eq_elt_dec</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.list_permut">Permut.list_permut</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.list_to_multiset">Permut.list_to_multiset</a> [definition, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PF87">PF87</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#pf:39">pf:39</a> [binder, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#pf:40">pf:40</a> [binder, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Schutte.Critical.html#phi">phi</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi">phi</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_RR">phi_mono_RR</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_psi">phi_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_6">phi_to_psi_6</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_5">phi_to_psi_5</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_4">phi_to_psi_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_3">phi_to_psi_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_2">phi_to_psi_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_1">phi_to_psi_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha">phi_to_psi.alpha</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi">phi_to_psi</a> [section, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_inj_r">phi_inj_r</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_weak_r">phi_mono_weak_r</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_r">phi_mono_r</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi_plus_fin">phi_of_psi_plus_fin</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_alpha_zero_gt_alpha">phi_alpha_zero_gt_alpha</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_principalR">phi_principalR</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_spec1">phi_spec1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_le_ge">phi_le_ge</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_le">phi_le</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_fix">phi_fix</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_any_cons">phi_of_any_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_nf">phi_nf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases">phi_cases</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases'">phi_cases'</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases_aux">phi_cases_aux</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi_succ">phi_of_psi_succ</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_alpha_zero">phi_alpha_zero</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_principal">phi_principal</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi">phi_to_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi">phi_of_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0">phi0</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0">phi0</a> [abbreviation, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_eq">phi0_eq</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_eq_bad">phi0_eq_bad</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_plus_mult">phi0_plus_mult</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono">phi0_mono</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono_strict_LT">phi0_mono_strict_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono_strict">phi0_mono_strict</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_ltR">phi0_ltR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_lt">phi0_lt</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_plus_lt_phi0R">phi0_mult_plus_lt_phi0R</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_plus_lt_phi0">phi0_mult_plus_lt_phi0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_lt_phi0">phi0_mult_lt_phi0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_ok">phi0_mult.f_ok</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_Sle">phi0_mult.f_Sle</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_mono">phi0_mult.f_mono</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.Halpha">phi0_mult.Halpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f">phi0_mult.f</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha">phi0_mult.alpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult">phi0_mult</a> [section, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#phi0_well_named">phi0_well_named</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_lt_epsilon0_R">phi0_lt_epsilon0_R</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_lt_epsilon0">phi0_lt_epsilon0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_of_limit">phi0_of_limit</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_sup">phi0_sup</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_alpha_phi0_beta">phi0_alpha_phi0_beta</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_positive">phi0_positive</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_inj">phi0_inj</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_R_weak">phi0_mono_R_weak</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_R">phi0_mono_R</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_weak">phi0_mono_weak</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono">phi0_mono</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_zero">phi0_zero</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_elim">phi0_elim</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_ordering">phi0_ordering</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#phi0_ref">phi0_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#phi0_mono">phi0_mono</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Phi0_def">Phi0_def</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#phi:198">phi:198</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#phi:214">phi:214</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#phi:29">phi:29</a> [binder, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#phi:69">phi:69</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#phi:76">phi:76</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#phi:85">phi:85</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi1_4IsPR">pi1_4IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi1_3IsPR">pi1_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi1_2IsPR">pi1_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_4IsPR">pi2_4IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_3IsPR">pi2_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_2IsPR">pi2_2IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi3_4IsPR">pi3_4IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi3_3IsPR">pi3_3IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi4_4IsPR">pi4_4IsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus">plus</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#plus">plus</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Plus">Plus</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#plus">plus</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus">plus</a> [definition, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#plusIndIsPR">plusIndIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#plusIsPR">plusIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_cons_cons_eqn">plus_cons_cons_eqn</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_smono_LT_r">plus_smono_LT_r</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_compat">plus_compat</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_left_absorb">plus_left_absorb</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_is_zero">plus_is_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_nf">plus_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_nf0">plus_nf0</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_cons_cons_rw3">plus_cons_cons_rw3</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_cons_cons_rw2">plus_cons_cons_rw2</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_cons_cons_rw1">plus_cons_cons_rw1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_zero_r">plus_zero_r</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_zero">plus_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#plus_alpha_mult_phi0">plus_alpha_mult_phi0</a> [lemma, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#plus_assoc">plus_assoc</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#plus_compat">plus_compat</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#plus_2">plus_2</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Schutte.AP.html#plus_lt_phi0">plus_lt_phi0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#plus_ref">plus_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#plus_comp">plus_comp</a> [instance, in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#plus_assoc">plus_assoc</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Plus_rw">Plus_rw</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#plus_mono_l_weak">plus_mono_l_weak</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#plus_mono_r">plus_mono_r</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#plus_nf">plus_nf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#plus_alpha_0">plus_alpha_0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Plus_">Plus_</a> [constructor, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_bi">plus_mono_bi</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_weak_l">plus_mono_weak_l</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc">plus_assoc</a> [instance, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc'">plus_assoc'</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc3">plus_assoc3</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc2">plus_assoc2</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc1">plus_assoc1</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_FF">plus_FF</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_limit">plus_limit</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_reg_r">plus_reg_r</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_r_weak">plus_mono_r_weak</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_of_succ">plus_of_succ</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_r">plus_mono_r</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_elim">plus_elim</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_ordering">plus_ordering</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#plus_correct">plus_correct</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.Monoid_instances.html#PMult">PMult</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.More_on_positive.html#positive_4step_ind">positive_4step_ind</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.AM.html#Pos_to_nat_diff_0">Pos_to_nat_diff_0</a> [lemma, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow_compat">Pos_pow_compat</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow_power">Pos_pow_power</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_ok">Pos_iter_ok</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_op_ok">Pos_iter_op_ok</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_op_ok_0">Pos_iter_op_ok_0</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow">Pos_pow</a> [definition, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.FirstSteps.html#Pos_bpow">Pos_bpow</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow_ok_R">Pos_bpow_ok_R</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow_ok">Pos_bpow_ok</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow">Pos_bpow</a> [definition, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_of_bpow">Pos_bpow_of_bpow</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_of_plus">Pos_bpow_of_plus</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_ok_R">Pos_bpow_ok_R</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_proper">Pos_bpow_proper</a> [instance, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_ok">Pos_bpow_ok</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow">Pos_bpow</a> [definition, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.More_on_positive.html#pos_div_eucl_quotient_lt">pos_div_eucl_quotient_lt</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_div_eucl_quotient_pos">pos_div_eucl_quotient_pos</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_gt_3">pos_gt_3</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_lt_wf">pos_lt_wf</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_lt_mul">pos_lt_mul</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_le_mul">pos_le_mul</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#Pos_to_nat_neq_0">Pos_to_nat_neq_0</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_eq_dec">pos_eq_dec</a> [definition, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Compatibility.html#Pos2Nat_morph">Pos2Nat_morph</a> [lemma, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.More_on_positive.html#Pos2Nat_le_n_pn">Pos2Nat_le_n_pn</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#Pos2Nat_le_1_p">Pos2Nat_le_1_p</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos2N_inj_add">pos2N_inj_add</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos2N_inj_lt">pos2N_inj_lt</a> [lemma, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Pow.html">Pow</a> [library]<br/>
+<a href="hydras.Prelude.More_Arith.html#power">power</a> [abbreviation, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.FirstSteps.html#power">power</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#power">power</a> [definition, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#power">power</a> [definition, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#power_of_1">power_of_1</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.FirstSteps.html#power_t">power_t</a> [definition, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#power_of_mult">power_of_mult</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_of_square">power_of_square</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_of_power">power_of_power</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_commute_with_x">power_commute_with_x</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_commute">power_commute</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_of_plus">power_of_plus</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_proper">power_proper</a> [instance, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_eq3">power_eq3</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_eq2">power_eq2</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_eq1">power_eq1</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#power_of_mult">power_of_mult</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_of_square">power_of_square</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_of_power_comm">power_of_power_comm</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_of_power">power_of_power</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_commute_with_x">power_commute_with_x</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_commute">power_commute</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_of_plus">power_of_plus</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_proper">power_proper</a> [instance, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_eq3">power_eq3</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_eq2">power_eq2</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_eq1">power_eq1</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_is_a_refinement">power_R_is_a_refinement</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_1">power_R_1</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_Mult">power_R_Mult</a> [lemma, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R">power_R</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Pow_variant.html#pow_17">pow_17</a> [definition, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#pow_17">pow_17</a> [definition, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow_variant.html">Pow_variant</a> [library]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3">pow3</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3IsPR">pow3IsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3Min">pow3Min</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3Monotone">pow3Monotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PO_lex_prod">PO_lex_prod</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO1">PO1</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO2">PO2</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO3">PO3</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO4">PO4</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO6">PO6</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO8">PO8</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO9">PO9</a> [lemma, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp">pp</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#ppE0">ppE0</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ppT1">ppT1</a> [inductive, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp_size">pp_size</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_omega">PP_omega</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_exp">PP_exp</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_mult">PP_mult</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_add">PP_add</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_fin">PP_fin</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp0">pp0</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.dickson.html#pq:68">pq:68</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PRcompose1">PRcompose1</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRcons">PRcons</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRcorrect">PRcorrect</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence">Precedence</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.A">Precedence.A</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.Lex">Precedence.Lex</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.Mul">Precedence.Mul</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec">Precedence.prec</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_transitive">Precedence.prec_transitive</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_antisym">Precedence.prec_antisym</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_dec">Precedence.prec_dec</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.status">Precedence.status</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.status_type">Precedence.status_type</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pred">pred</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#pred">pred</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Predecessor_unicity">Predecessor_unicity</a> [lemma, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#predIsPR">predIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html">predSuccUnicity</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#pred_nf">pred_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pred_LT">pred_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pred_of_limitR">pred_of_limitR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pred_of_limit">pred_of_limit</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pred_of_succ">pred_of_succ</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#pred_of_power">pred_of_power</a> [lemma, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#pred_of_succ">pred_of_succ</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#pred_of_limit">pred_of_limit</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#pred_of_cons'">pred_of_cons'</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#pred_of_cons">pred_of_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#preorder_equiv">preorder_equiv</a> [definition, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.model.html#preserveValue">preserveValue</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PReval">PReval</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRevalN">PRevalN</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Prf">Prf</a> [inductive, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#prf:11">prf:11</a> [binder, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#prf:121">prf:121</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#prf:129">prf:129</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.code.html#prf:45">prf:45</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#prf:47">prf:47</a> [binder, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Primitive_Recursive_Representable">Primitive_Recursive_Representable</a> [section, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.f">Primitive_recursion.f</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.h">Primitive_recursion.h</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.g">Primitive_recursion.g</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.z">Primitive_recursion.z</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.y">Primitive_recursion.y</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.x">Primitive_recursion.x</a> [variable, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion">Primitive_recursion</a> [section, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRec">PrimRec</a> [inductive, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html">primRec</a> [library]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html">PrimRecExamples</a> [library]<br/>
+<a href="Goedel.PRrepresentable.html#primRecFormula">primRecFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#primRecFunc">primRecFunc</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecPiFormulaHelp">primRecPiFormulaHelp</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecRepresentable">primRecRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecRepresentable1">primRecRepresentable1</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRecs">PrimRecs</a> [inductive, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecsFormula">primRecsFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecSigmaFormula">primRecSigmaFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecSigmaFormulaHelp">primRecSigmaFormulaHelp</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecSigmaRepresentable">primRecSigmaRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRecs_PrimRec_ind">PrimRecs_PrimRec_ind</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRecs_PrimRec_rec">PrimRecs_PrimRec_rec</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRec_PrimRecs_ind">PrimRec_PrimRecs_ind</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRec_PrimRecs_rec">PrimRec_PrimRecs_rec</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_2_S">PrimRec_2_S</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_2_0">PrimRec_2_0</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_1_S">PrimRec_1_S</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_1_0">PrimRec_1_0</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_0_S">PrimRec_0_S</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#PrimRec_0_0">PrimRec_0_0</a> [lemma, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html">prLogic</a> [library]<br/>
+<a href="hydras.Ackermann.primRec.html#PRnil">PRnil</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations">PRNotations</a> [module, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_3">PRNotations.pi1_3</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_2">PRNotations.pi1_2</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_1">PRNotations.pi1_1</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi2_3">PRNotations.pi2_3</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi2_2">PRNotations.pi2_2</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi3_3">PRNotations.pi3_3</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.PRcomp">PRNotations.PRcomp</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.PRrec">PRNotations.PRrec</a> [abbreviation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#0d92c10e9c6b3f86aa7104ae219ede74">[ _ ; _ ; .. ; _ ] (pr_scope)</a> [notation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#7270cfdd80e3adbb7f438dd90c04a50e">[ _ ] (pr_scope)</a> [notation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.::pr_scope:x_'::'_x">_ :: _ (pr_scope)</a> [notation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#progressive">progressive</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Progressive_Acc">Progressive_Acc</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projectionList">projectionList</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projectionListApplyParam">projectionListApplyParam</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projectionListId">projectionListId</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projectionListInd">projectionListInd</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projectionListPR">projectionListPR</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#projFormula">projFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projFunc">projFunc</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#projRepresentable">projRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#proj_le_max">proj_le_max</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH">ProofH</a> [section, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Example">ProofH.Example</a> [section, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Formula">ProofH.Formula</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Formulas">ProofH.Formulas</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.L">ProofH.L</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.System">ProofH.System</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Term">ProofH.Term</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Terms">ProofH.Terms</a> [variable, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf">ProofOfLexwf</a> [section, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.A">ProofOfLexwf.A</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.Accs">ProofOfLexwf.Accs</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.HwfA">ProofOfLexwf.HwfA</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA">ProofOfLexwf.ltA</a> [variable, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.Hgammaalpha">Proofs_of_unicity.Proofs.S2.Hgammaalpha</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.gamma">Proofs_of_unicity.Proofs.S2.gamma</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2">Proofs_of_unicity.Proofs.S2</a> [section, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.Halphagamma">Proofs_of_unicity.Proofs.S1.Halphagamma</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.gamma">Proofs_of_unicity.Proofs.S1.gamma</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1">Proofs_of_unicity.Proofs.S1</a> [section, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.Halphabeta">Proofs_of_unicity.Proofs.Halphabeta</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.beta">Proofs_of_unicity.Proofs.beta</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.alpha">Proofs_of_unicity.Proofs.alpha</a> [variable, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs">Proofs_of_unicity.Proofs</a> [section, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity">Proofs_of_unicity</a> [section, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta">Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta</a> [variable, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.gamma">Proofs_of_lt_succ_le.Proofs.S1.gamma</a> [variable, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1">Proofs_of_lt_succ_le.Proofs.S1</a> [section, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.Halphabeta">Proofs_of_lt_succ_le.Proofs.Halphabeta</a> [variable, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.beta">Proofs_of_lt_succ_le.Proofs.beta</a> [variable, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.alpha">Proofs_of_lt_succ_le.Proofs.alpha</a> [variable, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs">Proofs_of_lt_succ_le.Proofs</a> [section, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le">Proofs_of_lt_succ_le</a> [section, in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_dist.P">Proof_of_dist.P</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_dist">Proof_of_dist</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.IHbeta">Proof_of_mult_nf.Induction.IHbeta</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.Hbeta">Proof_of_mult_nf.Induction.Hbeta</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta">Proof_of_mult_nf.Induction.beta</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction">Proof_of_mult_nf.Induction</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P">Proof_of_mult_nf.P</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Halpha">Proof_of_mult_nf.Halpha</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha">Proof_of_mult_nf.alpha</a> [variable, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf">Proof_of_mult_nf</a> [section, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR.minPR">Proof_of_MinIsPR.minPR</a> [variable, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR">Proof_of_MinIsPR</a> [section, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.H0">Proof_of_oplus_lt1.H0</a> [variable, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.n">Proof_of_oplus_lt1.n</a> [variable, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a2">Proof_of_oplus_lt1.a2</a> [variable, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a1">Proof_of_oplus_lt1.a1</a> [variable, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1">Proof_of_oplus_lt1</a> [section, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_assoc">Proof_of_oplus_assoc</a> [section, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_comm">Proof_of_oplus_comm</a> [section, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_plus_nf">Proof_of_plus_nf</a> [section, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n">Proof_of_Ackn_PR.S_step.n</a> [variable, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step">Proof_of_Ackn_PR.S_step</a> [section, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR">Proof_of_Ackn_PR</a> [section, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HlargeA">Proof_of_4_5_2.HlargeA</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HAB1">Proof_of_4_5_2.HAB1</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HB">Proof_of_4_5_2.HB</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HA">Proof_of_4_5_2.HA</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.Halpha">Proof_of_4_5_2.Halpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha">Proof_of_4_5_2.alpha</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B2">Proof_of_4_5_2.B2</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1">Proof_of_4_5_2.B1</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A">Proof_of_4_5_2.A</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2">Proof_of_4_5_2</a> [section, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.IM">Proof_of_Lemma5.Alpha_positive.closedness.IM</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.CM">Proof_of_Lemma5.Alpha_positive.closedness.CM</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.HM">Proof_of_Lemma5.Alpha_positive.closedness.HM</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M">Proof_of_Lemma5.Alpha_positive.closedness.M</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness">Proof_of_Lemma5.Alpha_positive.closedness</a> [section, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma">Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta">Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded">Proof_of_Lemma5.Alpha_positive.Proof_unbounded</a> [section, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.IHalpha">Proof_of_Lemma5.Alpha_positive.IHalpha</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha_pos">Proof_of_Lemma5.Alpha_positive.alpha_pos</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha">Proof_of_Lemma5.Alpha_positive.alpha</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive">Proof_of_Lemma5.Alpha_positive</a> [section, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P">Proof_of_Lemma5.P</a> [variable, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5">Proof_of_Lemma5</a> [section, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Proof_of_nested_Ack_bound">Proof_of_nested_Ack_bound</a> [section, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Proof_of_mult_ref">Proof_of_mult_ref</a> [section, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter_cPair">Proof_of_FibIsPR.fib_iter_cPair</a> [variable, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step_cPair">Proof_of_FibIsPR.fib_step_cPair</a> [variable, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter">Proof_of_FibIsPR.fib_iter</a> [variable, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step">Proof_of_FibIsPR.fib_step</a> [variable, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR">Proof_of_FibIsPR</a> [section, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR.min_alt">Proof_of_MinIsPR.min_alt</a> [variable, in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR">Proof_of_MinIsPR</a> [section, in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Limit_case.Hbeta">Proof_of_H'_mono_l.Limit_case.Hbeta</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Limit_case">Proof_of_H'_mono_l.Limit_case</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.Hgamma">Proof_of_H'_mono_l.Succ_case.Hgamma</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.gamma">Proof_of_H'_mono_l.Succ_case.gamma</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case">Proof_of_H'_mono_l.Succ_case</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.H_alpha_beta">Proof_of_H'_mono_l.H_alpha_beta</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta">Proof_of_H'_mono_l.beta</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha">Proof_of_H'_mono_l.alpha</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l">Proof_of_H'_mono_l</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim">Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_limit">Proof_of_Abstract_Properties.The_induction.alpha_limit</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def">Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta">Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ">Proof_of_Abstract_Properties.The_induction.alpha_Succ</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.Halpha">Proof_of_Abstract_Properties.The_induction.Halpha</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha">Proof_of_Abstract_Properties.The_induction.alpha</a> [variable, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction">Proof_of_Abstract_Properties.The_induction</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties">Proof_of_Abstract_Properties</a> [section, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.g_alpha_beta">proof_of_associativity.g_alpha_beta</a> [variable, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.f_alpha_beta">proof_of_associativity.f_alpha_beta</a> [variable, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.beta">proof_of_associativity.beta</a> [variable, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.alpha">proof_of_associativity.alpha</a> [variable, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity">proof_of_associativity</a> [section, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.is_in_seq">Proof_of_lt_not_wf.seq_intro.is_in_seq</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.Rwf">Proof_of_lt_not_wf.seq_intro.Rwf</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R">Proof_of_lt_not_wf.seq_intro.R</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.seq">Proof_of_lt_not_wf.seq_intro.seq</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A">Proof_of_lt_not_wf.seq_intro.A</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro">Proof_of_lt_not_wf.seq_intro</a> [section, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.lt_wf">Proof_of_lt_not_wf.lt_wf</a> [variable, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf">Proof_of_lt_not_wf</a> [section, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.rpo.rpo.html#proof:202">proof:202</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#proof:206">proof:206</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties">Properties</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha.The_induction">Properties_of_f_alpha.The_induction</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha">Properties_of_f_alpha</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_limit.Hlim">Properties.The_induction.alpha_limit.Hlim</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_limit">Properties.The_induction.alpha_limit</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ.alpha_def">Properties.The_induction.alpha_Succ.alpha_def</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ.beta">Properties.The_induction.alpha_Succ.beta</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ">Properties.The_induction.alpha_Succ</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.Halpha">Properties.The_induction.Halpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha">Properties.The_induction.alpha</a> [variable, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction">Properties.The_induction</a> [section, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="additions.Euclidean_Chains.html#proper_statement">proper_statement</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Proper_A">Proper_A</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#proper_of_proper">proper_of_proper</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#proper_members">proper_members</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#proper_segment_of">proper_segment_of</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#proper_segment">proper_segment</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Proposition_2_3a">Proposition_2_3a</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.rpo.more_list.html#prop_map12_without_repetition">prop_map12_without_repetition</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#prop_map_without_repetition">prop_map_without_repetition</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PrO_Included">PrO_Included</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="Goedel.PRrepresentable.html">PRrepresentable</a> [library]<br/>
+<a href="Goedel.PRrepresentable.html#pr:120">pr:120</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pr:636">pr:636</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pr:641">pr:641</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.T2.html#psi">psi</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#psi_eq">psi_eq</a> [lemma, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#psi_term">psi_term</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_principal">psi_principal</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_lt_epsilon0">psi_lt_epsilon0</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_lt_head">psi_lt_head</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_le_cons">psi_le_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#psi:215">psi:215</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#ptwise_le">ptwise_le</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.AM.html#PUSH">PUSH</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PZero">PZero</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PZero">PZero</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="additions.Monoid_instances.html#P_mult_op">P_mult_op</a> [instance, in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#P_3">P_3</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#P_alpha">P_alpha</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#P_alpha_0">P_alpha_0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:190">P_dec:190</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:67">P_dec:67</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:63">P_dec:63</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:59">P_dec:59</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:55">P_dec:55</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:52">P_dec:52</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:49">P_dec:49</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:45">P_dec:45</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#P_dec:40">P_dec:40</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#P_dec">P_dec</a> [instance, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P_well_founded_induction_type">P_well_founded_induction_type</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.More_on_positive.html#p':2">p':2</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#p':22">p':22</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#p':40">p':40</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.rpo.dickson.html#p':66">p':66</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#p':7">p':7</a> [binder, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#P0">P0</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p0:395">p0:395</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p0:402">p0:402</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P0:46">P0:46</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p0:838">p0:838</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p0:841">p0:841</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P0:89">P0:89</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P0:95">P0:95</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.rpo.list_set.html#P1_dec:195">P1_dec:195</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:109">p1:109</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:114">p1:114</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:117">p1:117</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:122">p1:122</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:125">p1:125</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:128">p1:128</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1309">p1:1309</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:131">p1:131</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1313">p1:1313</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1315">p1:1315</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1318">p1:1318</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1320">p1:1320</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1323">p1:1323</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1326">p1:1326</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1329">p1:1329</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1332">p1:1332</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1335">p1:1335</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1338">p1:1338</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:134">p1:134</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1348">p1:1348</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1350">p1:1350</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1352">p1:1352</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1354">p1:1354</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1356">p1:1356</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p1:1358">p1:1358</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="hydras.Ackermann.codeList.html#p1:140">p1:140</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:143">p1:143</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:166">p1:166</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:169">p1:169</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:172">p1:172</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:175">p1:175</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:178">p1:178</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:181">p1:181</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:183">p1:183</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:185">p1:185</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:187">p1:187</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:189">p1:189</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:191">p1:191</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:193">p1:193</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="hydras.Ackermann.codeList.html#p1:222">p1:222</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:227">p1:227</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:230">p1:230</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:233">p1:233</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:236">p1:236</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:239">p1:239</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:242">p1:242</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:245">p1:245</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:248">p1:248</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:251">p1:251</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:254">p1:254</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:257">p1:257</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:260">p1:260</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:262">p1:262</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:264">p1:264</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:266">p1:266</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#p1:268">p1:268</a> [binder, in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#p2:1339">p2:1339</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1349">p2:1349</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1351">p2:1351</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1353">p2:1353</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1355">p2:1355</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1357">p2:1357</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p2:1359">p2:1359</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="additions.Addition_Chains.html#P87">P87</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
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+<a href="additions.More_on_positive.html#p:1">p:1</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
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+<a href="additions.FirstSteps.html#p:10">p:10</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:107">p:107</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:108">p:108</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:109">p:109</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#p:11">p:11</a> [binder, in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#p:11">p:11</a> [binder, in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="additions.More_on_positive.html#p:11">p:11</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#p:11">p:11</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#p:11">p:11</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:11">p:11</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#P:11">P:11</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:110">p:110</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#p:111">p:111</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:111">p:111</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#p:111">p:111</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:112">p:112</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:113">p:113</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#p:114">p:114</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:115">p:115</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:117">p:117</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1176">p:1176</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1179">p:1179</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.fol.html#p:118">p:118</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1182">p:1182</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1185">p:1185</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1188">p:1188</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:119">p:119</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#P:119">P:119</a> [binder, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="additions.Fib2.html#p:12">p:12</a> [binder, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.More_on_positive.html#p:12">p:12</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:12">p:12</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:121">p:121</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#P:121">P:121</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:123">p:123</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#P:124">P:124</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#p:124">p:124</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:127">p:127</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#p:127">p:127</a> [binder, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#p:128">p:128</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:129">p:129</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Fib2.html#p:13">p:13</a> [binder, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
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+<a href="additions.More_on_positive.html#p:13">p:13</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Trace_exercise.html#p:13">p:13</a> [binder, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Schutte.Addition.html#P:13">P:13</a> [binder, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Countable.html#p:13">p:13</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:130">p:130</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:131">p:131</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:133">p:133</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.fib.html#p:134">p:134</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1340">p:1340</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:135">p:135</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#p:137">p:137</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.fib.html#p:137">p:137</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="additions.FirstSteps.html#p:14">p:14</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:14">p:14</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#P:14">P:14</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
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+<a href="additions.fib.html#p:140">p:140</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#p:1413">p:1413</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#p:1415">p:1415</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:166">p:166</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:167">p:167</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:168">p:168</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:169">p:169</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:169">p:169</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#p:17">p:17</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.More_on_positive.html#p:17">p:17</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#p:17">p:17</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="additions.fib.html#p:17">p:17</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#P:17">P:17</a> [binder, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#p:17">p:17</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:170">p:170</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:171">p:171</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:172">p:172</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#p:172">p:172</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:173">p:173</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:174">p:174</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:175">p:175</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.rpo.rpo.html#p:175">p:175</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#p:176">p:176</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:176">p:176</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#p:176">p:176</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:177">p:177</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:178">p:178</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.rpo.rpo.html#p:178">p:178</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#p:179">p:179</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#p:18">p:18</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.Compatibility.html#p:18">p:18</a> [binder, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:18">p:18</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:180">p:180</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#p:180">p:180</a> [binder, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:180">p:180</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:182">p:182</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1820">p:1820</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1823">p:1823</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1826">p:1826</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1829">p:1829</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.term.html#p:183">p:183</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:183">p:183</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:1832">p:1832</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#p:184">p:184</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:185">p:185</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#p:186">p:186</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:187">p:187</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:189">p:189</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.rpo.list_set.html#P:189">P:189</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#P:19">P:19</a> [binder, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#p:19">p:19</a> [binder, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="Goedel.rosser.html#p:19">p:19</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:190">p:190</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:192">p:192</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:193">p:193</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:194">p:194</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Prelude.LibHyps_Experiments.html#p:2">p:2</a> [binder, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#p:2">p:2</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:20">p:20</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:203">p:203</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#p:206">p:206</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="hydras.Schutte.Critical.html#p:21">p:21</a> [binder, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#P:21">P:21</a> [binder, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="additions.fib.html#p:21">p:21</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#p:21">p:21</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.Addition_Chains.html#p:210">p:210</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:230">p:230</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#p:231">p:231</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:231">p:231</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:235">p:235</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:236">p:236</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:237">p:237</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:238">p:238</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:239">p:239</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:239">p:239</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.More_on_positive.html#p:24">p:24</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#p:24">p:24</a> [binder, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:240">p:240</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:241">p:241</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#P:241">P:241</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Addition_Chains.html#p:241">p:241</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:242">p:242</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:243">p:243</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:244">p:244</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:245">p:245</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:246">p:246</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:247">p:247</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:248">p:248</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:249">p:249</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:25">p:25</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#P:25">P:25</a> [binder, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Countable.html#p:25">p:25</a> [binder, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:250">p:250</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:251">p:251</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.rpo.more_list.html#P:252">P:252</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:252">p:252</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:254">p:254</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:255">p:255</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:255">p:255</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:256">p:256</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:257">p:257</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:258">p:258</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:259">p:259</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.AM.html#p:26">p:26</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.More_on_positive.html#p:26">p:26</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#P:26">P:26</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:261">p:261</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:262">p:262</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:263">p:263</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:263">p:263</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.rpo.more_list.html#P:264">P:264</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:268">p:268</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#p:269">p:269</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:269">p:269</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#p:269">p:269</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#p:27">p:27</a> [binder, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:27">p:27</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P:27">P:27</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:270">p:270</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:270">p:270</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:271">p:271</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:272">p:272</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:273">p:273</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:273">p:273</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:274">p:274</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:275">p:275</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P:275">P:275</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:276">p:276</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:276">p:276</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:277">p:277</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:278">p:278</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:279">p:279</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P:279">P:279</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#p:28">p:28</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:28">p:28</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#p:28">p:28</a> [binder, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#P:280">P:280</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:280">p:280</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:281">p:281</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:282">p:282</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:283">p:283</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:283">p:283</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:284">p:284</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:284">p:284</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#p:285">p:285</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Addition_Chains.html#p:285">p:285</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#p:286">p:286</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="additions.Addition_Chains.html#p:286">p:286</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:287">p:287</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Addition_Chains.html#p:287">p:287</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:289">p:289</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P:289">P:289</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:29">p:29</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#p:29">p:29</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:290">p:290</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#p:291">p:291</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:291">p:291</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#P:291">P:291</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:292">p:292</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:293">p:293</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:294">p:294</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:296">p:296</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#P:299">P:299</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="additions.Strategies.html#p:3">p:3</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
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+<a href="additions.BinaryStrat.html#p:3">p:3</a> [binder, in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
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+<a href="Goedel.rosserPA.html#p:30">p:30</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:304">p:304</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:305">p:305</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:306">p:306</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#p:32">p:32</a> [binder, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:381">p:381</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#P:381">P:381</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:382">p:382</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:382">p:382</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:383">p:383</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:384">p:384</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:384">p:384</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:385">p:385</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:386">p:386</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:387">p:387</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:388">p:388</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:389">p:389</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#p:39">p:39</a> [binder, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Compatibility.html#p:39">p:39</a> [binder, in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:39">p:39</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.More_on_positive.html#p:39">p:39</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Dichotomy.html#p:39">p:39</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:390">p:390</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:390">p:390</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:391">p:391</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:392">p:392</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:395">p:395</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:396">p:396</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.Euclidean_Chains.html#p:397">p:397</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#p:398">p:398</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="additions.Fib2.html#P:4">P:4</a> [binder, in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.More_on_positive.html#p:4">p:4</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#p:4">p:4</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#p:4">p:4</a> [binder, in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="additions.Dichotomy.html#p:4">p:4</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#p:40">p:40</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:40">p:40</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="additions.More_on_positive.html#p:40">p:40</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.Naive.html#p:40">p:40</a> [binder, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#P:40">P:40</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.Dichotomy.html#p:40">p:40</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:401">p:401</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:402">p:402</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:403">p:403</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:404">p:404</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:405">p:405</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:409">p:409</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:41">p:41</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#P:41">P:41</a> [binder, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
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+<a href="additions.More_on_positive.html#p:41">p:41</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:414">p:414</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:415">p:415</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:416">p:416</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:417">p:417</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="additions.Dichotomy.html#p:42">p:42</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#P:427">P:427</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="additions.Dichotomy.html#p:43">p:43</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#p:44">p:44</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="additions.Pow_variant.html#p:45">p:45</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:468">p:468</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:469">p:469</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:47">p:47</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#p:47">p:47</a> [binder, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.fol.html#p:47">p:47</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:470">p:470</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:471">p:471</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#P:471">P:471</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#p:472">p:472</a> [binder, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:474">p:474</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.PRrepresentable.html#p:475">p:475</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:475">p:475</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:475">p:475</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:476">p:476</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:477">p:477</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:478">p:478</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:478">p:478</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Schutte.Correctness_E0.html#p:48">p:48</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="additions.FirstSteps.html#p:48">p:48</a> [binder, in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#p:48">p:48</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.rpo.list_set.html#P:48">P:48</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:480">p:480</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#p:481">p:481</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:482">p:482</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:483">p:483</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:484">p:484</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:484">p:484</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:485">p:485</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#p:487">p:487</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:488">p:488</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:489">p:489</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:49">p:49</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#P:49">P:49</a> [binder, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:492">p:492</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:493">p:493</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:494">p:494</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:495">p:495</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:496">p:496</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:497">p:497</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:498">p:498</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="additions.Strategies.html#p:5">p:5</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="additions.More_on_positive.html#p:5">p:5</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#p:5">p:5</a> [binder, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#p:5">p:5</a> [binder, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="additions.Dichotomy.html#p:5">p:5</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#p:50">p:50</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#P:502">P:502</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:505">p:505</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P:505">P:505</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:51">p:51</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P:51">P:51</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#P:51">P:51</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
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+<a href="hydras.Ackermann.code.html#p:52">p:52</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:521">p:521</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#p:528">p:528</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Ackermann.primRec.html#p:607">p:607</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#P:607">P:607</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#p:608">p:608</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:61">p:61</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#P:61">P:61</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#p:610">p:610</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#p:612">p:612</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#P:612">P:612</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#p:615">p:615</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#P:615">P:615</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.list_set.html#P:62">P:62</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#p:622">p:622</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="additions.Pow_variant.html#p:63">p:63</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="additions.Pow.html#p:63">p:63</a> [binder, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.More_on_positive.html#p:63">p:63</a> [binder, in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#p:630">p:630</a> [binder, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="additions.fib.html#p:64">p:64</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.rpo.dickson.html#p:64">p:64</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
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+<a href="hydras.MoreAck.Ack.html#p:65">p:65</a> [binder, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#p:650">p:650</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Pow_variant.html#p:66">p:66</a> [binder, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:679">p:679</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="hydras.Ackermann.checkPrf.html#p:682">p:682</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#p:683">p:683</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
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+<a href="additions.fib.html#p:70">p:70</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="additions.Naive.html#p:9">p:9</a> [binder, in <a href="additions.Naive.html">additions.Naive</a>]<br/>
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+<a href="additions.Dichotomy.html#p:9">p:9</a> [binder, in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
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+<a href="additions.Pow.html#p:91">p:91</a> [binder, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.fib.html#p:91">p:91</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<a href="additions.fib.html#p:96">p:96</a> [binder, in <a href="additions.fib.html">additions.fib</a>]<br/>
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+<br/><br/><a id="global_R"></a><h2>R </h2>
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+<a href="Goedel.PRrepresentable.html#rec:148">rec:148</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#rec:15">rec:15</a> [binder, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rec:159">rec:159</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.fol.html#rec:160">rec:160</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#rec:166">rec:166</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rec:242">rec:242</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rec:248">rec:248</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#rec:29">rec:29</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="Goedel.rosser.html#rec:30">rec:30</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#rec:32">rec:32</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#rec:34">rec:34</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="Goedel.codeSysPrf.html#rec:36">rec:36</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#rec:38">rec:38</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#rec:39">rec:39</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#rec:44">rec:44</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.codeSysPrf.html#rec:46">rec:46</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rec:463">rec:463</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rec:471">rec:471</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.codeSysPrf.html#rec:48">rec:48</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.rosser.html#rec:49">rec:49</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#rec:54">rec:54</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:540">rec:540</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:543">rec:543</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:547">rec:547</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:550">rec:550</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:552">rec:552</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:554">rec:554</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:556">rec:556</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:558">rec:558</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.rosser.html#rec:56">rec:56</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:560">rec:560</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.rosser.html#rec:58">rec:58</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#rec:580">rec:580</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#rec:581">rec:581</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#rec:582">rec:582</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#rec:583">rec:583</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:591">rec:591</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:593">rec:593</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:595">rec:595</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:597">rec:597</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:599">rec:599</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#rec:6">rec:6</a> [binder, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:60">rec:60</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:601">rec:601</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:603">rec:603</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:605">rec:605</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:607">rec:607</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:62">rec:62</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#rec:62">rec:62</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:620">rec:620</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:622">rec:622</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:624">rec:624</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:626">rec:626</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:628">rec:628</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:630">rec:630</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:632">rec:632</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:634">rec:634</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:636">rec:636</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.rosser.html#rec:64">rec:64</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:69">rec:69</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#rec:7">rec:7</a> [binder, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:74">rec:74</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:79">rec:79</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:795">rec:795</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#rec:8">rec:8</a> [binder, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:801">rec:801</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:808">rec:808</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#rec:81">rec:81</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#rec:812">rec:812</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#rec:816">rec:816</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:84">rec:84</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:86">rec:86</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:88">rec:88</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#rec:9">rec:9</a> [binder, in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="Goedel.rosserPA.html#rec:95">rec:95</a> [binder, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddExists">reduceAddExists</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddExistsOneWay">reduceAddExistsOneWay</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddForalls">reduceAddForalls</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceAnd">reduceAnd</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceCloseList">reduceCloseList</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceExist">reduceExist</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceForall">reduceForall</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceIff">reduceIff</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceIfThenElse">reduceIfThenElse</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceImp">reduceImp</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceNot">reduceNot</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceOr">reduceOr</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceSub">reduceSub</a> [lemma, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#reduceSubAll">reduceSubAll</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#reduce_lt_plus">reduce_lt_plus</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.more_list.html#reduce_assoc_list">reduce_assoc_list</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#reduce1stCompose">reduce1stCompose</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#reduce2ndCompose">reduce2ndCompose</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_eq">Refinement_proof.E_eq</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_one">Refinement_proof.E_one</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_op">Refinement_proof.E_op</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.A">Refinement_proof.A</a> [variable, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof">Refinement_proof</a> [section, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refinesPred">refinesPred</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refinesRel">refinesRel</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines0">refines0</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines1">refines1</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines1_R">refines1_R</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines2">refines2</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines2_R">refines2_R</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="Goedel.goedel1.html#reflection">reflection</a> [abbreviation, in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="hydras.Ackermann.model.html#rel">rel</a> [projection, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Relations">Relations</a> [projection, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#RelDecision">RelDecision</a> [record, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#RelDecision">RelDecision</a> [inductive, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#RelDecision_Inverse_fun">RelDecision_Inverse_fun</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RElim">RElim</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RElim">RElim</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_surj_i">rel_surj_i</a> [constructor, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_surjection">rel_surjection</a> [inductive, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_inj_i">rel_inj_i</a> [constructor, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_injection">rel_injection</a> [inductive, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inv">rel_inv</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_bij_i">rel_bij_i</a> [constructor, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_bijection">rel_bijection</a> [inductive, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inj_i">rel_inj_i</a> [constructor, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_injection">rel_injection</a> [inductive, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inj">rel_inj</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_onto">rel_onto</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_functional">rel_functional</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_codomain">rel_codomain</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_domain">rel_domain</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Countable.html#rel_enumerates">rel_enumerates</a> [definition, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#rel_numbers">rel_numbers</a> [definition, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rem">rem</a> [abbreviation, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Remaining">Remaining</a> [constructor, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.rpo.more_list.html#remove">remove</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#remove_first_sumand">remove_first_sumand</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.more_list.html#remove_list">remove_list</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#remove_heads_r_free">remove_heads_r_free</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#remove_heads_r">remove_heads_r</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem0">Rem0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem01">Rem01</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem02">Rem02</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem03">Rem03</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem04">Rem04</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem05">Rem05</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem06">Rem06</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem08">Rem08</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem09">Rem09</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem1">rem1</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem1">Rem1</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem1">Rem1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem10">Rem10</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem11">Rem11</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem2">rem2</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem2">Rem2</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem2">Rem2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem3">rem3</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem3">Rem3</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Rem3">Rem3</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Rem4">Rem4</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem6">rem6</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem61">rem61</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem62">rem62</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#rep">rep</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#replaceCompose2">replaceCompose2</a> [lemma, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm">ReplaceFormulaTerm</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermIsPR">ReplaceFormulaTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermMonotone">ReplaceFormulaTermMonotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermSub">ReplaceFormulaTermSub</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement">Replacement</a> [section, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Formula">Replacement.Formula</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Formulas">Replacement.Formulas</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.L">Replacement.L</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.SysPrf">Replacement.SysPrf</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.System">Replacement.System</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Term">Replacement.Term</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Terms">Replacement.Terms</a> [variable, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm">ReplaceTermsTerm</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermIsPR">ReplaceTermsTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermMonotone">ReplaceTermsTermMonotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermSub">ReplaceTermsTermSub</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm">ReplaceTermTerm</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermIsPR">ReplaceTermTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermMonotone">ReplaceTermTermMonotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTerm">ReplaceTermTermsTerm</a> [definition, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTermIsPR">ReplaceTermTermsTermIsPR</a> [instance, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTermMonotone">ReplaceTermTermsTermMonotone</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermSub">ReplaceTermTermSub</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.rpo.term.html#replace_at_pos_list:338">replace_at_pos_list:338</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#replace_at_pos_list:160">replace_at_pos_list:160</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Representable">Representable</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable">Representable</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentableAlternate">RepresentableAlternate</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableAlternate">RepresentableAlternate</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible">RepresentableExpressible</a> [section, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.closedT1">RepresentableExpressible.closedT1</a> [variable, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.nn1">RepresentableExpressible.nn1</a> [variable, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.T">RepresentableExpressible.T</a> [variable, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalfHelp">RepresentableHalfHelp</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf1">RepresentableHalf1</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf1Alternate">RepresentableHalf1Alternate</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf2">RepresentableHalf2</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf2Alternate">RepresentableHalf2Alternate</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentableHelp">RepresentableHelp</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHelp">RepresentableHelp</a> [definition, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentablesHelp">RepresentablesHelp</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Representable_ext">Representable_ext</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable_ext">Representable_ext</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable2Expressible">Representable2Expressible</a> [lemma, in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.rosser.html#RepresentsInSelf">RepresentsInSelf</a> [definition, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#rep:66">rep:66</a> [binder, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restrict">restrict</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.onType.html#restrict">restrict</a> [definition, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#restrict">restrict</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restrict">restrict</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.R">restricted_recursion.R</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.E">restricted_recursion.E</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.A">restricted_recursion.A</a> [variable, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion">restricted_recursion</a> [section, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.R">restricted_recursion.R</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.P">restricted_recursion.P</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.A">restricted_recursion.A</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion">restricted_recursion</a> [section, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Restriction.html">Restriction</a> [library]<br/>
+<a href="hydras.Prelude.Restriction.html#restriction_fwd">restriction_fwd</a> [lemma, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restrict_build">restrict_build</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#rest:3">rest:3</a> [binder, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#rest:63">rest:63</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.AM.html#result_equiv_equiv">result_equiv_equiv</a> [instance, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#result_equiv_success">result_equiv_success</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#result_equiv_fail">result_equiv_fail</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#result_equiv">result_equiv</a> [inductive, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#Return">Return</a> [constructor, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Return_proper">Return_proper</a> [instance, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="Goedel.rosser.html">rosser</a> [library]<br/>
+<a href="Goedel.rosserPA.html">rosserPA</a> [library]<br/>
+<a href="Goedel.rosserPA.html#Rosser'sIncompleteness">Rosser'sIncompleteness</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#Rosser'sIncompleteness">Rosser'sIncompleteness</a> [lemma, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.expressT2">Rosser's_Incompleteness.expressT2</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.expressT1">Rosser's_Incompleteness.expressT1</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.freeVarRepT">Rosser's_Incompleteness.freeVarRepT</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0">Rosser's_Incompleteness.v0</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT">Rosser's_Incompleteness.repT</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.extendsPA">Rosser's_Incompleteness.extendsPA</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.T">Rosser's_Incompleteness.T</a> [variable, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness">Rosser's_Incompleteness</a> [section, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.expressT2">Rosser's_Incompleteness.expressT2</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.expressT1">Rosser's_Incompleteness.expressT1</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT">Rosser's_Incompleteness.freeVarRepT</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.v0">Rosser's_Incompleteness.v0</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.repT">Rosser's_Incompleteness.repT</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN">Rosser's_Incompleteness.extendsNN</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.T">Rosser's_Incompleteness.T</a> [variable, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness">Rosser's_Incompleteness</a> [section, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#round">round</a> [projection, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round">round</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds">rounds</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_free_equiv2">rounds_free_equiv2</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_free_equiv1">rounds_free_equiv1</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_trans">rounds_trans</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds_n">rounds_n</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds_1">rounds_1</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#round_decr">round_decr</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_n_inv">round_n_inv</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_n_remove_h">round_n_remove_h</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_plus_trans">round_plus_trans</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#round_S">round_S</a> [lemma, in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_t">round_t</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_spec">round_spec</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_star_intro">round_star_intro</a> [lemma, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_star">round_star</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_plus">round_plus</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_n">round_n</a> [definition, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RP">RP</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RP">RP</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO">RPO</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html">rpo</a> [library]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_7_1">rpo_7_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_6_4">rpo_6_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_6_1">rpo_6_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_4">rpo_5_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_1">rpo_5_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_3">rpo_5_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_2">rpo_5_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_4">rpo_4_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_1">rpo_4_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_3">rpo_4_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_2">rpo_4_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_4">rpo_3_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_1">rpo_3_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_3">rpo_3_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_2">rpo_3_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_4">rpo_2_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_1">rpo_2_1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_3">rpo_2_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_2">rpo_2_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_trans">rpo_trans</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Eq">RPO.Eq</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_mul">RPO.List_mul</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_eq">RPO.List_eq</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_gt">RPO.List_gt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.LP">RPO.LP</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Lt">RPO.Lt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.P">RPO.P</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo">RPO.rpo</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_add_context">RPO.rpo_add_context</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_subst">RPO.rpo_subst</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_trans">RPO.rpo_trans</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_closure">RPO.rpo_closure</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_mul">RPO.rpo_mul</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_lex">RPO.rpo_lex</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_eq">RPO.rpo_eq</a> [inductive, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Subterm">RPO.Subterm</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.T">RPO.T</a> [module, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_eq_mul">RPO.Top_eq_mul</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_eq_lex">RPO.Top_eq_lex</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_gt">RPO.Top_gt</a> [constructor, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.wf_rpo">RPO.wf_rpo</a> [axiom, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#rt:9">rt:9</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Rwf">Rwf</a> [lemma, in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given.R">R_given.R</a> [variable, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given.A">R_given.A</a> [variable, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given">R_given</a> [section, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee_domain">R_nat_elaguee_domain</a> [lemma, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee_fun">R_nat_elaguee_fun</a> [lemma, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee">R_nat_elaguee</a> [definition, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv_inj">R_inv_inj</a> [lemma, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv">R_inv</a> [definition, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv_surj">R_inv_surj</a> [lemma, in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R_pred_Sn">R_pred_Sn</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R_inc_rpo">R_inc_rpo</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#r':32">r':32</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#r':38">r':38</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#R':550">R':550</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#R':564">R':564</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#r':61">r':61</a> [binder, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#r':90">r':90</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R0">R0</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R0">R0</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="gaia_hydras.onType.html#R0">R0</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R0">R0</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R00">R00</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R00">R00</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R01">R01</a> [instance, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R01">R01</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R02">R02</a> [instance, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R02">R02</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R03">R03</a> [instance, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R1">R1</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#R1">R1</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#R1">R1</a> [lemma, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#R1">R1</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#R1">R1</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R1">R1</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#R1">R1</a> [lemma, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="gaia_hydras.onType.html#R1">R1</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R1">R1</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R1">R1</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R1">R1</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R1">R1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#R1">R1</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R1">R1</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_mem_head">R1_mem_head</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_R2">R1_R2</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_hcons_mult">R1_hcons_mult</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_hcons">R1_hcons</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R1_decr">R1_decr</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R1_decr_0">R1_decr_0</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_remove_r_i">R1_remove_r_i</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_app">R1_app</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_iff">R1_iff</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R1_aux">R1_aux</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R1_intro">R1_intro</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R12">R12</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R13">R13</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R14">R14</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R15">R15</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R16">R16</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R17">R17</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R18">R18</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R19">R19</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Wf_transparent.html#R1:2">R1:2</a> [binder, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r1:41">r1:41</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r1:49">r1:49</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r1:57">r1:57</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r1:65">r1:65</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R2">R2</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#R2">R2</a> [instance, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#R2">R2</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R2">R2</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#R2">R2</a> [lemma, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R2">R2</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R2">R2</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R2">R2</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R2">R2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2">R2</a> [inductive, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#r2i">r2i</a> [definition, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R2_decr">R2_decr</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R2_decr_0">R2_decr_0</a> [lemma, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R2_app">R2_app</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R2_ind2">R2_ind2</a> [definition, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R2_iff">R2_iff</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2_intro_2">R2_intro_2</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2_intro">R2_intro</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R20">R20</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R21">R21</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R22">R22</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R23">R23</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R24">R24</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R25">R25</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R26">R26</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R27">R27</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R28">R28</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R29">R29</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Wf_transparent.html#R2:3">R2:3</a> [binder, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r2:42">r2:42</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r2:50">r2:50</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r2:58">r2:58</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#r2:66">r2:66</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R3">R3</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R3">R3</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#R3">R3</a> [lemma, in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R3">R3</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R3">R3</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R3">R3</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R3">R3</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R3">R3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R3'">R3'</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R30">R30</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R31">R31</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R31_0">R31_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R4">R4</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R4">R4</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R4">R4</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R4">R4</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R4">R4</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R4">R4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R4_3_4">R4_3_4</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R4_3_3">R4_3_3</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R4_3_2">R4_3_2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R4_3_1">R4_3_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R4'">R4'</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R4''">R4''</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R42">R42</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R5">R5</a> [lemma, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R5">R5</a> [lemma, in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R5">R5</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R5">R5</a> [lemma, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R5">R5</a> [lemma, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R5">R5</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R5">R5</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R6">R6</a> [lemma, in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R6">R6</a> [lemma, in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R6">R6</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R7">R7</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.rpo.dickson.html#R:1">R:1</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.Ackermann.PA.html#r:10">r:10</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#R:10">R:10</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.rpo.dickson.html#R:10">R:10</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="additions.Strategies.html#r:11">r:11</a> [binder, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#R:11">R:11</a> [binder, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.rpo.closure.html#R:11">R:11</a> [binder, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:110">R:110</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#r:114">r:114</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:121">R:121</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:128">R:128</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#R:130">R:130</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#r:131">r:131</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:133">R:133</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Gamma0.T2.html#r:137">r:137</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:138">R:138</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.model.html#r:14">r:14</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#r:14">r:14</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#R:14">R:14</a> [binder, in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="Goedel.codeSysPrf.html#r:14">r:14</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:143">R:143</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#r:148">r:148</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#r:15">r:15</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#R:16">R:16</a> [binder, in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.rpo.closure.html#R:16">R:16</a> [binder, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#R:16">R:16</a> [binder, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#R:166">R:166</a> [binder, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#r:17">r:17</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#r:17">r:17</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.dickson.html#R:17">R:17</a> [binder, in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
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+<a href="additions.Addition_Chains.html#R:184">R:184</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#r:185">r:185</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.rpo.closure.html#R:19">R:19</a> [binder, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#r:193">r:193</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
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+<a href="hydras.Ackermann.code.html#R:8">R:8</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#r:8">r:8</a> [binder, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Ackermann.model.html#r:85">r:85</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#r:88">r:88</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#r:89">r:89</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="global_S"></a><h2>S </h2>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#s">s</a> [definition, in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.rpo.decidable_set.html#S">S</a> [module, in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S">S</a> [module, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#s">s</a> [definition, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#SA1">SA1</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#SA2">SA2</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#SA:17">SA:17</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#SA:19">SA:19</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#SB:18">SB:18</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#SB:20">SB:20</a> [binder, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Schutte.Schutte.html">Schutte</a> [library]<br/>
+<a href="hydras.Schutte.Schutte_basics.html">Schutte_basics</a> [library]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">schutte_cnf_counter_example</a> [library]<br/>
+<a href="Goedel.rosserPA.html#searchProof">searchProof</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#searchProof">searchProof</a> [lemma, in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#searchXYIsPR">searchXYIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="Goedel.goedel2.html#SecondIncompletness">SecondIncompletness</a> [lemma, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment">segment</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment_lt_closed">segment_lt_closed</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment_the_ordering_segment">segment_the_ordering_segment</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment_unbounded">segment_unbounded</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment_lt">segment_lt</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#self_lt_free">self_lt_free</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#self_lt_standard">self_lt_standard</a> [lemma, in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="additions.AM.html#Semantics">Semantics</a> [section, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Semantics.A">Semantics.A</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Semantics.mul">Semantics.mul</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Semantics.one">Semantics.one</a> [variable, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled">Semibundled</a> [module, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.dec_dec">Semibundled.dec_dec</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.TotalDec">Semibundled.TotalDec</a> [record, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.TotalDecPreOrder">Semibundled.TotalDecPreOrder</a> [record, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec_total">Semibundled.total_dec_total</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec_pre_order">Semibundled.total_dec_pre_order</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec">Semibundled.total_dec</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Sentence">Sentence</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Sentence">Sentence</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Sentence">Sentence</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#seq">seq</a> [definition, in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences">Sequences</a> [section, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.A">Sequences.A</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.R">Sequences.R</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.W">Sequences.W</a> [variable, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#seq_mono_inj">seq_mono_inj</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#seq_mono_intro">seq_mono_intro</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#seq_mono">seq_mono</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Countable.html#seq_range_countable">seq_range_countable</a> [lemma, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#seq_range">seq_range</a> [definition, in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#seq:22">seq:22</a> [binder, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#set_eq">set_eq</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift">shift</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift_unshift">shift_unshift</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift_no_zero">shift_no_zero</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift_interval">shift_interval</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift_iota_from">shift_iota_from</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigA:279">SigA:279</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigA:282">SigA:282</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigA:297">SigA:297</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigB:280">SigB:280</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigB:283">SigB:283</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#SigB:298">SigB:298</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.rpo.term.html#sigma1:113">sigma1:113</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma1:122">sigma1:122</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma1:291">sigma1:291</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma1:300">sigma1:300</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma1:302">sigma1:302</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma2:114">sigma2:114</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma2:123">sigma2:123</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma2:292">sigma2:292</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma2:301">sigma2:301</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma2:303">sigma2:303</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:102">sigma:102</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:111">sigma:111</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:119">sigma:119</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:126">sigma:126</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:128">sigma:128</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:184">sigma:184</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:280">sigma:280</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:289">sigma:289</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:297">sigma:297</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:307">sigma:307</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:309">sigma:309</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#sigma:315">sigma:315</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#sigma:363">sigma:363</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#sigma:86">sigma:86</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Signature">Signature</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="additions.Euclidean_Chains.html#signature">signature</a> [inductive, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#signature_measure">signature_measure</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#signature_exponent">signature_exponent</a> [definition, in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.AC">Signature.AC</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.arity">Signature.arity</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.arity_type">Signature.arity_type</a> [inductive, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.C">Signature.C</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.eq_symbol_dec">Signature.eq_symbol_dec</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.Free">Signature.Free</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.symb">Signature.symb</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#sig_eq_intro">sig_eq_intro</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#simple_last_app1">simple_last_app1</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#simple_last_app">simple_last_app</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#simple_last_correct">simple_last_correct</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#simple_last">simple_last</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html">Simple_LexProd</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#single_nf">single_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#single_nf">single_nf</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.rpo.term.html#size_acc:203">size_acc:203</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#size_acc:20">size_acc:20</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.more_list.html#size_acc:54">size_acc:54</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size1:57">size1:57</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size2:58">size2:58</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size:37">size:37</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size:42">size:42</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size:47">size:47</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#size:51">size:51</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.list_permut.html#size:69">size:69</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.more_list.html#size:73">size:73</a> [binder, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.list_permut.html#size:74">size:74</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="additions.fib.html#slowexp">slowexp</a> [definition, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#smallestExists">smallestExists</a> [lemma, in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#small_h">small_h</a> [definition, in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#small_lemma">small_lemma</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#SmNotPR">SmNotPR</a> [lemma, in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#smono_Sle">smono_Sle</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.rpo.rpo.html#sn_tt:288">sn_tt:288</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#sn_tt:285">sn_tt:285</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#sn_tt:282">sn_tt:282</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#some">some</a> [definition, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#some_pos'">some_pos'</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#some_pos">some_pos</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.rpo.more_list.html#some_nb_occ_Sn">some_nb_occ_Sn</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#some_pos">some_pos</a> [definition, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort">sort</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted">sorted</a> [definition, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_suffix">sorted_ge_suffix</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_max_2">sorted_max_2</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_cut">sorted_cut</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_max_1">sorted_max_1</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_not_in_tail">sorted_not_in_tail</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_In">sorted_In</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_prefix">sorted_ge_prefix</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_iff">sorted_ge_iff</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_iff0">sorted_ge_iff0</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Sorted_mono">Sorted_mono</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_head">sorted_head</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_tail'">sorted_tail'</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_tail">sorted_tail</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_le">sorted_le</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_inv_gt">sorted_inv_gt</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_not_In">sorted_ge_not_In</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_trans">sorted_ge_trans</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_Forall">sorted_ge_Forall</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_cons">sorted_ge_cons</a> [constructor, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_one">sorted_ge_one</a> [constructor, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_nil">sorted_ge_nil</a> [constructor, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge">sorted_ge</a> [inductive, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted_lt">sorted_lt</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted_lt_lt_2">sorted_lt_lt_2</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted_lt_lt">sorted_lt_lt</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted_tail">sorted_tail</a> [lemma, in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_spec">sort_spec</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_correct">sort_correct</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_rel">sort_rel</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_fun_t">sort_fun_t</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html">Sort_spec</a> [library]<br/>
+<a href="hydras.rpo.more_list.html#split_list_app_cons">split_list_app_cons</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#split_list">split_list</a> [definition, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split_permutation">split_permutation</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split_decr">split_decr</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split'_permutation">split'_permutation</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split'_aux_eq">split'_aux_eq</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split'_decr">split'_decr</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split'_aux_length_fst">split'_aux_length_fst</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#split'_aux_length_preserve">split'_aux_length_preserve</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#sp_mergesort_OK">sp_mergesort_OK</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#sp_merge_sort">sp_merge_sort</a> [definition, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="additions.AM.html#SQR">SQR</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR">sqrtIsPR</a> [section, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.P">sqrtIsPR.P</a> [variable, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n">sqrtIsPR.Proof_isqrt.n</a> [variable, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt">sqrtIsPR.Proof_isqrt</a> [section, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrt_correct">sqrt_correct</a> [lemma, in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="additions.Pow_variant.html#sqr_eqn">sqr_eqn</a> [lemma, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#sqr_eqn">sqr_eqn</a> [lemma, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.AM.html#squares:6">squares:6</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html">ssrnat_extracts</a> [library]<br/>
+<a href="hydras.Ackermann.fol.html#ss:113">ss:113</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:120">ss:120</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:125">ss:125</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:130">ss:130</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:28">ss:28</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:30">ss:30</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.code.html#ss:31">ss:31</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#ss:34">ss:34</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#ss:37">ss:37</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:37">ss:37</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#ss:4">ss:4</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:44">ss:44</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#ss:51">ss:51</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:51">ss:51</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:59">ss:59</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:66">ss:66</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:68">ss:68</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:73">ss:73</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:78">ss:78</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ss:78">ss:78</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:83">ss:83</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:94">ss:94</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ss:97">ss:97</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#stable">stable</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#stable_test">stable_test</a> [definition, in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#stable_mergesort_OK">stable_mergesort_OK</a> [lemma, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#stable_merge_sort">stable_merge_sort</a> [definition, in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="additions.AM.html#stack">stack</a> [definition, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Stack_equiv_equiv">Stack_equiv_equiv</a> [instance, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Stack_equiv_refl">Stack_equiv_refl</a> [instance, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#stack_equivn">stack_equivn</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#stack_equiv0">stack_equiv0</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#stack_equiv">stack_equiv</a> [inductive, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#standard">standard</a> [instance, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GPaths.html#standard_gnaw_iota_from">standard_gnaw_iota_from</a> [lemma, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#standard_gnaw">standard_gnaw</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#standard_path">standard_path</a> [definition, in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#standard_battle_head">standard_battle_head</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#standard_incl_free">standard_incl_free</a> [lemma, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_2_zero">standard_gnaw_2_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_to_zero">standard_path_to_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_to_zero">standard_gnaw_to_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_pathS_app">standard_pathS_app</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_equiv_2">standard_path_equiv_2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_equiv_1">standard_path_equiv_1</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_nf">standard_gnaw_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_to_path">standard_gnaw_to_path</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_S_zero">standard_gnaw_S_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_compose">standard_path_compose</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_lt2">standard_path_lt2</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_zero">standard_path_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_to_nf">standard_path_to_nf</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_LE">standard_path_LE</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_plus">standard_gnaw_plus</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_iota_from">standard_gnaw_iota_from</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw_zero">standard_gnaw_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_path_to">standard_path_path_to</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_pathS_path_toS">standard_pathS_path_toS</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_origin">standard_path_origin</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_toS_zero">standard_path_toS_zero</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_to_le_inv">standard_path_to_le_inv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_toS_le_inv">standard_path_toS_le_inv</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_unshift">standard_path_unshift</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_unshift_0">standard_path_unshift_0</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_shift">standard_path_shift</a> [lemma, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path">standard_path</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_to">standard_path_to</a> [inductive, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_pathS">standard_pathS</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_toS">standard_path_toS</a> [inductive, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw">standard_gnaw</a> [definition, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#state">state</a> [record, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html">STDPP_compat</a> [library]<br/>
+<a href="hydras.Epsilon0.Paths.html#stdS_S">stdS_S</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#stdS_1">stdS_1</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#std_S">std_S</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#std_1">std_1</a> [constructor, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step">step</a> [section, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#step">step</a> [inductive, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps">steps</a> [inductive, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps_rounds">steps_rounds</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps_S">steps_S</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps1">steps1</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step_rounds">step_rounds</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#step_to_battle">step_to_battle</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.F">step.F</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.Hn">step.Hn</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.n">step.n</a> [variable, in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step1">step1</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step2">step2</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step3">step3</a> [constructor, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#sto">sto</a> [instance, in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#sto">sto</a> [instance, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:12">sto:12</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:17">sto:17</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:24">sto:24</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:30">sto:30</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:34">sto:34</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:37">sto:37</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:40">sto:40</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:48">sto:48</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:5">sto:5</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#sto:8">sto:8</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="additions.Strategies.html">Strategies</a> [library]<br/>
+<a href="additions.Strategies.html#Strategy">Strategy</a> [record, in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_T1limit">strict_lub_T1limit</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_unique">strict_lub_unique</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_maj">strict_lub_maj</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_lub">strict_lub_lub</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub">strict_lub</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#strict_mono_iterate_S">strict_mono_iterate_S</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#strict_mono">strict_mono</a> [definition, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#strict_mono">strict_mono</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#strict_lub_ref">strict_lub_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Strict_lex">Strict_lex</a> [instance, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.Monoid_def.html#string_op">string_op</a> [instance, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Schutte.Critical.html#strongly_critical">strongly_critical</a> [definition, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Strong_Recursion">Strong_Recursion</a> [section, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#St:110">St:110</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="Goedel.PRrepresentable.html#subAddExistsNice">subAddExistsNice</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#subAddForallsNice">subAddForallsNice</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.subAll.html">subAll</a> [library]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllCloseFrom">subAllCloseFrom</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllCloseFrom1">subAllCloseFrom1</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllFormula">subAllFormula</a> [definition, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllFormulaId">subAllFormulaId</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllFormula_ext">subAllFormula_ext</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#subAllnVars1">subAllnVars1</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#subAllnVars2">subAllnVars2</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllSubAllFormula">subAllSubAllFormula</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllSubAllTerm">subAllSubAllTerm</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllSubAllTerms">subAllSubAllTerms</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllSubFormula">subAllSubFormula</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerm">subAllTerm</a> [definition, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTermId">subAllTermId</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerms">subAllTerms</a> [definition, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTermsId">subAllTermsId</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerms_ext">subAllTerms_ext</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerm_ext">subAllTerm_ext</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars">SubAllVars</a> [section, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars.L">SubAllVars.L</a> [variable, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars.subAllCloseFrom">SubAllVars.subAllCloseFrom</a> [section, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaAnd">subFormulaAnd</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaDepth">subFormulaDepth</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaEqual">subFormulaEqual</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subFormulaExch">subFormulaExch</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaExist">subFormulaExist</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaExist2">subFormulaExist2</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaForall">subFormulaForall</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaForall2">subFormulaForall2</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaId">subFormulaId</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaIff">subFormulaIff</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaIfThenElse">subFormulaIfThenElse</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaImp">subFormulaImp</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="Goedel.PRrepresentable.html#subFormulaMinimize">subFormulaMinimize</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subFormulaNil">subFormulaNil</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaNot">subFormulaNot</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subFormulaNTE">subFormulaNTE</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subFormulaNTEHelp">subFormulaNTEHelp</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaOr">subFormulaOr</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subFormulaRelation">subFormulaRelation</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subFormulaTrans">subFormulaTrans</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.model.html#subInterpFormula">subInterpFormula</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#subInterpFormula1">subInterpFormula1</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#subInterpFormula2">subInterpFormula2</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#subInterpRel">subInterpRel</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#subInterpTerm">subInterpTerm</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#subNNHelp">subNNHelp</a> [lemma, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON">SubON</a> [record, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_least">SubON_least</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_limit">SubON_limit</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_successor">SubON_successor</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_inj">SubON_inj</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_mono">SubON_mono</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties">SubON_properties</a> [section, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_op">SubON_same_op</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_fun">SubON_same_fun</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_cst">SubON_same_cst</a> [definition, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_onto">SubON_onto</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_incl">SubON_incl</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_compare">SubON_compare</a> [projection, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.subProp.html">subProp</a> [library]<br/>
+<a href="Goedel.fixPoint.html#subStar">subStar</a> [definition, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#subStarIsPR">subStarIsPR</a> [instance, in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substExHC">substExHC</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF">substF</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF2">substF2</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF3">substF3</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF4">substF4</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF5">substF5</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF6">substF6</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF7">substF7</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF8">substF8</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF9">substF9</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormula">substituteFormula</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaForall">substituteFormulaForall</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaForallNice">substituteFormulaForallNice</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaHelp">substituteFormulaHelp</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaImp">substituteFormulaImp</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaImpNice">substituteFormulaImpNice</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaNot">substituteFormulaNot</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaNotNice">substituteFormulaNotNice</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteTerm">substituteTerm</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteTerms">substituteTerms</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf">Substitution_Properties.SysPrf</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms">Substitution_Properties.Terms</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Term">Substitution_Properties.Term</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.System">Substitution_Properties.System</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Formulas">Substitution_Properties.Formulas</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula">Substitution_Properties.Formula</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.L">Substitution_Properties.L</a> [variable, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties">Substitution_Properties</a> [section, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substT">substT</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substTs">substTs</a> [definition, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subSubAllFormula">subSubAllFormula</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subSubAllTerm">subSubAllTerm</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subSubAllTerms">subSubAllTerms</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#subSubFormula">subSubFormula</a> [lemma, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#subSubTerm">subSubTerm</a> [lemma, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#subSubTerms">subSubTerms</a> [lemma, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm">subterm</a> [inductive, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subTermApply">subTermApply</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermExch">subTermExch</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subTermId">subTermId</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermNil">subTermNil</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermsExch">subTermsExch</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subTermsId">subTermsId</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermsNil">subTermsNil</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermsTrans">subTermsTrans</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#subTermTrans">subTermTrans</a> [lemma, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subTermVar1">subTermVar1</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#subTermVar2">subTermVar2</a> [lemma, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_lt">subterm_lt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_trans">subterm_trans</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_c">subterm_c</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_b">subterm_b</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_a">subterm_a</a> [constructor, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subToSubAll">subToSubAll</a> [lemma, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#subWithEquals">subWithEquals</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#subWithEqualsTerm">subWithEqualsTerm</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#subWithEqualsTerms">subWithEqualsTerms</a> [lemma, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ">succ</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#succ">succ</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#succ">succ</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ">succ</a> [section, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ">succ</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ">succ</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ">Succ</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succb">succb</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#succb">succb</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succb">succb</a> [definition, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succb_correct">succb_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succb_Succ">Succb_Succ</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succb_not_lim">succb_not_lim</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Successor">Successor</a> [definition, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Successor_succ">Successor_succ</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Successor_not">Successor_not</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Successor_inv">Successor_inv</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Successor_unicity">Successor_unicity</a> [lemma, in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Successor_succ">Successor_succ</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Successor_inv4">Successor_inv4</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Successor_inv3">Successor_inv3</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Successor_inv2">Successor_inv2</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Successor_inv1">Successor_inv1</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="Goedel.PRrepresentable.html#succFormula">succFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#succFunc">succFunc</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#succIsPR">succIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="Goedel.PRrepresentable.html#succRepresentable">succRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_rw1">succ_rw1</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_cons'">succ_cons'</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_cons">succ_cons</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_lt_limit">succ_lt_limit</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_not_limit">succ_not_limit</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_compat">succ_compat</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_compatS">succ_compatS</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_injective">succ_injective</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_monomorphism">succ_monomorphism</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_strict_monoR">succ_strict_monoR</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_mono">succ_mono</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_strict_mono_LT">succ_strict_mono_LT</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_strict_mono">succ_strict_mono</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_of_plus_finite">succ_of_plus_finite</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_is_plus_one">succ_is_plus_one</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_nf">succ_nf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_lt_a_phi0_b">succ_lt_a_phi0_b</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_is_succ">succ_is_succ</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ_not_zero">succ_not_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#succ_is_plus_1">succ_is_plus_1</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#succ_ok">succ_ok</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#succ_finite">succ_finite</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ_correct">succ_correct</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#succ_ref">succ_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_zero_diff">succ_zero_diff</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_injection">succ_injection</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_monoR">succ_monoR</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_mono">succ_mono</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_ok">succ_ok</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_spec">succ_spec</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#succ_correct">succ_correct</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_of_cons">Succ_of_cons</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_rw">Succ_rw</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_lt_T1limit">Succ_lt_T1limit</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_not_T1limit">Succ_not_T1limit</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_not_Zero">Succ_not_Zero</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_Succb">Succ_Succb</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ_inj">Succ_inj</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#succ_rounds">succ_rounds</a> [lemma, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#succ_step">succ_step</a> [constructor, in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succ_limit_dec">succ_limit_dec</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succ_nf">succ_nf</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succ_as_plus">succ_as_plus</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succ_of_cons">succ_of_cons</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#succ_lt_le">succ_lt_le</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Succ_">Succ_</a> [constructor, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Schutte.Addition.html#succ_is_plus_1">succ_is_plus_1</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.beta">succ.beta</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f">succ.f</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_ok">succ.f_ok</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_Sle">succ.f_Sle</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_mono">succ.f_mono</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.Hbeta">succ.Hbeta</a> [variable, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#sumToN">sumToN</a> [definition, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#sumToNIsPR">sumToNIsPR</a> [instance, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#sumToN1">sumToN1</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#sumToN2">sumToN2</a> [lemma, in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup">sup</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#SuperbigStep">SuperbigStep</a> [lemma, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Schutte.AP.html#supM_gt0">supM_gt0</a> [lemma, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_members_disj">sup_members_disj</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_members_not_succ">sup_members_not_succ</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_members_succ">sup_members_succ</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_eq_intro">sup_eq_intro</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_mono">sup_mono</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_of_leq">sup_of_leq</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_least_upper_bound">sup_least_upper_bound</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_upper_bound">sup_upper_bound</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_ok1">sup_ok1</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_spec_unicity">sup_spec_unicity</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_unicity">sup_unicity</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_exists">sup_exists</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#sup_spec">sup_spec</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#sup_M_in_B">sup_M_in_B</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Su:208">Su:208</a> [binder, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="additions.AM.html#SWAP">SWAP</a> [constructor, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#swapIsPR">swapIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#switchIsPR">switchIsPR</a> [instance, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#switchPR">switchPR</a> [definition, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#switch5IsPR">switch5IsPR</a> [lemma, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#sysExtend">sysExtend</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#sysExtend">sysExtend</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#sysExtend">sysExtend</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#SysPrf">SysPrf</a> [definition, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SysPrf">SysPrf</a> [abbreviation, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#SysPrf">SysPrf</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#SysPrf">SysPrf</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#SysPrf_rephrase">SysPrf_rephrase</a> [lemma, in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#System">System</a> [abbreviation, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#System">System</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#System">System</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#System">System</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#System">System</a> [abbreviation, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.fol.html#System">System</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#System">System</a> [abbreviation, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#sysWeaken">sysWeaken</a> [lemma, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#sysWeaken">sysWeaken</a> [lemma, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#sysWeaken">sysWeaken</a> [lemma, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#S_pred_rw">S_pred_rw</a> [lemma, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#S_">S_</a> [abbreviation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#s_decr">s_decr</a> [lemma, in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="additions.AM.html#s':117">s':117</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':13">s':13</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':144">s':144</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':148">s':148</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':16">s':16</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':19">s':19</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#s':199">s':199</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#s':202">s':202</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':21">s':21</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':21">s':21</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':28">s':28</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':289">s':289</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':290">s':290</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':291">s':291</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':292">s':292</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':299">s':299</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':300">s':300</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':31">s':31</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':37">s':37</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#s':40">s':40</a> [binder, in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.O2H.html#s':43">s':43</a> [binder, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':49">s':49</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="additions.AM.html#s':51">s':51</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':53">s':53</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.rpo.rpo.html#s':53">s':53</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':58">s':58</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#s':62">s':62</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':63">s':63</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#s':66">s':66</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.AM.html#s':69">s':69</a> [binder, in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#s':75">s':75</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#s':8">s':8</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#s':82">s':82</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#s':84">s':84</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.decidable_set.html#S.A">S.A</a> [axiom, in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.cardinal">S.cardinal</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.cardinal_subset">S.cardinal_subset</a> [axiom, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.DS">S.DS</a> [module, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.elt">S.elt</a> [definition, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.decidable_set.html#S.eq_A_dec">S.eq_A_dec</a> [axiom, in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#ta1:398">ta1:398</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#tb1:356">tb1:356</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tb1:360">tb1:360</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tb1:366">tb1:366</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tb1:374">tb1:374</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tb1:380">tb1:380</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tb1:383">tb1:383</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="hydras.Ackermann.LNT.html#Term">Term</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Term">Term</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#Term">Term</a> [inductive, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Term">Term</a> [abbreviation, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.rpo.term.html">term</a> [library]<br/>
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+<a href="hydras.Ackermann.subAll.html#Terms">Terms</a> [abbreviation, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Terms">Terms</a> [definition, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Terms">Terms</a> [definition, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Terms">Terms</a> [abbreviation, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#Terms">Terms</a> [inductive, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Terms">Terms</a> [abbreviation, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#terms_eqdec">terms_eqdec</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Terms_Term_rec_full">Terms_Term_rec_full</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Terms_Term_rec">Terms_Term_rec</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Terms_Term_ind">Terms_Term_ind</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#term_eqdec">term_eqdec</a> [lemma, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Term_Terms_rec_full">Term_Terms_rec_full</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Term_Terms_rec">Term_Terms_rec</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="hydras.rpo.term.html#Term.apply_subst">Term.apply_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.direct_subterm">Term.direct_subterm</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.term.html#Term.map_subst">Term.map_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.term.html#Term.replace_at_pos_is_replace_at_pos2">Term.replace_at_pos_is_replace_at_pos2</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos_is_replace_at_pos1">Term.replace_at_pos_is_replace_at_pos1</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.term.html#Term.replace_at_pos_list">Term.replace_at_pos_list</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos">Term.replace_at_pos</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size">Term.size</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_subterm_at_pos">Term.size_subterm_at_pos</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_direct_subterm">Term.size_direct_subterm</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_ge_one">Term.size_ge_one</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_unfold">Term.size_unfold</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.substitution">Term.substitution</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subst_comp_is_subst_comp">Term.subst_comp_is_subst_comp</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subst_comp_is_subst_comp_aux1">Term.subst_comp_is_subst_comp_aux1</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subst_comp">Term.subst_comp</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos">Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subterm_at_pos">Term.subterm_at_pos</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.symbol">Term.symbol</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Term">Term.Term</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term">Term.term</a> [inductive, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Term_eq_dec">Term.Term_eq_dec</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec8">Term.term_rec8</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec7">Term.term_rec7</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec4">Term.term_rec4</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec3">Term.term_rec3</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec2">Term.term_rec2</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Var">Term.Var</a> [constructor, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.variable">Term.variable</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_apply_subst">Term.well_formed_apply_subst</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_subst">Term.well_formed_subst</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_list">Term.well_formed_list</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_fold">Term.well_formed_fold</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_unfold">Term.well_formed_unfold</a> [axiom, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed">Term.well_formed</a> [definition, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.X">Term.X</a> [module, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#test">test</a> [definition, in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#the">the</a> [definition, in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Theorem_4_5">Theorem_4_5</a> [lemma, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#Theorem_4_5">Theorem_4_5</a> [lemma, in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#the_least_ok">the_least_ok</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_least_unicity">the_least_unicity</a> [lemma, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_least">the_least</a> [definition, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.About_WO">the_context.About_WO</a> [section, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.Lt">the_context.Lt</a> [variable, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.M">the_context.M</a> [variable, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context">the_context</a> [section, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="additions.Addition_Chains.html#the_exponent">the_exponent</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#the_exponent_nat">the_exponent_nat</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#the_ordering_segment">the_ordering_segment</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#TH_packed">TH_packed</a> [lemma, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#th_In">th_In</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#TH_13_6R">TH_13_6R</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#TH_13_6">TH_13_6</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th_13_5_2">Th_13_5_2</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th_13_5_1">Th_13_5_1</a> [lemma, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#th_14_6">th_14_6</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#th_14_5">th_14_5</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Th13_8_1">Th13_8_1</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Th13_8">Th13_8</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.alpha_">Th13_5.verso.U_fixed.alpha_</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_inc_A">Th13_5.verso.U_fixed.U_inc_A</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_den">Th13_5.verso.U_fixed.U_den</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_non_empty">Th13_5.verso.U_fixed.U_non_empty</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U">Th13_5.verso.U_fixed.U</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed">Th13_5.verso.U_fixed</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.B_closed">Th13_5.verso.B_closed</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso">Th13_5.verso</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.U">Th13_5.recto.M_fixed.U</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.den">Th13_5.recto.M_fixed.den</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.ne">Th13_5.recto.M_fixed.ne</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.inc">Th13_5.recto.M_fixed.inc</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.M">Th13_5.recto.M_fixed.M</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed">Th13_5.recto.M_fixed</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.f_cont">Th13_5.recto.f_cont</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto">Th13_5.recto</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.f_ord">Th13_5.f_ord</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.f">Th13_5.f</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.B">Th13_5.B</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.A">Th13_5.A</a> [variable, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5">Th13_5</a> [section, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Times_">Times_</a> [constructor, in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.rpo.rpo.html#tj:174">tj:174</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#tj:177">tj:177</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#tj:180">tj:180</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Tnil">Tnil</a> [constructor, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Too_far">Too_far</a> [constructor, in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Total">Total</a> [record, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Total">Total</a> [inductive, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#totalness">totalness</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#totalness">totalness</a> [constructor, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#TotalPreOrder">TotalPreOrder</a> [record, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Total_Inverse_fun">Total_Inverse_fun</a> [instance, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#total_pre_order_total">total_pre_order_total</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#total_pre_order_pre">total_pre_order_pre</a> [projection, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Total_Reflexive">Total_Reflexive</a> [instance, in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2">tower2</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2IsPR">tower2IsPR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt_PR">tower2_alt_PR</a> [instance, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt_ok">tower2_alt_ok</a> [lemma, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt">tower2_alt</a> [definition, in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy">Toy</a> [module, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.A">Toy.A</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.a">Toy.a</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Adds">Toy.Adds</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.arityF">Toy.arityF</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.arityR">Toy.arityR</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.a_">Toy.a_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.A_">Toy.A_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.B">Toy.B</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.b">Toy.b</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.b_">Toy.b_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.B_">Toy.B_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.C">Toy.C</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.C_">Toy.C_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.ded1">Toy.ded1</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.ded2">Toy.ded2</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.ded3">Toy.ded3</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.Drinkers_theorem.f">Toy.Drinkers_theorem.f</a> [variable, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.D0">Toy.D0</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.D01">Toy.D01</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.eq_refl">Toy.eq_refl</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.Fun">Toy.Fun</a> [inductive, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.f_">Toy.f_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F1">Toy.F1</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F2">Toy.F2</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F3">Toy.F3</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F4">Toy.F4</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F5">Toy.F5</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.F6">Toy.F6</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.g">Toy.g</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.g_">Toy.g_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.h">Toy.h</a> [abbreviation, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.h_">Toy.h_</a> [constructor, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.L">Toy.L</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Ldec">Toy.Ldec</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.MP'">Toy.MP'</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.OnSubstF.F">Toy.OnSubstF.F</a> [variable, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.peirce">Toy.peirce</a> [lemma, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Peirce">Toy.Peirce</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.PeirceProof">Toy.PeirceProof</a> [section, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.PrfContrex7">Toy.PrfContrex7</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.PrfEx1">Toy.PrfEx1</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.PrfEx6">Toy.PrfEx6</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#Toy.Unnamed_thm1">Toy.Unnamed_thm1</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="Goedel.rosserPA.html#Tprf2T'prf">Tprf2T'prf</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.goedel2.html#Tprf2T'prf">Tprf2T'prf</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
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+<a href="hydras.Hydra.O2H.html#trace_to_std">trace_to_std</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
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+<a href="hydras.Hydra.O2H.html#trace_to_round_plus">trace_to_round_plus</a> [lemma, in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="additions.Trace_exercise.html#trace_mult">trace_mult</a> [definition, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="additions.Trace_exercise.html#trace_compare">trace_compare</a> [definition, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#trace_toS">trace_toS</a> [constructor, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
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+<a href="hydras.Prelude.Simple_LexProd.html#Trans_lex">Trans_lex</a> [instance, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#TreeT1K">TreeT1K</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Tree2T1">Tree2T1</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#trichotomy">trichotomy</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_aux">tricho_aux</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_7">tricho_lt_7</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_5">tricho_lt_5</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_4'">tricho_lt_4'</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_4">tricho_lt_4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_3">tricho_lt_3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_2'">tricho_lt_2'</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#tricho_lt_2">tricho_lt_2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.rpo.html#triple:184">triple:184</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#triple:186">triple:186</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Tr:100">Tr:100</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#ts:10">ts:10</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:100">ts:100</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:102">ts:102</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:105">ts:105</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:105">ts:105</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:106">ts:106</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:107">ts:107</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:108">ts:108</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:108">ts:108</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#ts:11">ts:11</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.PA.html#ts:11">ts:11</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:110">ts:110</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:112">ts:112</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:112">ts:112</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:115">ts:115</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:117">ts:117</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:119">ts:119</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:119">ts:119</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:12">ts:12</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:12">ts:12</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#ts:1255">ts:1255</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:1261">ts:1261</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:1263">ts:1263</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="hydras.Ackermann.subProp.html#ts:128">ts:128</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:129">ts:129</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#ts:13">ts:13</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:14">ts:14</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#ts:149">ts:149</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#ts:15">ts:15</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#ts:16">ts:16</a> [binder, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:16">ts:16</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.model.html#ts:161">ts:161</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:17">ts:17</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#ts:17">ts:17</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:18">ts:18</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#ts:18">ts:18</a> [binder, in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:180">ts:180</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:183">ts:183</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:186">ts:186</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
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+<a href="hydras.Ackermann.LNN2LNT.html#ts:21">ts:21</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:212">ts:212</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:217">ts:217</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#ts:22">ts:22</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.code.html#ts:22">ts:22</a> [binder, in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:222">ts:222</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:23">ts:23</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#ts:23">ts:23</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:23">ts:23</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#ts:25">ts:25</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#ts:27">ts:27</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#ts:77">ts:77</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ts:77">ts:77</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:78">ts:78</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#ts:78">ts:78</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:80">ts:80</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:81">ts:81</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#ts:82">ts:82</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#ts:82">ts:82</a> [binder, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:82">ts:82</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:86">ts:86</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:874">ts:874</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:878">ts:878</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:882">ts:882</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ts:91">ts:91</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:914">ts:914</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:919">ts:919</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ts:924">ts:924</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:93">ts:93</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ts:93">ts:93</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ts:96">ts:96</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#ts:97">ts:97</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#ts:98">ts:98</a> [binder, in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Two">Two</a> [instance, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.BinaryStrat.html#two">two</a> [definition, in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="additions.BinaryStrat.html#Two_strat">Two_strat</a> [instance, in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#type:64">type:64</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#type:792">type:792</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#type:840">type:840</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#type:843">type:843</a> [binder, in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.rpo.closure.html#t_trans">t_trans</a> [constructor, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.rpo.closure.html#t_step">t_step</a> [constructor, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#t_0_nil">t_0_nil</a> [lemma, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="Goedel.rosserPA.html#T'">T'</a> [definition, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.goedel2.html#T'">T'</a> [definition, in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.rosserPA.html#T'prf2Tprf">T'prf2Tprf</a> [lemma, in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t'':115">t'':115</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t':114">t':114</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.Trace_exercise.html#t':16">t':16</a> [binder, in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="additions.Addition_Chains.html#t':165">t':165</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#t':178">t':178</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.rpo.rpo.html#t':322">t':322</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#t':323">t':323</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Gamma0.T2.html#t':70">t':70</a> [binder, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="additions.Addition_Chains.html#t':86">t':86</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t':89">t':89</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t':91">t':91</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t':93">t':93</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#t0_empty">t0_empty</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:171">t0:171</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t0:18">t0:18</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#t0:18">t0:18</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:187">t0:187</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t0:20">t0:20</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:202">t0:202</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:203">t0:203</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:219">t0:219</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#t0:22">t0:22</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:225">t0:225</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t0:24">t0:24</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:243">t0:243</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:250">t0:250</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t0:26">t0:26</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:266">t0:266</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:293">t0:293</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:299">t0:299</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:315">t0:315</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:331">t0:331</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:339">t0:339</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:347">t0:347</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:363">t0:363</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t0:379">t0:379</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:383">t0:383</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#t0:40">t0:40</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.fol.html#t0:437">t0:437</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#t0:44">t0:44</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:84">t0:84</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:861">t0:861</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:864">t0:864</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:885">t0:885</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:888">t0:888</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.PA.html#t0:9">t0:9</a> [binder, in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:901">t0:901</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:904">t0:904</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:928">t0:928</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#t0:931">t0:931</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1">T1</a> [inductive, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1">T1</a> [abbreviation, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html">T1</a> [library]<br/>
+<a href="hydras.Epsilon0.T1.html#T1add">T1add</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1addA">T1addA</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1addAssoc">T1addAssoc</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1add_not_monotonous_l">T1add_not_monotonous_l</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html">T1Bridge</a> [library]<br/>
+<a href="gaia_hydras.T1Choice.html#T1Choice">T1Choice</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html">T1Choice</a> [library]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1compare">T1compare</a> [instance, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1compare_correct">T1compare_correct</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1eqE">T1eqE</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1eq_h2g">T1eq_h2g</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1eq_refl">T1eq_refl</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#T1F_">T1F_</a> [definition, in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1is_succ_iff">T1is_succ_iff</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1is_succ_def">T1is_succ_def</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1is_succ">T1is_succ</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1is_succ_ref">T1is_succ_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1leOrderMixin">T1leOrderMixin</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1le_asym">T1le_asym</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1le_iff">T1le_iff</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1limit">T1limit</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1limit_succ_tail">T1limit_succ_tail</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1limit_not_zero">T1limit_not_zero</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1limit_succ">T1limit_succ</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1limit_cases">T1limit_cases</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1limit_ref">T1limit_ref</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#T1limit_canonS_not_zero">T1limit_canonS_not_zero</a> [lemma, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#T1limit_plus">T1limit_plus</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#T1limit_phi0">T1limit_phi0</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#T1limit_Omega_term">T1limit_Omega_term</a> [lemma, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1ltE">T1ltE</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1lt_iff">T1lt_iff</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1max">T1max</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1maxE">T1maxE</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1min">T1min</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1minE">T1minE</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1mixin">T1mixin</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1mul">T1mul</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1mulE4">T1mulE4</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1mulE5">T1mulE5</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1mul_not_monotonous">T1mul_not_monotonous</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1mul_cons_cons_E">T1mul_cons_cons_E</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1mul_a0E">T1mul_a0E</a> [lemma, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1nat">T1nat</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.GCanon.html#T1nf_canon">T1nf_canon</a> [lemma, in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1omega">T1omega</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1orderType">T1orderType</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1pp">T1pp</a> [definition, in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GHydra.html#T1toH">T1toH</a> [definition, in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1_wf">T1_wf</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1_eq_dec">T1_eq_dec</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#t1_strorder">t1_strorder</a> [instance, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T1_le_Mixin">T1_le_Mixin</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Gamma0.T2.html#T1_to_T2">T1_to_T2</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#t1_0">t1_0</a> [definition, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T1_to_T2_monoR">T1_to_T2_monoR</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T1_to_T2_mono">T1_to_T2_mono</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T1_to_T2_R">T1_to_T2_R</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T1_to_T2_lt">T1_to_T2_lt</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T1_to_T2_inj">T1_to_T2_inj</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#t1_0">t1_0</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html">T1_ltNotWf</a> [library]<br/>
+<a href="gaia_hydras.T1Choice.html#T12Tree">T12Tree</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#T12Tree_inj">T12Tree_inj</a> [lemma, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.rpo.rpo.html#t1:106">t1:106</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#t1:113">t1:113</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:118">t1:118</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:121">t1:121</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t1:14">t1:14</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t1:14">t1:14</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.rpo.list_set.html#t1:14">t1:14</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t1:144">t1:144</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t1:16">t1:16</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.rpo.term.html#t1:197">t1:197</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:206">t1:206</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t1:208">t1:208</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:230">t1:230</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:236">t1:236</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t1:238">t1:238</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:240">t1:240</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:244">t1:244</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:252">t1:252</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:256">t1:256</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:27">t1:27</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:275">t1:275</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#t1:31">t1:31</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="Goedel.PRrepresentable.html#t1:36">t1:36</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#t1:40">t1:40</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#t1:445">t1:445</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t1:49">t1:49</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t1:53">t1:53</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t1:53">t1:53</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.list_permut.html#t1:6">t1:6</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#t1:61">t1:61</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:63">t1:63</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t1:65">t1:65</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t1:68">t1:68</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:684">t1:684</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t1:69">t1:69</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t1:76">t1:76</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t1:81">t1:81</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t1:94">t1:94</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.T2.html#T2">T2</a> [inductive, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#T2">T2</a> [abbreviation, in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Gamma0.T2.html">T2</a> [library]<br/>
+<a href="gaia_hydras.T2Bridge.html">T2Bridge</a> [library]<br/>
+<a href="hydras.Gamma0.T2.html#t2_length_aux">t2_length_aux</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#t2_length">t2_length</a> [definition, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size_psi">T2_size_psi</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size4">T2_size4</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size3">T2_size3</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size2">T2_size2</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size1">T2_size1</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size">T2_size</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_2_term">T2_2_term</a> [definition, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.rpo.html#t2:108">t2:108</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#t2:115">t2:115</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:119">t2:119</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:122">t2:122</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#t2:145">t2:145</a> [binder, in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.rpo.term.html#t2:15">t2:15</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#t2:15">t2:15</a> [binder, in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#t2:198">t2:198</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:207">t2:207</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t2:209">t2:209</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:229">t2:229</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:237">t2:237</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t2:239">t2:239</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:241">t2:241</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:243">t2:243</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:253">t2:253</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:257">t2:257</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:261">t2:261</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:263">t2:263</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:276">t2:276</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.term.html#t2:28">t2:28</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#t2:32">t2:32</a> [binder, in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="Goedel.PRrepresentable.html#t2:37">t2:37</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#t2:41">t2:41</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#t2:446">t2:446</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t2:50">t2:50</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t2:52">t2:52</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t2:54">t2:54</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t2:62">t2:62</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:64">t2:64</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t2:66">t2:66</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t2:67">t2:67</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:685">t2:685</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.list_permut.html#t2:7">t2:7</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="Goedel.codeSysPrf.html#t2:70">t2:70</a> [binder, in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.rpo.term.html#t2:77">t2:77</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#t2:82">t2:82</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t2:95">t2:95</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t3:120">t3:120</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#t3:123">t3:123</a> [binder, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#t:1">t:1</a> [binder, in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#t:1">t:1</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
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+<a href="hydras.Ackermann.codeSubTerm.html#t:96">t:96</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#t:96">t:96</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#T:96">T:96</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#T:97">T:97</a> [binder, in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#t:97">t:97</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#t:98">t:98</a> [binder, in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#t:98">t:98</a> [binder, in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#T:98">T:98</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#t:98">t:98</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.rpo.rpo.html#t:98">t:98</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#t:99">t:99</a> [binder, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#T:99">T:99</a> [binder, in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#T:99">T:99</a> [binder, in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#T:99">T:99</a> [binder, in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#t:99">t:99</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="global_U"></a><h2>U </h2>
+<a href="hydras.Ackermann.model.html#U">U</a> [projection, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.rpo.term.html#ui:187">ui:187</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#ui:325">ui:325</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#ui:328">ui:328</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#ui:369">ui:369</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#unbounded">unbounded</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Unbounded">Unbounded</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Unbounded_Cr">Unbounded_Cr</a> [lemma, in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Unbounded_not_countable">Unbounded_not_countable</a> [lemma, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Addition.html#Unbounded_ge">Unbounded_ge</a> [lemma, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#undecidable">undecidable</a> [abbreviation, in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#unique_decomposition">unique_decomposition</a> [lemma, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#unique_decomposition">unique_decomposition</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm9">Unnamed_thm9</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm8">Unnamed_thm8</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm7">Unnamed_thm7</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm6">Unnamed_thm6</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm5">Unnamed_thm5</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm5">Unnamed_thm5</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm4">Unnamed_thm4</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm3">Unnamed_thm3</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm2">Unnamed_thm2</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm1">Unnamed_thm1</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="additions.Monoid_def.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm1">Unnamed_thm1</a> [definition, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm1">Unnamed_thm1</a> [definition, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm2">Unnamed_thm2</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm1">Unnamed_thm1</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm0">Unnamed_thm0</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm">Unnamed_thm</a> [definition, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift">unshift</a> [definition, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_pred">unshift_pred</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_shift">unshift_shift</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_interval_pred">unshift_interval_pred</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_interval">unshift_interval</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_iota_from">unshift_iota_from</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_not_nil">unshift_not_nil</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_app">unshift_app</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_but_last">unshift_but_last</a> [lemma, in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.model.html#updateValue">updateValue</a> [definition, in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Schutte.Lub.html#upper_bound">upper_bound</a> [definition, in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Usual">Usual</a> [module, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Usual.beta">Usual.beta</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#ed2bf37be4fac66de48eeab40fec4772">Usual.β</a> [abbreviation, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u':693">u':693</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.rpo.html#u1:107">u1:107</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#u1:114">u1:114</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#u1:121">u1:121</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u1:89">u1:89</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u1:97">u1:97</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#u2:109">u2:109</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#u2:116">u2:116</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#u2:122">u2:122</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u2:90">u2:90</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u2:98">u2:98</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u3:91">u3:91</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u3:99">u3:99</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u4:100">u4:100</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#u4:92">u4:92</a> [binder, in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1202">u:1202</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1213">u:1213</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.rpo.rpo.html#u:123">u:123</a> [binder, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1238">u:1238</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1242">u:1242</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1245">u:1245</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1265">u:1265</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1281">u:1281</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:130">u:130</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#U:136">U:136</a> [binder, in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:136">u:136</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:137">u:137</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#u:143">u:143</a> [binder, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:143">u:143</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.rpo.term.html#u:147">u:147</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#U:15">U:15</a> [binder, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.rpo.term.html#u:150">u:150</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:150">u:150</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1512">u:1512</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1515">u:1515</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#u:1518">u:1518</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#u:155">u:155</a> [binder, in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#u:161">u:161</a> [binder, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.rpo.term.html#u:162">u:162</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsCorrect1">wellFormedTermsCorrect1</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsCorrect2">wellFormedTermsCorrect2</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsIsPR">wellFormedTermsIsPR</a> [instance, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTerms">wellFormedTermTerms</a> [definition, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTermsCorrect2">wellFormedTermTermsCorrect2</a> [lemma, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTermsIsPR">wellFormedTermTermsIsPR</a> [instance, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.rpo.term.html#well_formed_list:270">well_formed_list:270</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#well_formed_list:90">well_formed_list:90</a> [binder, in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#well_founded_restriction_rect">well_founded_restriction_rect</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#well_founded_restriction">well_founded_restriction</a> [definition, in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.rpo.more_list.html#well_founded_length">well_founded_length</a> [lemma, in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.Formula">Well_Formed_Term.Well_Formed_Formula.Formula</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2">Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1">Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR">Well_Formed_Term.Well_Formed_Formula.codeArityR</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula">Well_Formed_Term.Well_Formed_Formula</a> [section, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms">Well_Formed_Term.Terms</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Term">Well_Formed_Term.Term</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityFIsCorrect2">Well_Formed_Term.codeArityFIsCorrect2</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityFIsCorrect1">Well_Formed_Term.codeArityFIsCorrect1</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF">Well_Formed_Term.codeArityF</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L">Well_Formed_Term.L</a> [variable, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term">Well_Formed_Term</a> [section, in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#well_order">well_order</a> [projection, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded.R">well_founded.R</a> [variable, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded">well_founded</a> [section, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded_rpo">well_founded_rpo</a> [lemma, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#well_founded_P">well_founded_P</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html">Well_Orders</a> [library]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#WF">WF</a> [instance, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#wf">wf</a> [projection, in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#WF">WF</a> [instance, in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum">Wfsum</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.extension">Wfsum.extension</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f">Wfsum.f</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_correct">Wfsum.f_correct</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_eqn">Wfsum.f_eqn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec_simp">Wfsum.f_spec_simp</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec">Wfsum.f_spec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f1">Wfsum.f1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f2">Wfsum.f2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.lt_dec">Wfsum.lt_dec</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.psum">Wfsum.psum</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.psum_exten">Wfsum.psum_exten</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#WfVariant">WfVariant</a> [record, in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html">WfVariant</a> [library]<br/>
+<a href="additions.Wf_transparent.html#wf_inverse_image_transparent">wf_inverse_image_transparent</a> [lemma, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#wf_incl_transparent">wf_incl_transparent</a> [lemma, in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#wf_measure">wf_measure</a> [lemma, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="gaia_hydras.onType.html#wf_ltn">wf_ltn</a> [lemma, in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.rpo.closure.html#wf_trans">wf_trans</a> [lemma, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_correct">Wf_ex3.f_correct</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_eqn">Wf_ex3.f_eqn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f">Wf_ex3.f</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f2">Wf_ex3.f2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f1">Wf_ex3.f1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.odd_dec">Wf_ex3.odd_dec</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0c">Wf_ex3.f0c</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0b">Wf_ex3.f0b</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0a">Wf_ex3.f0a</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp2">Wf_ex3.f_spec_simp2</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp1">Wf_ex3.f_spec_simp1</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp">Wf_ex3.f_spec_simp</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec">Wf_ex3.f_spec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte_wf">Wf_ex3.lte_wf</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte">Wf_ex3.lte</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3">Wf_ex3</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_correct">Wf_ex.f_correct</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_eqn">Wf_ex.f_eqn</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f">Wf_ex.f</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f2">Wf_ex.f2</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f1">Wf_ex.f1</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f0">Wf_ex.f0</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec_simp">Wf_ex.f_spec_simp</a> [lemma, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec">Wf_ex.f_spec</a> [definition, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex">Wf_ex</a> [module, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#wf_lexico">wf_lexico</a> [lemma, in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.rpo.rpo.html#wf_lex">wf_lex</a> [lemma, in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="additions.Wf_transparent.html">Wf_transparent</a> [library]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Wf:101">Wf:101</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Wf:111">Wf:111</a> [binder, in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Wf:146">Wf:146</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Wf:156">Wf:156</a> [binder, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.wConsistent.html#wInconsistent">wInconsistent</a> [definition, in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="additions.fib.html#with_matrices.R">with_matrices.R</a> [variable, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#with_matrices">with_matrices</a> [section, in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#WO">WO</a> [record, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#WO_ord">WO_ord</a> [instance, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#WO_nat">WO_nat</a> [instance, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Wo:19">Wo:19</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#WO:54">WO:54</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#WO:59">WO:59</a> [binder, in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="Goedel.PRrepresentable.html#w:101">w:101</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#w:122">w:122</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#w:1237">w:1237</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="Goedel.PRrepresentable.html#w:129">w:129</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#w:1388">w:1388</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#w:1392">w:1392</a> [binder, in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#w:14">w:14</a> [binder, in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="Goedel.PRrepresentable.html#w:140">w:140</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#w:151">w:151</a> [binder, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#w:1513">w:1513</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#w:1516">w:1516</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#w:1519">w:1519</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#w:300">w:300</a> [binder, in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#w:402">w:402</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#w:410">w:410</a> [binder, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#w:60">w:60</a> [binder, in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#w:705">w:705</a> [binder, in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<br/><br/><a id="global_Z"></a><h2>Z </h2>
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+<a href="hydras.Epsilon0.T1.html#zero">zero</a> [constructor, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Gamma0.T2.html#zero">zero</a> [constructor, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#zeroFormula">zeroFormula</a> [definition, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#zeroFunc">zeroFunc</a> [constructor, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#zeroRepresentable">zeroRepresentable</a> [lemma, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#zero_limit_succ_dec">zero_limit_succ_dec</a> [definition, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#z:88">z:88</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.rpo.closure.html#z:9">z:9</a> [binder, in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
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+<a href="hydras.Epsilon0.Large_Sets.html#z:98">z:98</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.fol.html#z:98">z:98</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#z:98">z:98</a> [binder, in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#z:99">z:99</a> [binder, in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.fol.html#z:99">z:99</a> [binder, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Addition_Chains.html#z:99">z:99</a> [binder, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<br/><br/><a id="global__"></a><h2>_ </h2>
+<a href="hydras.Schutte.AP.html#_phi0">_phi0</a> [definition, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#_omega">_omega</a> [definition, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#_Omega">_Omega</a> [abbreviation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#_f">_f</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#_A">_A</a> [definition, in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="global_*"></a><h2>other </h2>
+<a href="hydras.Ackermann.fol.html#:fol::x_'&lt;&gt;'_x">fol:_ &lt;&gt; _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#99fc41bec617cc902ba1cfab8f026785">fol:_ = _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#:fol::x_'&lt;-&gt;'_x">fol:_ &lt;-&gt; _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#:fol::x_'-&gt;'_x">fol:_ -&gt; _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#64a970f6f4c4d1914c59e379c64a481b">fol:_ ∧ _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#146ef165103f02c940f12358e2d84d5c">fol:_ ∨ _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#299c44a9fb7c7425e5e57147661f89e4">fol:( _ )</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#b8de2539629ab8cd49874e6c530636b7">fol:{ _ }</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#b9c468757fc602e0e6e4d7f82378fa40">fol:~ _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#80736841a977d43b8bf9e594dd2212d2">fol:∀ _ , _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#0e2109d6ebbe119a5bf3c8b38478024d">fol:∃ _ , _</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#9b335537bf17336701138ce9fa631c98">_ o+ _ (cantor_scope)</a> [notation, in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#bae485d6f56c6bf81e7b6584053562d1">_ O+ _ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#c28e4e4c810917276f0357982f044110">_ * _ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#c17b9ed2f114699671bc84e75652badb">E0_omega^ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#829ecb68b8b34ee91d22b1ffdd27a15c">_ + _ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#b8cab4266e2151a748b18e18aa1598f9">_ o&lt;= _ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#::E0_scope:x_'o&lt;'_x">_ o&lt; _ (E0_scope)</a> [notation, in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51">_ ^ _ (M_scope)</a> [notation, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#0ad68a69d54362abc023eb138c77275f">_ ^b _ (M_scope)</a> [notation, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#765592cb239fbb605ff794a9833fef51">_ ^ _ (M_scope)</a> [notation, in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#765592cb239fbb605ff794a9833fef51">_ ^ _ (M_scope)</a> [notation, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#0ad68a69d54362abc023eb138c77275f">_ ^b _ (M_scope)</a> [notation, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#765592cb239fbb605ff794a9833fef51">_ ^ _ (M_scope)</a> [notation, in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Monoid_def.html#e1cb395850f224a43e9550833db388b3">_ * _ (M_scope)</a> [notation, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-4'">_ .-4 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-3'">_ .-3 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-2'">_ .-2 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#::nat_scope:x_'.-1'">_ .-1 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#f78f3853e8d6bffffe9b6bc1ce727fd3">_ .+5 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#854d640e6c55155f8eda90636a67ef35">_ .+4 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#331a87f8f0f3c9336601c27f1e458dc3">_ .+3 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#bc63483175fac06bcd5541a5e9093b18">_ .+2 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Prelude.ssrnat_extracts.html#bda89d73ec4a8f23ae92b565ffb5aaa6">_ .+1 (nat_scope)</a> [notation, in <a href="hydras.Prelude.ssrnat_extracts.html">hydras.Prelude.ssrnat_extracts</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'''_x">_ &lt;-&gt;' _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'''''_x">_ &lt;-&gt;'' _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#d0a47be881e3150d2a9af078bfd6971b">_ /\' _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#c559e70a5c5fc9dc7c84ae2eb6b09bac">_ \/' _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#8823e980df46f52eb8bcd9d68ef4ed03">_ * _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#ed08876ea67a642780c9e14210615148">_ + _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#4bb2f2560044148bb04f513fa721bd27">allH _ .. _ , _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#14879ad85c55a476ba57071753760440">exH _ .. _ , _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#c1d299e5a32db34ea906aad92603b7d6">v# _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;&gt;'_x">_ &lt;&gt; _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41">_ = _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#::nt_scope:x_'&lt;-&gt;'_x">_ &lt;-&gt; _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#798e0981258b01fbb1da8c3e84f3db7e">~ _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#::nt_scope:x_'-&gt;'_x">_ -&gt; _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#281ad9f5c20247dcb66364deeca521cd">_ /\ _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#215352f85b2d50eebfec8d16206295aa">_ \/ _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#fed6c14e49eff7036666eb0f91269e41">_ = _ (nt_scope)</a> [notation, in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#4a4f2f510af1c6a5e896f63b5eead422">_ * _ (ON_scope)</a> [notation, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#a050364cdd242546ff34b16b41a206b9">_ + _ (ON_scope)</a> [notation, in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#147b23beaf5fa5d05316eba5f8182d20">_ o?= _ (ON_scope)</a> [notation, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#9e4e840fd05a61043174d8c81c890780">_ o&lt;= _ (ON_scope)</a> [notation, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#::ON_scope:x_'o&lt;'_x">_ o&lt; _ (ON_scope)</a> [notation, in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#3f1bc0399e791f84036bad45d79e1159">_ ^ _ (ppT1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d1152fe3a7d79cfdcdc4303f763052d6">_ * _ (ppT1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#72f214ae79555f1ca58c18aadc6d29d1">_ + _ (ppT1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.AP.html#6b5cc3fabae1095df9f5452df6b2c0df">omega^ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#abe2b27667df40e2c3f74853287f2063">|_| _ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#13c3c4e605a8e463a22f054036895bb7">_ &lt;= _ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#::schutte_scope:x_'&lt;'_x">_ &lt; _ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Addition.html#c07b3f4290a4e84a9a38136928201701">_ * _ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#ef17d4fde9f4f9a265266bf8f6c56f86">_ + _ (schutte_scope)</a> [notation, in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.Monoid_def.html#2596316c6f2ff161cb03e25fd8914390">_ == _ (type_scope)</a> [notation, in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Addition_Chains.html#2596316c6f2ff161cb03e25fd8914390">_ == _ (type_scope)</a> [notation, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x_'t1&lt;'_x">_ t1&lt; _ t1&lt; _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#f097afe6f86551e3dd19af72feb179da">_ t1&lt;= _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#::t1_scope:x_'t1&lt;'_x">_ t1&lt; _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#6d507108061d062f40e7ec3dfffa8239">_ ^ _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#::t1_scope:x_'-'_x">_ - _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ec5016939656baf2f20020d4a10c3866">_ * _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#f1825676a507995dc4cc5d57158dc2ac">_ + _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#d9b48d1d60bd5708755ece6f90c63f3a">\F _ (t1_scope)</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#700fc75f60b054fa3b8ffac2d7cf60a3">_ t2&lt;= _ (T2_scope)</a> [notation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#::T2_scope:x_'t2&lt;'_x">_ t2&lt; _ (T2_scope)</a> [notation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#19285cf1708f341b0870d51774920007">[ _ , _ ] (T2_scope)</a> [notation, in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#43589bbdaffd7406c3f135d34490f8e8">_ + _ (T2_scope)</a> [notation, in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#:::x_'&lt;_phi0'_x">_ &lt;_phi0 _</a> [notation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#:::x_'&lt;'_x">_ &lt; _</a> [notation, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#b8382ebd4d1ecdc4b5385bb249fc75b4">_ &lt;= _</a> [notation, in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#:::x_'''&lt;s'''_x">_ '&lt;s' _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#:::x_'&lt;:'_x">_ &lt;: _</a> [notation, in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#:::x_'&gt;&gt;s'_x">_ &gt;&gt;s _</a> [notation, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#:::x_'&gt;&gt;'_x">_ &gt;&gt; _</a> [notation, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#25e51f6c9a7647f5a562b237127007c4">_ &lt;&lt;= _</a> [notation, in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#3cf9144baff8c6bf36bff5a2c3dc2aab">_ o+ _</a> [notation, in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#0a87cec6db386ad672945a4ad4643c14">_ =x= _</a> [notation, in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Addition_Chains.html#5870a17092f4a8078b9692b4c6687556">_ &lt;--- _ times _ ; _</a> [notation, in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#01d24958106a67451f21f2873a331e59">_ -*-&gt; _</a> [notation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#4b2603e3c2ae6b73edb417c5b09dc498">_ -+-&gt; _</a> [notation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#:::x_'-1-&gt;'_x">_ -1-&gt; _</a> [notation, in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#027d47fdf302b28789c559f47c303a6a">f[ _ ]f</a> [notation, in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.NewNotations.html#:::'SetAdds'_x_x_'..'_x">SetAdds _ _ .. _</a> [notation, in <a href="hydras.Ackermann.NewNotations.html">hydras.Ackermann.NewNotations</a>]<br/>
+<a href="hydras.Ackermann.NewNotations.html#:::'SetEnum'_x_x_'..'_x">SetEnum _ _ .. _</a> [notation, in <a href="hydras.Ackermann.NewNotations.html">hydras.Ackermann.NewNotations</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#:::'v_apply'_x_x">v_apply _ _</a> [notation, in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="gaia_hydras.HydraGaia_Examples.html#ab410a966ac148e9b78c65c6cdf301fd">α</a> [definition, in <a href="gaia_hydras.HydraGaia_Examples.html">gaia_hydras.HydraGaia_Examples</a>]<br/>
+<a href="Goedel.PRrepresentable.html#dc5233cb1d950ecad15b1e9b2514f665">β</a> [abbreviation, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="gaia_hydras.HydraGaia_Examples.html#dc5233cb1d950ecad15b1e9b2514f665">β</a> [definition, in <a href="gaia_hydras.HydraGaia_Examples.html">gaia_hydras.HydraGaia_Examples</a>]<br/>
+<a href="Goedel.PRrepresentable.html#cb118be89dab27ceb8f4a738e65da758">βR</a> [abbreviation, in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3">ω</a> [abbreviation, in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<br/><br/><hr/>
+<h1>Notation Index</h1>
+<a id="notation_A"></a><h2>A </h2>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'-'_x">_ - _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#07c58f65e5105c31dc2f9d874ea182dd">_ + _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&gt;'_x">_ &gt; _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#752c7e761a34d95cac9b695b31aec25e">_ &gt;= _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#aac3646df53c7919a108e9925f5268d7">_ &lt;= _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::ak_scope:x_'&lt;'_x">_ &lt; _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#7fdc35ce1b6f711389e0dbafa7795fb3">\F _ (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#c590f386bdc613e117a7531b9fbcc613">[ _ , _ , _ ] (ak_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.::cantor_scope:x_'&lt;'_x">_ &lt; _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="notation_C"></a><h2>C </h2>
+<a href="gaia_hydras.nfwfgaia.html#663739e56a48dc47ea1955ef538eefc5">_ ^ _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#48d91ed92bfa929de85077cab5690e4f">_ * _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'-'_x">_ - _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#33ed9a5b2d2229e33bbdf1ef87454c0d">_ + _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&gt;'_x">_ &gt; _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#81970e39e8d1381cef754a4956da95e7">_ &gt;= _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#0ccb34c09c66b2db1577192ef152bbf9">_ &lt;= _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.::cantor_scope:x_'&lt;'_x">_ &lt; _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#511f6a25e6d530048c45997becda2220">\F _ (cantor_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="notation_D"></a><h2>D </h2>
+<a href="additions.FirstSteps.html#072addf1f952576a9d67e98c63e20939">_ ^ _</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#0f83c338fb6cc901ce3fda5df0657ac3">_ * _</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.:::'1'">1</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#56ea2d5e19744d3f0cba9664d875d7ec">_ &lt;= _</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs.:::x_'&lt;'_x">_ &lt; _</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<br/><br/><a id="notation_F"></a><h2>F </h2>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'''_x">_ &lt;-&gt;' _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'''''_x">_ &lt;-&gt;'' _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#bb04b0324214688cf30460e944283f4c">_ /\' _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#e4a489994216e94595670248e88226a2">_ \/' _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;&gt;'_x">_ &lt;&gt; _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0">_ = _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ab366141b54751aa5ea1077b9b863e32">allH _ .. _ , _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#0b83cac238622d86e6ca74792daf53c5">exH _ .. _ , _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#a12ef5915927c1f20da57c369b8b6c71">v# _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'&lt;-&gt;'_x">_ &lt;-&gt; _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#7648da2bbc0fe930b661dc530c384448">~ _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.::fol_scope:x_'-&gt;'_x">_ -&gt; _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#4ba889964d74201d72d870be149fb4e5">_ /\ _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#95d02b143473b1bf1f30676a3f27357c">_ \/ _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#3dd041be98fa842518e2776cd5aaa2a0">_ = _ (fol_scope)</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<br/><br/><a id="notation_G"></a><h2>G </h2>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'-'_x">_ - _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#513cd892ae2a37a8e1be319c411cb220">_ + _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&gt;'_x">_ &gt; _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#a8834fd6e3c1640c213de7020e0510e2">_ &gt;= _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#bed9251d191f4ad1afb65df9e03eb72e">_ &lt;= _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.::g0_scope:x_'&lt;'_x">_ &lt; _ (g0_scope)</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="additions.Monoid_instances.html#M2_def.:::'1'">1</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
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+<a href="hydras.Ackermann.codeList.html#Hrecs:130">Hrecs:130</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:133">Hrecs:133</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:136">Hrecs:136</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:139">Hrecs:139</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:14">Hrecs:14</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:142">Hrecs:142</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:145">Hrecs:145</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:147">Hrecs:147</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:149">Hrecs:149</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:15">Hrecs:15</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:151">Hrecs:151</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:153">Hrecs:153</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:155">Hrecs:155</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:16">Hrecs:16</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:165">Hrecs:165</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:168">Hrecs:168</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:17">Hrecs:17</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:171">Hrecs:171</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:174">Hrecs:174</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:177">Hrecs:177</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:18">Hrecs:18</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:180">Hrecs:180</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Hrecs:19">Hrecs:19</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:19">Hrecs:19</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:197">Hrecs:197</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:199">Hrecs:199</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:2">Hrecs:2</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:20">Hrecs:20</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:201">Hrecs:201</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:203">Hrecs:203</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:205">Hrecs:205</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:207">Hrecs:207</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Hrecs:21">Hrecs:21</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:21">Hrecs:21</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:213">Hrecs:213</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:218">Hrecs:218</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:22">Hrecs:22</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:221">Hrecs:221</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:226">Hrecs:226</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:229">Hrecs:229</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:23">Hrecs:23</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:232">Hrecs:232</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:235">Hrecs:235</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:238">Hrecs:238</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:24">Hrecs:24</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:241">Hrecs:241</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:244">Hrecs:244</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:247">Hrecs:247</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:25">Hrecs:25</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:250">Hrecs:250</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:253">Hrecs:253</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:256">Hrecs:256</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:259">Hrecs:259</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:26">Hrecs:26</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:27">Hrecs:27</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:276">Hrecs:276</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:278">Hrecs:278</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:28">Hrecs:28</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:280">Hrecs:280</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:282">Hrecs:282</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:284">Hrecs:284</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:286">Hrecs:286</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:29">Hrecs:29</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:30">Hrecs:30</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:31">Hrecs:31</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:32">Hrecs:32</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:34">Hrecs:34</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:36">Hrecs:36</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:41">Hrecs:41</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:42">Hrecs:42</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:44">Hrecs:44</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:46">Hrecs:46</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:46">Hrecs:46</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:48">Hrecs:48</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:49">Hrecs:49</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:5">Hrecs:5</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:54">Hrecs:54</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:57">Hrecs:57</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:60">Hrecs:60</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:63">Hrecs:63</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:66">Hrecs:66</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:69">Hrecs:69</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Hrecs:7">Hrecs:7</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:7">Hrecs:7</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:72">Hrecs:72</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:74">Hrecs:74</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:76">Hrecs:76</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:78">Hrecs:78</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:80">Hrecs:80</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:82">Hrecs:82</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:84">Hrecs:84</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:86">Hrecs:86</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:9">Hrecs:9</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:90">Hrecs:90</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:92">Hrecs:92</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:94">Hrecs:94</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#Hrecs:96">Hrecs:96</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hR:28">hR:28</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs':140">hs':140</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs':142">hs':142</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs:139">hs:139</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#hs:141">hs:141</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hs:8">hs:8</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hs:9">hs:9</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:18">Htrans:18</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:21">Htrans:21</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#Htrans:25">Htrans:25</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Hvar:2">Hvar:2</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Hvar:2">Hvar:2</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#Hwf:130">Hwf:130</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Hwf:21">Hwf:21</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#HX:142">HX:142</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#HX:145">HX:145</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:27">Hx:27</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:31">Hx:31</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Hx:38">Hx:38</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#HX:61">HX:61</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
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+<br/><br/><hr/>
+<h1>Module Index</h1>
+<a id="module_A"></a><h2>A </h2>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann">Ackermann</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="additions.Demo.html#Alt">Alt</a> [in <a href="additions.Demo.html">additions.Demo</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt">Alt</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alt">Alt</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
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+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec">Alt.Eps0_prec</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig">Alt.Eps0_sig</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars">Alt.Vars</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2">Alt2</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt3">Alt3</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<br/><br/><a id="module_B"></a><h2>B </h2>
+<a href="additions.FirstSteps.html#Bad">Bad</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Bad">Bad</a> [in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad">Bad</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html#Bad">Bad</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#BadSubst">BadSubst</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#BadSubstF2">BadSubstF2</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad2">Bad2</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad3">Bad3</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad4">Bad4</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="module_C"></a><h2>C </h2>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal">CantorOrdinal</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Compat815.html#Compat815">Compat815</a> [in <a href="hydras.Prelude.Compat815.html">hydras.Prelude.Compat815</a>]<br/>
+<br/><br/><a id="module_D"></a><h2>D </h2>
+<a href="additions.Monoid_def.html#Demo">Demo</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof">Direct_proof</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<br/><br/><a id="module_E"></a><h2>E </h2>
+<a href="hydras.Hydra.Hydra_Examples.html#Examples">Examples</a> [in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#Examples">Examples</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Examples">Examples</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="module_F"></a><h2>F </h2>
+<a href="gaia_hydras.GHessenberg.html#FixpointDef">FixpointDef</a> [in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations">FolNotations</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<br/><br/><a id="module_G"></a><h2>G </h2>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0">Gamma0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_rpo">Gamma0_rpo</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_alg">Gamma0_alg</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec">Gamma0_prec</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig">Gamma0_sig</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0">G0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="module_I"></a><h2>I </h2>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad">iota_demo.Bad</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo">iota_demo</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="module_L"></a><h2>L </h2>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations">LispAbbreviations</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#LNT">LNT</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO">LO</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<br/><br/><a id="module_M"></a><h2>M </h2>
+<a href="hydras.rpo.term.html#Make">Make</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make">Make</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make">Make</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make">Make</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make">Make</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DecVar">Make.DecVar</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.DS">Make.DS</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.DS">Make.DS</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.DS">Make.DS</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.F">Make.F</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.LP">Make.LP</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.LP">Make.LP</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.LP">Make.LP</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.P">Make.P</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.T">Make.T</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term_eq_dec">Make.Term_eq_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.VSet">Make.VSet</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.X">Make.X</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#MoreExamples">MoreExamples</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.FirstSteps.html#M2">M2</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2">M2</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="module_N"></a><h2>N </h2>
+<a href="hydras.Ackermann.LNN.html#NNnotations">NNnotations</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="additions.Naive.html#N_mod">N_mod</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="module_O"></a><h2>O </h2>
+<a href="gaia_hydras.onType.html#ONDef">ONDef</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports">ONDef.Exports</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO">OO</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#OpaqueWf">OpaqueWf</a> [in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<br/><br/><a id="module_P"></a><h2>P </h2>
+<a href="hydras.rpo.list_permut.html#Permut">Permut</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.DS">Permut.DS</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence">Precedence</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations">PRNotations</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="module_R"></a><h2>R </h2>
+<a href="hydras.rpo.rpo.html#RPO">RPO</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.LP">RPO.LP</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.P">RPO.P</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.T">RPO.T</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<br/><br/><a id="module_S"></a><h2>S </h2>
+<a href="hydras.rpo.decidable_set.html#S">S</a> [in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S">S</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled">Semibundled</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.rpo.term.html#Signature">Signature</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.DS">S.DS</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<br/><br/><a id="module_T"></a><h2>T </h2>
+<a href="hydras.rpo.term.html#Term">Term</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.F">Term.F</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Term_eq_dec">Term.Term_eq_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.X">Term.X</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy">Toy</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<br/><br/><a id="module_U"></a><h2>U </h2>
+<a href="Goedel.PRrepresentable.html#Usual">Usual</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<br/><br/><a id="module_V"></a><h2>V </h2>
+<a href="hydras.rpo.term.html#Variables">Variables</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Vars">Vars</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="module_W"></a><h2>W </h2>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum">Wfsum</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3">Wf_ex3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex">Wf_ex</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="module_Z"></a><h2>Z </h2>
+<a href="additions.Naive.html#Z">Z</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><hr/>
+<h1>Variable Index</h1>
+<a id="variable_A"></a><h2>A </h2>
+<a href="hydras.Epsilon0.Hprime.html#Abstract_Properties.alpha">Abstract_Properties.alpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun.Rf">AB_given.rel_to_fun.Rf</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given.Rf">AB_given.f_given.Rf</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given.f">AB_given.f_given.f</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.DB">AB_given.DB</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.DA">AB_given.DA</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.Hb">AB_given.Hb</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.Ha">AB_given.Ha</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.B">AB_given.B</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.A">AB_given.A</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.Hm">Ack_Properties.Induc_step.Hm</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step.m">Ack_Properties.Induc_step.m</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.P">Ack_Properties.P</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded.R">Alt.well_founded.R</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.denM">AP_closed.denM</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.inhM">AP_closed.inhM</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.OM">AP_closed.OM</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed.M">AP_closed.M</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.n">AP_Unbounded.ksi_fixed.n</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.lt_ksi">AP_Unbounded.ksi_fixed.lt_ksi</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed.ksi">AP_Unbounded.ksi_fixed.ksi</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.beta">AP_Unbounded.beta</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.seq">AP_Unbounded.seq</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.alpha">AP_Unbounded.alpha</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A_def.m">A_def.m</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given.LtA">A_given.LtA</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given.A">A_given.A</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<br/><br/><a id="variable_B"></a><h2>B </h2>
+<a href="hydras.MoreAck.BadSubst.html#BadExample.F">BadExample.F</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#BadExample.F1">BadExample.F1</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.CounterExample.mul">Bad.CounterExample.mul</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.CounterExample.one">Bad.CounterExample.one</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.m">battle_length_notPR.m</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.h">battle_length_notPR.h</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR.alpha">battle_length_notPR.alpha</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length.l">Battle_length.l</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length.Hk">Battle_length.Hk</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length.k">Battle_length.k</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length.Halpha">Battle_length.Halpha</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length.alpha">Battle_length.alpha</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_eq">binary_power_proof.E_eq</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_one">binary_power_proof.E_one</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.E_op">binary_power_proof.E_op</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof.A">binary_power_proof.A</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded.m_decrease">Bounded.m_decrease</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction.B">building_ordering_function_by_induction.B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.g_1">building_ordering_function_1.g_1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta_B">building_ordering_function_1.beta_fixed.beta_B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed.beta">building_ordering_function_1.beta_fixed.beta</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.H_B">building_ordering_function_1.H_B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.B">building_ordering_function_1.B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="variable_C"></a><h2>C </h2>
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+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedFormula">Check_Proof.wellFormedFormula</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.wellFormedTerm">Check_Proof.wellFormedTerm</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Formula">Check_Proof.Formula</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Terms">Check_Proof.Terms</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.Term">Check_Proof.Term</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeRInj">Check_Proof.codeRInj</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeFInj">Check_Proof.codeFInj</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityRIsCorrect2">Check_Proof.codeArityRIsCorrect2</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityRIsCorrect1">Check_Proof.codeArityRIsCorrect1</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityFIsCorrect2">Check_Proof.codeArityFIsCorrect2</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityFIsCorrect1">Check_Proof.codeArityFIsCorrect1</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityR">Check_Proof.codeArityR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.codeArityF">Check_Proof.codeArityF</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof.L">Check_Proof.L</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.codeFormula">close.codeFormula</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.Formula">close.Formula</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close.L">close.L</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Terms">Code_Free_Vars.Terms</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.Term">Code_Free_Vars.Term</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars.System">Code_Free_Vars.System</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
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+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula.L">Code_Substitute_Formula.L</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Prf">Code_Term_Formula_Proof.Prf</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Terms">Code_Term_Formula_Proof.Terms</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Term">Code_Term_Formula_Proof.Term</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
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+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formulas">Code_Term_Formula_Proof.Formulas</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.Formula">Code_Term_Formula_Proof.Formula</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cR">Code_Term_Formula_Proof.LcodeDef.cR</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef.cF">Code_Term_Formula_Proof.LcodeDef.cF</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.L">Code_Term_Formula_Proof.L</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
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+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.u">Composition.u</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.v">Composition.v</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.x">Composition.x</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.y">Composition.y</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition.z">Composition.z</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.m">Constant_to_standard_Proof.m</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.gamma">Constant_to_standard_Proof.gamma</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l">Constant_to_standard_Proof.l</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.l_def">Constant_to_standard_Proof.l_def</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.P">Constant_to_standard_Proof.P</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.t">Constant_to_standard_Proof.t</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Hpa">Constant_to_standard_Proof.Hpa</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Hpos">Constant_to_standard_Proof.Hpos</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof.Halpha">Constant_to_standard_Proof.Halpha</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#Correctness.L">Correctness.L</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.P">Correctness.P</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.Q">Correctness.Q</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness.R">Correctness.R</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="hydras.Schutte.Countable.html#Countable.U">Countable.U</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
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+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example.Hwf">Counter_Example.Hwf</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
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+<br/><br/><a id="variable_D"></a><h2>D </h2>
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+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem.Formulas">Deduction_Theorem.Formulas</a> [in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
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+<a href="additions.FirstSteps.html#Definitions.A">Definitions.A</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
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+<a href="additions.FirstSteps.html#Definitions.mult">Definitions.mult</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions.one">Definitions.one</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
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+<a href="additions.Compatibility.html#Equivalence.one">Equivalence.one</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.op">Equivalence.op</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence.pos_iter_M">Equivalence.pos_iter_M</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#essai.n">essai.n</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Examples.f">Examples.f</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f1">Examples.f1</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f2">Examples.f2</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f2'">Examples.f2'</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f3">Examples.f3</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.f4">Examples.f4</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples.t1">Examples.t1</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs.f">Exs.f</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs.fdiv2">Exs.fdiv2</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs.ffib">Exs.ffib</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="variable_F"></a><h2>F </h2>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.consTermsHelp">First_Order_Logic.Formula_Decidability.consTermsHelp</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.nilTermsHelp">First_Order_Logic.Formula_Decidability.nilTermsHelp</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability.language_eqdec">First_Order_Logic.Formula_Decidability.language_eqdec</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.L">First_Order_Logic.L</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.A">Fix.A</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.E">Fix.E</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.F">Fix.F</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.Hwf">Fix.Hwf</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.P">Fix.P</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.R">Fix.R</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix.Rwf">Fix.Rwf</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.lt_depth">Fol_Properties.lt_depth</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.L">Fol_Properties.L</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.Hd">F_monotony_l.case_lt.Hd</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.Hlt">F_monotony_l.case_lt.Hlt</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt.n">F_monotony_l.case_lt.n</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_eq.Heq">F_monotony_l.case_eq.Heq</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.H'_beta_alpha">F_monotony_l.H'_beta_alpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.beta">F_monotony_l.beta</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.alpha">F_monotony_l.alpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.H">F_alpha_notPR.H</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.case_lt.Halpha">F_alpha_notPR.case_lt.Halpha</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.alpha">F_alpha_notPR.alpha</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<br/><br/><a id="variable_G"></a><h2>G </h2>
+<a href="additions.AM.html#Gamma.All_OK.c">Gamma.All_OK.c</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Gamma.All_OK.n">Gamma.All_OK.n</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Gamma.gamma">Gamma.gamma</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Gamma.gamma">Gamma.gamma</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage.R">General_Relations.elagage.to_nat_elagage.R</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R_surj">General_Relations.surjection2injection.R_surj</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection.R">General_Relations.surjection2injection.R</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R_inj">General_Relations.injection2surjection.R_inj</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection.R">General_Relations.injection2surjection.R</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.DB">General_Relations.DB</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.DA">General_Relations.DA</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.B">General_Relations.B</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.A">General_Relations.A</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.R">General_Relations.Definitions.R</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DB">General_Relations.Definitions.DB</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.DA">General_Relations.Definitions.DA</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.B">General_Relations.Definitions.B</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions.A">General_Relations.Definitions.A</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.A">Generic.A</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split_permutation">Generic.Merging.Correctness.merge_sort.split_permutation</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split_decr">Generic.Merging.Correctness.merge_sort.split_decr</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort.split">Generic.Merging.Correctness.merge_sort.split</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.le">Generic.Merging.le</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT2">Goedel's_1st_Incompleteness.expressT2</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.expressT1">Goedel's_1st_Incompleteness.expressT1</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.freeVarRepT">Goedel's_1st_Incompleteness.freeVarRepT</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.v0">Goedel's_1st_Incompleteness.v0</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.repT">Goedel's_1st_Incompleteness.repT</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.extendsNN">Goedel's_1st_Incompleteness.extendsNN</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness.T">Goedel's_1st_Incompleteness.T</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.HBL3">Goedel's_2nd_Incompleteness.HBL3</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.HBL2">Goedel's_2nd_Incompleteness.HBL2</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT2">Goedel's_2nd_Incompleteness.expressT2</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.expressT1">Goedel's_2nd_Incompleteness.expressT1</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.freeVarRepT">Goedel's_2nd_Incompleteness.freeVarRepT</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.v0">Goedel's_2nd_Incompleteness.v0</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.repT">Goedel's_2nd_Incompleteness.repT</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.extendsPA">Goedel's_2nd_Incompleteness.extendsPA</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.T">Goedel's_2nd_Incompleteness.T</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<br/><br/><a id="variable_H"></a><h2>H </h2>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hn">Hypos.Hn</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hnp">Hypos.Hnp</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.Hp">Hypos.Hp</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.R">Hypos.R</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.search_toggle">Hypos.search_toggle</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Hypos.spec">Hypos.spec</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.IHalpha">H'_F.IHalpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.alpha">H'_F.alpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F.P">H'_F.P</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.N">H'_omega_cube_3.N</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3.f">H'_omega_cube_3.f</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons.i">H'_cons.i</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons.alpha">H'_cons.alpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<br/><br/><a id="variable_I"></a><h2>I </h2>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.m">ImpossibilityProof.m</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.mh">ImpossibilityProof.mh</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.mu">ImpossibilityProof.mu</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof.nfMu">ImpossibilityProof.nfMu</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.iota">Impossibility_Proof.iota</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof.m">Impossibility_Proof.m</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.small_h">Impossibility_Proof.small_h</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.big_h">Impossibility_Proof.big_h</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.iota">Impossibility_Proof.iota</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof.m">Impossibility_Proof.m</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.m">Impossibility1.m</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.mDecr">Impossibility1.mDecr</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.x">Impossibility1.x</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1.y">Impossibility1.y</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.b">Inclusion_ij.b</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.Hij">Inclusion_ij.Hij</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.j">Inclusion_ij.j</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij.i">Inclusion_ij.i</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step.Hn">inductive_step.Hn</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step.n">inductive_step.n</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.Rof">Inverse_Image_transp.Rof</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.f">Inverse_Image_transp.f</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.R">Inverse_Image_transp.R</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.B">Inverse_Image_transp.B</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp.A">Inverse_Image_transp.A</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f_b">inversion_of_bijection.f_b</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inv_spec">inversion_of_bijection.inv_spec</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.f">inversion_of_bijection.f</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DB">inversion_of_bijection.DB</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.DA">inversion_of_bijection.DA</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.inhA">inversion_of_bijection.inhA</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.B">inversion_of_bijection.B</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection.A">inversion_of_bijection.A</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<br/><br/><a id="variable_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hq_prop">KFK_proof.Hq_prop</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hbr_prop">KFK_proof.Hbr_prop</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hq">KFK_proof.Hq</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.Hbr">KFK_proof.Hbr</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.fq">KFK_proof.fq</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.kbr">KFK_proof.kbr</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.r">KFK_proof.r</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.q">KFK_proof.q</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proof.b">KFK_proof.b</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#KP.h'">KP.h'</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<br/><br/><a id="variable_L"></a><h2>L </h2>
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+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.H">Lemma_4_4_Proof.H</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Hs">Lemma_4_4_Proof.Hs</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.s">Lemma_4_4_Proof.s</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.n">Lemma_4_4_Proof.n</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Hbeta">Lemma_4_4_Proof.Hbeta</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.Halpha">Lemma_4_4_Proof.Halpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.beta">Lemma_4_4_Proof.beta</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof.alpha">Lemma_4_4_Proof.alpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H4">Lemma_4_3_Proof.H4</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H1">Lemma_4_3_Proof.H1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H0">Lemma_4_3_Proof.H0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n2">Lemma_4_3_Proof.n2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n1">Lemma_4_3_Proof.n1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.n0">Lemma_4_3_Proof.n0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.nf2">Lemma_4_3_Proof.nf2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.nf1">Lemma_4_3_Proof.nf1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.H00">Lemma_4_3_Proof.H00</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.beta">Lemma_4_3_Proof.beta</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof.alpha">Lemma_4_3_Proof.alpha</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex1">LExamples.ex1</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex2">LExamples.ex2</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.ex3">LExamples.ex3</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.L">LExamples.L</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.P">LExamples.P</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples.Q">LExamples.Q</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.A">lexprod.A</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.B">lexprod.B</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leA">lexprod.leA</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod.leB">lexprod.leB</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.f">lim.f</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.H">lim.H</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.Hlim">lim.Hlim</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.Hnf">lim.Hnf</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim.lambda">lim.lambda</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove.Aeq_dec">List_Remove.Aeq_dec</a> [in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove.A">List_Remove.A</a> [in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="Goedel.fixPoint.html#LNN_FixPoint.codeFormula">LNN_FixPoint.codeFormula</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#LNT_FixPoint.codeFormula">LNT_FixPoint.codeFormula</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
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+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Prf">Logic_Rules.Prf</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Term">Logic_Rules.Term</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.L">Logic_Rules.L</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.rpo.term.html#Make.Recursion.Pl">Make.Recursion.Pl</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.Hlim">MoreOrderType.Succ_no_limit.Hlim</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
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+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.s">MoreOrderType.Succ_no_limit.s</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.y">MoreOrderType.Succ_no_limit.y</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit.x">MoreOrderType.Succ_no_limit.x</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.T">MoreOrderType.T</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.SysPrf">More_Logic_Rules.SysPrf</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Prf">More_Logic_Rules.Prf</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Terms">More_Logic_Rules.Terms</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Term">More_Logic_Rules.Term</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.System">More_Logic_Rules.System</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formulas">More_Logic_Rules.Formulas</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.Formula">More_Logic_Rules.Formula</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules.L">More_Logic_Rules.L</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_eq">M_given.E_eq</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_one">M_given.E_one</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.E_op">M_given.E_op</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.A">M_given.A</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#M_given.E_one">M_given.E_one</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#M_given.A">M_given.A</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.rt">M2_def.rt</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.mult">M2_def.mult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.plus">M2_def.plus</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.one">M2_def.one</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.zero">M2_def.zero</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def.A">M2_def.A</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.A">M2.Definitions.A</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.mult">M2.Definitions.mult</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.one">M2.Definitions.one</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.plus">M2.Definitions.plus</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.rt">M2.Definitions.rt</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions.zero">M2.Definitions.zero</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.rt">M2.M2_Definitions.rt</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.mult">M2.M2_Definitions.mult</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.plus">M2.M2_Definitions.plus</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.one">M2.M2_Definitions.one</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.zero">M2.M2_Definitions.zero</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions.A">M2.M2_Definitions.A</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<br/><br/><a id="variable_N"></a><h2>N </h2>
+<a href="additions.Monoid_instances.html#Nmodulo.m">Nmodulo.m</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmodulo.m_gt_1">Nmodulo.m_gt_1</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Addition_Chains.html#non_optimality_proof.binary_opt">non_optimality_proof.binary_opt</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Naive.html#N_mod.m_fixed.m">N_mod.m_fixed.m</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="variable_O"></a><h2>O </h2>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.disp">ONDef.Exports.Lemmas.disp</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas.U">ONDef.Exports.Lemmas.U</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing.cT">ONDef.Packing.cT</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing.disp">ONDef.Packing.disp</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteDef.i">onFiniteDef.i</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S2.H">On_alpha.S2.H</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.Hbeta">On_alpha.S1.Hbeta</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.beta">On_alpha.S1.beta</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1.H">On_alpha.S1.H</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.HnonZero">On_alpha.HnonZero</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.Halpha">On_alpha.Halpha</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.alpha">On_alpha.alpha</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.unique_P">On_Iota.unique_P</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.inhA">On_Iota.inhA</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.P">On_Iota.P</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota.A">On_Iota.A</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.O2">ordering_function_unicity.O2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.O1">ordering_function_unicity.O1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f2">ordering_function_unicity.f2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.f1">ordering_function_unicity.f1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A2">ordering_function_unicity.A2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.A1">ordering_function_unicity.A1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity.B">ordering_function_unicity.B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.O2">ordering_function_unicity_1.O2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.O1">ordering_function_unicity_1.O1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.f2">ordering_function_unicity_1.f2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.f1">ordering_function_unicity_1.f1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.A2">ordering_function_unicity_1.A2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.A1">ordering_function_unicity_1.A1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1.B">ordering_function_unicity_1.B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="variable_P"></a><h2>P </h2>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi.alpha">phi_to_psi.alpha</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_ok">phi0_mult.f_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_Sle">phi0_mult.f_Sle</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f_mono">phi0_mult.f_mono</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.Halpha">phi0_mult.Halpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.f">phi0_mult.f</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult.alpha">phi0_mult.alpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.f">Primitive_recursion.f</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.h">Primitive_recursion.h</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.g">Primitive_recursion.g</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.z">Primitive_recursion.z</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.y">Primitive_recursion.y</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion.x">Primitive_recursion.x</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Formula">ProofH.Formula</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Formulas">ProofH.Formulas</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.L">ProofH.L</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.System">ProofH.System</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Term">ProofH.Term</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Terms">ProofH.Terms</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.A">ProofOfLexwf.A</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.Accs">ProofOfLexwf.Accs</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.HwfA">ProofOfLexwf.HwfA</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf.ltA">ProofOfLexwf.ltA</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.Hgammaalpha">Proofs_of_unicity.Proofs.S2.Hgammaalpha</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2.gamma">Proofs_of_unicity.Proofs.S2.gamma</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.Halphagamma">Proofs_of_unicity.Proofs.S1.Halphagamma</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1.gamma">Proofs_of_unicity.Proofs.S1.gamma</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.Halphabeta">Proofs_of_unicity.Proofs.Halphabeta</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.beta">Proofs_of_unicity.Proofs.beta</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.alpha">Proofs_of_unicity.Proofs.alpha</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta">Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1.gamma">Proofs_of_lt_succ_le.Proofs.S1.gamma</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.Halphabeta">Proofs_of_lt_succ_le.Proofs.Halphabeta</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.beta">Proofs_of_lt_succ_le.Proofs.beta</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.alpha">Proofs_of_lt_succ_le.Proofs.alpha</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_dist.P">Proof_of_dist.P</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.IHbeta">Proof_of_mult_nf.Induction.IHbeta</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.Hbeta">Proof_of_mult_nf.Induction.Hbeta</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction.beta">Proof_of_mult_nf.Induction.beta</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.P">Proof_of_mult_nf.P</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Halpha">Proof_of_mult_nf.Halpha</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.alpha">Proof_of_mult_nf.alpha</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR.minPR">Proof_of_MinIsPR.minPR</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.H0">Proof_of_oplus_lt1.H0</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.n">Proof_of_oplus_lt1.n</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a2">Proof_of_oplus_lt1.a2</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1.a1">Proof_of_oplus_lt1.a1</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step.n">Proof_of_Ackn_PR.S_step.n</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HlargeA">Proof_of_4_5_2.HlargeA</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HAB1">Proof_of_4_5_2.HAB1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HB">Proof_of_4_5_2.HB</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.HA">Proof_of_4_5_2.HA</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.Halpha">Proof_of_4_5_2.Halpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.alpha">Proof_of_4_5_2.alpha</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B2">Proof_of_4_5_2.B2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.B1">Proof_of_4_5_2.B1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2.A">Proof_of_4_5_2.A</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.IM">Proof_of_Lemma5.Alpha_positive.closedness.IM</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.CM">Proof_of_Lemma5.Alpha_positive.closedness.CM</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.HM">Proof_of_Lemma5.Alpha_positive.closedness.HM</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness.M">Proof_of_Lemma5.Alpha_positive.closedness.M</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma">Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta">Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.IHalpha">Proof_of_Lemma5.Alpha_positive.IHalpha</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha_pos">Proof_of_Lemma5.Alpha_positive.alpha_pos</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.alpha">Proof_of_Lemma5.Alpha_positive.alpha</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.P">Proof_of_Lemma5.P</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter_cPair">Proof_of_FibIsPR.fib_iter_cPair</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step_cPair">Proof_of_FibIsPR.fib_step_cPair</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_iter">Proof_of_FibIsPR.fib_iter</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR.fib_step">Proof_of_FibIsPR.fib_step</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR.min_alt">Proof_of_MinIsPR.min_alt</a> [in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Limit_case.Hbeta">Proof_of_H'_mono_l.Limit_case.Hbeta</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.Hgamma">Proof_of_H'_mono_l.Succ_case.Hgamma</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case.gamma">Proof_of_H'_mono_l.Succ_case.gamma</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.H_alpha_beta">Proof_of_H'_mono_l.H_alpha_beta</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.beta">Proof_of_H'_mono_l.beta</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.alpha">Proof_of_H'_mono_l.alpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim">Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def">Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta">Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.Halpha">Proof_of_Abstract_Properties.The_induction.Halpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha">Proof_of_Abstract_Properties.The_induction.alpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.g_alpha_beta">proof_of_associativity.g_alpha_beta</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.f_alpha_beta">proof_of_associativity.f_alpha_beta</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.beta">proof_of_associativity.beta</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity.alpha">proof_of_associativity.alpha</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.is_in_seq">Proof_of_lt_not_wf.seq_intro.is_in_seq</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.Rwf">Proof_of_lt_not_wf.seq_intro.Rwf</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.R">Proof_of_lt_not_wf.seq_intro.R</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.seq">Proof_of_lt_not_wf.seq_intro.seq</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro.A">Proof_of_lt_not_wf.seq_intro.A</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.lt_wf">Proof_of_lt_not_wf.lt_wf</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_limit.Hlim">Properties.The_induction.alpha_limit.Hlim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ.alpha_def">Properties.The_induction.alpha_Succ.alpha_def</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ.beta">Properties.The_induction.alpha_Succ.beta</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.Halpha">Properties.The_induction.Halpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha">Properties.The_induction.alpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<br/><br/><a id="variable_R"></a><h2>R </h2>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_eq">Refinement_proof.E_eq</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_one">Refinement_proof.E_one</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.E_op">Refinement_proof.E_op</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#Refinement_proof.A">Refinement_proof.A</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Formula">Replacement.Formula</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Formulas">Replacement.Formulas</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.L">Replacement.L</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.SysPrf">Replacement.SysPrf</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.System">Replacement.System</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Term">Replacement.Term</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement.Terms">Replacement.Terms</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.closedT1">RepresentableExpressible.closedT1</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.nn1">RepresentableExpressible.nn1</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible.T">RepresentableExpressible.T</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.R">restricted_recursion.R</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.E">restricted_recursion.E</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion.A">restricted_recursion.A</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.R">restricted_recursion.R</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.P">restricted_recursion.P</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion.A">restricted_recursion.A</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.expressT2">Rosser's_Incompleteness.expressT2</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.expressT1">Rosser's_Incompleteness.expressT1</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.freeVarRepT">Rosser's_Incompleteness.freeVarRepT</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.v0">Rosser's_Incompleteness.v0</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.repT">Rosser's_Incompleteness.repT</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.extendsPA">Rosser's_Incompleteness.extendsPA</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness.T">Rosser's_Incompleteness.T</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.expressT2">Rosser's_Incompleteness.expressT2</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.expressT1">Rosser's_Incompleteness.expressT1</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.freeVarRepT">Rosser's_Incompleteness.freeVarRepT</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.v0">Rosser's_Incompleteness.v0</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.repT">Rosser's_Incompleteness.repT</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.extendsNN">Rosser's_Incompleteness.extendsNN</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness.T">Rosser's_Incompleteness.T</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given.R">R_given.R</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given.A">R_given.A</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<br/><br/><a id="variable_S"></a><h2>S </h2>
+<a href="additions.AM.html#Semantics.A">Semantics.A</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Semantics.mul">Semantics.mul</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Semantics.one">Semantics.one</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.A">Sequences.A</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.R">Sequences.R</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences.W">Sequences.W</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.P">sqrtIsPR.P</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt.n">sqrtIsPR.Proof_isqrt.n</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.F">step.F</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.Hn">step.Hn</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step.n">step.n</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars.L">SubAllVars.L</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.SysPrf">Substitution_Properties.SysPrf</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Terms">Substitution_Properties.Terms</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Term">Substitution_Properties.Term</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.System">Substitution_Properties.System</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Formulas">Substitution_Properties.Formulas</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.Formula">Substitution_Properties.Formula</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties.L">Substitution_Properties.L</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.beta">succ.beta</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f">succ.f</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_ok">succ.f_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_Sle">succ.f_Sle</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.f_mono">succ.f_mono</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ.Hbeta">succ.Hbeta</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#S1.H">S1.H</a> [in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#S1.instance_H">S1.instance_H</a> [in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.Hlambda">S1.Limit.Hlambda</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.IHlambda">S1.Limit.IHlambda</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.lambda">S1.Limit.lambda</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.S3.Hn">S1.Limit.S3.Hn</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.S3.n">S1.Limit.S3.n</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.P">S1.P</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor.alpha">S1.Successor.alpha</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor.H_alpha">S1.Successor.H_alpha</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.Hn">S1.Successor.S2.Hn</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor.S2.n">S1.Successor.S2.n</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#S2.a">S2.a</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#S2.A">S2.A</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#S2.b">S2.b</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#S2.B">S2.B</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#S2.R_true">S2.R_true</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html#S256.mod256">S256.mod256</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><a id="variable_T"></a><h2>T </h2>
+<a href="hydras.rpo.term.html#Term.DoubleRecursion.Pl2">Term.DoubleRecursion.Pl2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.DoubleRecursion.P2">Term.DoubleRecursion.P2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Recursion.P">Term.Recursion.P</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Recursion.Pl">Term.Recursion.Pl</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.Lt">the_context.Lt</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.M">the_context.M</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.alpha_">Th13_5.verso.U_fixed.alpha_</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_inc_A">Th13_5.verso.U_fixed.U_inc_A</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_den">Th13_5.verso.U_fixed.U_den</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U_non_empty">Th13_5.verso.U_fixed.U_non_empty</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed.U">Th13_5.verso.U_fixed.U</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.B_closed">Th13_5.verso.B_closed</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.U">Th13_5.recto.M_fixed.U</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.den">Th13_5.recto.M_fixed.den</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.ne">Th13_5.recto.M_fixed.ne</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.inc">Th13_5.recto.M_fixed.inc</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed.M">Th13_5.recto.M_fixed.M</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.f_cont">Th13_5.recto.f_cont</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.f_ord">Th13_5.f_ord</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.f">Th13_5.f</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.B">Th13_5.B</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.A">Th13_5.A</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Drinkers_theorem.f">Toy.Drinkers_theorem.f</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.OnSubstF.F">Toy.OnSubstF.F</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Translate_Proof.AxiomsOK">Translate_Proof.AxiomsOK</a> [in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Translate_Proof.V">Translate_Proof.V</a> [in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Translate_Proof.U">Translate_Proof.U</a> [in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.induc">trans_proof.induc</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.H23">trans_proof.H23</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.H12">trans_proof.H12</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.n3">trans_proof.n3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.n2">trans_proof.n2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.n1">trans_proof.n1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.c3">trans_proof.c3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.b3">trans_proof.b3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.a3">trans_proof.a3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.c2">trans_proof.c2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.b2">trans_proof.b2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.a2">trans_proof.a2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.c1">trans_proof.c1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.b1">trans_proof.b1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof.a1">trans_proof.a1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="variable_V"></a><h2>V </h2>
+<a href="hydras.Prelude.WfVariant.html#Variants.E">Variants.E</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#Variants.lt">Variants.lt</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#Variants.m">Variants.m</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#Variants.T">Variants.T</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#Variants.tr">Variants.tr</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<br/><br/><a id="variable_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.Formula">Well_Formed_Term.Well_Formed_Formula.Formula</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2">Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1">Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula.codeArityR">Well_Formed_Term.Well_Formed_Formula.codeArityR</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Terms">Well_Formed_Term.Terms</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Term">Well_Formed_Term.Term</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityFIsCorrect2">Well_Formed_Term.codeArityFIsCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityFIsCorrect1">Well_Formed_Term.codeArityFIsCorrect1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.codeArityF">Well_Formed_Term.codeArityF</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.L">Well_Formed_Term.L</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded.R">well_founded.R</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.fib.html#with_matrices.R">with_matrices.R</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<br/><br/><hr/>
+<h1>Library Index</h1>
+<a id="library_A"></a><h2>A </h2>
+<a href="hydras.MoreAck.Ack.html">Ack</a> <br/>
+<a href="hydras.MoreAck.AckNotPR.html">AckNotPR</a> <br/>
+<a href="hydras.Schutte.Addition.html">Addition</a> <br/>
+<a href="additions.Addition_Chains.html">Addition_Chains</a> <br/>
+<a href="additions.AM.html">AM</a> <br/>
+<a href="hydras.Schutte.AP.html">AP</a> <br/>
+<br/><br/><a id="library_B"></a><h2>B </h2>
+<a href="hydras.MoreAck.BadSubst.html">BadSubst</a> <br/>
+<a href="hydras.Hydra.Battle_length.html">Battle_length</a> <br/>
+<a href="hydras.Hydra.BigBattle.html">BigBattle</a> <br/>
+<a href="additions.BinaryStrat.html">BinaryStrat</a> <br/>
+<br/><br/><a id="library_C"></a><h2>C </h2>
+<a href="hydras.Epsilon0.Canon.html">Canon</a> <br/>
+<a href="hydras.Ackermann.checkPrf.html">checkPrf</a> <br/>
+<a href="hydras.rpo.closure.html">closure</a> <br/>
+<a href="hydras.Schutte.CNF.html">CNF</a> <br/>
+<a href="hydras.Ackermann.code.html">code</a> <br/>
+<a href="hydras.Ackermann.codeFreeVar.html">codeFreeVar</a> <br/>
+<a href="hydras.Ackermann.codeList.html">codeList</a> <br/>
+<a href="hydras.Ackermann.codeNatToTerm.html">codeNatToTerm</a> <br/>
+<a href="hydras.Ackermann.codePA.html">codePA</a> <br/>
+<a href="hydras.Ackermann.codeSubFormula.html">codeSubFormula</a> <br/>
+<a href="hydras.Ackermann.codeSubTerm.html">codeSubTerm</a> <br/>
+<a href="Goedel.codeSysPrf.html">codeSysPrf</a> <br/>
+<a href="hydras.Prelude.Comparable.html">Comparable</a> <br/>
+<a href="additions.Compatibility.html">Compatibility</a> <br/>
+<a href="hydras.Prelude.Compat815.html">Compat815</a> <br/>
+<a href="hydras.Schutte.Correctness_E0.html">Correctness_E0</a> <br/>
+<a href="hydras.Schutte.Countable.html">Countable</a> <br/>
+<a href="hydras.Ackermann.cPair.html">cPair</a> <br/>
+<a href="hydras.Schutte.Critical.html">Critical</a> <br/>
+<br/><br/><a id="library_D"></a><h2>D </h2>
+<a href="hydras.rpo.decidable_set.html">decidable_set</a> <br/>
+<a href="hydras.Prelude.DecPreOrder.html">DecPreOrder</a> <br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html">DecPreOrder_Instances</a> <br/>
+<a href="hydras.Ackermann.Deduction.html">Deduction</a> <br/>
+<a href="additions.Demo.html">Demo</a> <br/>
+<a href="additions.Demo_power.html">Demo_power</a> <br/>
+<a href="additions.Dichotomy.html">Dichotomy</a> <br/>
+<a href="hydras.rpo.dickson.html">dickson</a> <br/>
+<br/><br/><a id="library_E"></a><h2>E </h2>
+<a href="hydras.Epsilon0.Epsilon0.html">Epsilon0</a> <br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html">Epsilon0rpo</a> <br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html">Epsilon0_Needed_Std</a> <br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html">Epsilon0_Needed_Free</a> <br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html">Epsilon0_Needed_Generic</a> <br/>
+<a href="additions.Euclidean_Chains.html">Euclidean_Chains</a> <br/>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html">Example_3PlusOmega</a> <br/>
+<a href="hydras.Ackermann.expressible.html">expressible</a> <br/>
+<a href="hydras.MoreAck.expressibleExamples.html">expressibleExamples</a> <br/>
+<a href="hydras.Prelude.Exp2.html">Exp2</a> <br/>
+<a href="hydras.Ackermann.extEqualNat.html">extEqualNat</a> <br/>
+<a href="hydras.Epsilon0.E0.html">E0</a> <br/>
+<br/><br/><a id="library_F"></a><h2>F </h2>
+<a href="additions.fib.html">fib</a> <br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html">FibonacciPR</a> <br/>
+<a href="additions.Fib2.html">Fib2</a> <br/>
+<a href="additions.FirstSteps.html">FirstSteps</a> <br/>
+<a href="hydras.Prelude.First_toggle.html">First_toggle</a> <br/>
+<a href="Goedel.fixPoint.html">fixPoint</a> <br/>
+<a href="hydras.Ackermann.fol.html">fol</a> <br/>
+<a href="hydras.MoreAck.FolExamples.html">FolExamples</a> <br/>
+<a href="hydras.Ackermann.folLogic.html">folLogic</a> <br/>
+<a href="hydras.Ackermann.folLogic2.html">folLogic2</a> <br/>
+<a href="hydras.Ackermann.folLogic3.html">folLogic3</a> <br/>
+<a href="hydras.Ackermann.folProof.html">folProof</a> <br/>
+<a href="hydras.Ackermann.folProp.html">folProp</a> <br/>
+<a href="hydras.Ackermann.folReplace.html">folReplace</a> <br/>
+<a href="hydras.Prelude.Fuel.html">Fuel</a> <br/>
+<a href="hydras.solutions_exercises.F_3.html">F_3</a> <br/>
+<a href="hydras.Epsilon0.F_alpha.html">F_alpha</a> <br/>
+<a href="hydras.Epsilon0.F_omega.html">F_omega</a> <br/>
+<br/><br/><a id="library_G"></a><h2>G </h2>
+<a href="gaia_hydras.GaiaToHydra.html">GaiaToHydra</a> <br/>
+<a href="hydras.Gamma0.Gamma0.html">Gamma0</a> <br/>
+<a href="gaia_hydras.GCanon.html">GCanon</a> <br/>
+<a href="hydras.solutions_exercises.ge_omega_iff.html">ge_omega_iff</a> <br/>
+<a href="gaia_hydras.GF_alpha.html">GF_alpha</a> <br/>
+<a href="gaia_hydras.GHessenberg.html">GHessenberg</a> <br/>
+<a href="gaia_hydras.GHprime.html">GHprime</a> <br/>
+<a href="gaia_hydras.GHydra.html">GHydra</a> <br/>
+<a href="gaia_hydras.GLarge_Sets.html">GLarge_Sets</a> <br/>
+<a href="gaia_hydras.GL_alpha.html">GL_alpha</a> <br/>
+<a href="Goedel.goedel1.html">goedel1</a> <br/>
+<a href="Goedel.goedel2.html">goedel2</a> <br/>
+<a href="gaia_hydras.GPaths.html">GPaths</a> <br/>
+<a href="gaia_hydras.GPrelude.html">GPrelude</a> <br/>
+<a href="hydras.Schutte.GRelations.html">GRelations</a> <br/>
+<br/><br/><a id="library_H"></a><h2>H </h2>
+<a href="hydras.Epsilon0.Hessenberg.html">Hessenberg</a> <br/>
+<a href="hydras.Epsilon0.Hprime.html">Hprime</a> <br/>
+<a href="gaia_hydras.HydraGaia_Examples.html">HydraGaia_Examples</a> <br/>
+<a href="hydras.Hydra.Hydra_Termination.html">Hydra_Termination</a> <br/>
+<a href="hydras.Hydra.Hydra_Extraction.html">Hydra_Extraction</a> <br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html">Hydra_Lemmas</a> <br/>
+<a href="hydras.Hydra.Hydra_Definitions.html">Hydra_Definitions</a> <br/>
+<a href="hydras.Hydra.Hydra_Examples.html">Hydra_Examples</a> <br/>
+<a href="hydras.Hydra.Hydra_Theorems.html">Hydra_Theorems</a> <br/>
+<br/><br/><a id="library_I"></a><h2>I </h2>
+<a href="hydras.solutions_exercises.isqrt.html">isqrt</a> <br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html">is_F_monotonous</a> <br/>
+<a href="hydras.Prelude.Iterates.html">Iterates</a> <br/>
+<a href="hydras.MoreAck.Iterate_compat.html">Iterate_compat</a> <br/>
+<br/><br/><a id="library_K"></a><h2>K </h2>
+<a href="hydras.Hydra.KP_example.html">KP_example</a> <br/>
+<br/><br/><a id="library_L"></a><h2>L </h2>
+<a href="hydras.Ackermann.Languages.html">Languages</a> <br/>
+<a href="hydras.Epsilon0.Large_Sets.html">Large_Sets</a> <br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html">Large_Sets_Examples</a> <br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html">LibHyps_Experiments</a> <br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html">Limit_Infinity</a> <br/>
+<a href="hydras.Ackermann.ListExt.html">ListExt</a> <br/>
+<a href="hydras.rpo.list_permut.html">list_permut</a> <br/>
+<a href="hydras.rpo.list_set.html">list_set</a> <br/>
+<a href="hydras.Ackermann.LNN.html">LNN</a> <br/>
+<a href="hydras.MoreAck.LNN_Examples.html">LNN_Examples</a> <br/>
+<a href="hydras.Ackermann.LNN2LNT.html">LNN2LNT</a> <br/>
+<a href="hydras.Ackermann.LNT.html">LNT</a> <br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html">lt_succ_le</a> <br/>
+<a href="hydras.Schutte.Lub.html">Lub</a> <br/>
+<a href="hydras.Epsilon0.L_alpha.html">L_alpha</a> <br/>
+<br/><br/><a id="library_M"></a><h2>M </h2>
+<a href="hydras.Prelude.Merge_Sort.html">Merge_Sort</a> <br/>
+<a href="hydras.solutions_exercises.MinPR.html">MinPR</a> <br/>
+<a href="hydras.solutions_exercises.MinPR2.html">MinPR2</a> <br/>
+<a href="hydras.Ackermann.misc.html">misc</a> <br/>
+<a href="hydras.Ackermann.model.html">model</a> <br/>
+<a href="additions.Monoid_instances.html">Monoid_instances</a> <br/>
+<a href="additions.Monoid_def.html">Monoid_def</a> <br/>
+<a href="hydras.Prelude.MoreDecidable.html">MoreDecidable</a> <br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html">MoreEpsilonIota</a> <br/>
+<a href="hydras.Prelude.MoreLibHyps.html">MoreLibHyps</a> <br/>
+<a href="hydras.Prelude.MoreLists.html">MoreLists</a> <br/>
+<a href="hydras.Prelude.MoreOrders.html">MoreOrders</a> <br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html">MorePRExamples</a> <br/>
+<a href="hydras.Prelude.MoreVectors.html">MoreVectors</a> <br/>
+<a href="hydras.Prelude.More_Arith.html">More_Arith</a> <br/>
+<a href="hydras.rpo.more_list.html">more_list</a> <br/>
+<a href="additions.More_on_positive.html">More_on_positive</a> <br/>
+<a href="hydras.solutions_exercises.MultisetWf.html">MultisetWf</a> <br/>
+<br/><br/><a id="library_N"></a><h2>N </h2>
+<a href="additions.Naive.html">Naive</a> <br/>
+<a href="hydras.Ackermann.NewNotations.html">NewNotations</a> <br/>
+<a href="gaia_hydras.nfwfgaia.html">nfwfgaia</a> <br/>
+<a href="hydras.Ackermann.NN.html">NN</a> <br/>
+<a href="hydras.Ackermann.NNtheory.html">NNtheory</a> <br/>
+<a href="hydras.Ackermann.NN2PA.html">NN2PA</a> <br/>
+<br/><br/><a id="library_O"></a><h2>O </h2>
+<a href="hydras.OrdinalNotations.OmegaOmega.html">OmegaOmega</a> <br/>
+<a href="hydras.Hydra.Omega_Small.html">Omega_Small</a> <br/>
+<a href="hydras.Hydra.Omega2_Small.html">Omega2_Small</a> <br/>
+<a href="hydras.solutions_exercises.OnCodeList.html">OnCodeList</a> <br/>
+<a href="gaia_hydras.onType.html">onType</a> <br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html">ON_Generic</a> <br/>
+<a href="hydras.OrdinalNotations.ON_mult.html">ON_mult</a> <br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html">ON_Finite</a> <br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">ON_Omega_plus_omega</a> <br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html">ON_Omega</a> <br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html">ON_Omega2</a> <br/>
+<a href="hydras.OrdinalNotations.ON_plus.html">ON_plus</a> <br/>
+<a href="gaia_hydras.ON_gfinite.html">ON_gfinite</a> <br/>
+<a href="hydras.OrdinalNotations.ON_O.html">ON_O</a> <br/>
+<a href="hydras.Schutte.Ordering_Functions.html">Ordering_Functions</a> <br/>
+<a href="hydras.Prelude.OrdNotations.html">OrdNotations</a> <br/>
+<a href="hydras.Hydra.O2H.html">O2H</a> <br/>
+<br/><br/><a id="library_P"></a><h2>P </h2>
+<a href="hydras.Ackermann.PA.html">PA</a> <br/>
+<a href="hydras.Ackermann.PAconsistent.html">PAconsistent</a> <br/>
+<a href="hydras.Schutte.PartialFun.html">PartialFun</a> <br/>
+<a href="hydras.Ackermann.PAtheory.html">PAtheory</a> <br/>
+<a href="hydras.Epsilon0.Paths.html">Paths</a> <br/>
+<a href="additions.Pow.html">Pow</a> <br/>
+<a href="additions.Pow_variant.html">Pow_variant</a> <br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html">predSuccUnicity</a> <br/>
+<a href="hydras.Ackermann.primRec.html">primRec</a> <br/>
+<a href="hydras.MoreAck.PrimRecExamples.html">PrimRecExamples</a> <br/>
+<a href="hydras.Ackermann.prLogic.html">prLogic</a> <br/>
+<a href="Goedel.PRrepresentable.html">PRrepresentable</a> <br/>
+<br/><br/><a id="library_R"></a><h2>R </h2>
+<a href="hydras.Prelude.Restriction.html">Restriction</a> <br/>
+<a href="Goedel.rosser.html">rosser</a> <br/>
+<a href="Goedel.rosserPA.html">rosserPA</a> <br/>
+<a href="hydras.rpo.rpo.html">rpo</a> <br/>
+<br/><br/><a id="library_S"></a><h2>S </h2>
+<a href="hydras.Schutte.Schutte.html">Schutte</a> <br/>
+<a href="hydras.Schutte.Schutte_basics.html">Schutte_basics</a> <br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">schutte_cnf_counter_example</a> <br/>
+<a href="hydras.Prelude.Simple_LexProd.html">Simple_LexProd</a> <br/>
+<a href="hydras.Prelude.Sort_spec.html">Sort_spec</a> <br/>
+<a href="hydras.Prelude.ssrnat_extracts.html">ssrnat_extracts</a> <br/>
+<a href="hydras.Prelude.STDPP_compat.html">STDPP_compat</a> <br/>
+<a href="additions.Strategies.html">Strategies</a> <br/>
+<a href="hydras.Ackermann.subAll.html">subAll</a> <br/>
+<a href="hydras.Ackermann.subProp.html">subProp</a> <br/>
+<br/><br/><a id="library_T"></a><h2>T </h2>
+<a href="hydras.rpo.term.html">term</a> <br/>
+<a href="additions.Trace_exercise.html">Trace_exercise</a> <br/>
+<a href="hydras.Epsilon0.T1.html">T1</a> <br/>
+<a href="gaia_hydras.T1Bridge.html">T1Bridge</a> <br/>
+<a href="gaia_hydras.T1Choice.html">T1Choice</a> <br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html">T1_ltNotWf</a> <br/>
+<a href="hydras.Gamma0.T2.html">T2</a> <br/>
+<a href="gaia_hydras.T2Bridge.html">T2Bridge</a> <br/>
+<br/><br/><a id="library_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wConsistent.html">wConsistent</a> <br/>
+<a href="hydras.Ackermann.wellFormed.html">wellFormed</a> <br/>
+<a href="hydras.Schutte.Well_Orders.html">Well_Orders</a> <br/>
+<a href="hydras.Prelude.WfVariant.html">WfVariant</a> <br/>
+<a href="additions.Wf_transparent.html">Wf_transparent</a> <br/>
+<br/><br/><hr/>
+<h1>Lemma Index</h1>
+<a id="lemma_A"></a><h2>A </h2>
+<a href="hydras.Ackermann.LNN.html#absurd1">absurd1</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#absurd1">absurd1</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#absurd1">absurd1</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Accs_all">Accs_all</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.O2H.html#acc_from_to_round_plus">acc_from_to_round_plus</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Acc_nil">Acc_nil</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="additions.Wf_transparent.html#Acc_inverse_image">Acc_inverse_image</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#Acc_lemma">Acc_lemma</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from_LT">acc_from_LT</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from_trans">acc_from_trans</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.rpo.closure.html#acc_trans">acc_trans</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#acc_rec">acc_rec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#acc_imp">acc_imp</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.addC_CE">Ackermann.addC_CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_normal">Ackermann.add_normal</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le4">Ackermann.add_le4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le3">Ackermann.add_le3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_inj">Ackermann.add_inj</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le2">Ackermann.add_le2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_le1">Ackermann.add_le1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_to_cons">Ackermann.add_to_cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_fin_omega">Ackermann.add_fin_omega</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add_int">Ackermann.add_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.add1Nfin">Ackermann.add1Nfin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.all_zeroE">Ackermann.all_zeroE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr2CE">Ackermann.ap_pr2CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr4">Ackermann.ap_pr4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr3">Ackermann.ap_pr3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr2">Ackermann.ap_pr2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr1">Ackermann.ap_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_limit">Ackermann.ap_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.ap_pr0">Ackermann.ap_pr0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.conc1">Ackermann.conc1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.fincP">Ackermann.fincP</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.finite_ltP">Ackermann.finite_ltP</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.fooCE">Ackermann.fooCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.le_succ_succE">Ackermann.le_succ_succE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.le_succ_succ">Ackermann.le_succ_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_pr">Ackermann.limit_pr</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_pr1">Ackermann.limit_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit1">Ackermann.limit1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit10">Ackermann.limit10</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit11">Ackermann.limit11</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12a">Ackermann.limit12a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b1">Ackermann.limit12b1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b2">Ackermann.limit12b2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b3">Ackermann.limit12b3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b4">Ackermann.limit12b4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12b5">Ackermann.limit12b5</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit2">Ackermann.limit2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit3">Ackermann.limit3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit4">Ackermann.limit4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit5">Ackermann.limit5</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit6">Ackermann.limit6</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit7">Ackermann.limit7</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit8">Ackermann.limit8</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit9">Ackermann.limit9</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_4">Ackermann.lt_succ_le_4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_3">Ackermann.lt_succ_le_3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_2">Ackermann.lt_succ_le_2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_le_1">Ackermann.lt_succ_le_1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_succE">Ackermann.lt_succ_succE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_succ_succ">Ackermann.lt_succ_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_omega2">Ackermann.lt_omega2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_omega1">Ackermann.lt_omega1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_epsilon0">Ackermann.lt_epsilon0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_not_wf">Ackermann.lt_not_wf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.minus_le">Ackermann.minus_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.minus_lt">Ackermann.minus_lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_add">Ackermann.nf_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_sub">Ackermann.nf_sub</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_pred">Ackermann.nf_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_succ">Ackermann.nf_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_finite">Ackermann.nf_finite</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_omega">Ackermann.nf_omega</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_split">Ackermann.nf_split</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_Wf">Ackermann.nf_Wf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_consE">Ackermann.nf_consE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_cons_cons">Ackermann.nf_cons_cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_int">Ackermann.nf_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_psi">Ackermann.nf_psi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.nf_0">Ackermann.nf_0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_compose">Ackermann.normal_compose</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_limit">Ackermann.normal_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal_id">Ackermann.normal_id</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_minus_one">Ackermann.omega_minus_one</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_least_inf2">Ackermann.omega_least_inf2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega_least_inf1">Ackermann.omega_least_inf1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE">Ackermann.phiE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_3">Ackermann.phiE_3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_2">Ackermann.phiE_2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiE_1">Ackermann.phiE_1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phiL">Ackermann.phiL</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi3v">Ackermann.phi3v</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_succ">Ackermann.pred_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_lt">Ackermann.pred_lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.pred_le">Ackermann.pred_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ_pr1">Ackermann.psi_succ_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_lt2">Ackermann.psi_lt2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_lt1">Ackermann.psi_lt1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_prop">Ackermann.size_prop</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_prop1">Ackermann.size_prop1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_psi">Ackermann.size_psi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_d">Ackermann.size_d</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_c">Ackermann.size_c</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_b">Ackermann.size_b</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size_a">Ackermann.size_a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_add">Ackermann.split_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_le">Ackermann.split_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_pred">Ackermann.split_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_succ">Ackermann.split_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_is_succ">Ackermann.split_is_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_limit">Ackermann.split_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.split_finite">Ackermann.split_finite</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1rCE">Ackermann.sub_pr1rCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1r">Ackermann.sub_pr1r</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_nz">Ackermann.sub_nz</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr1">Ackermann.sub_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_pr">Ackermann.sub_pr</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_le1">Ackermann.sub_le1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_1bCE">Ackermann.sub_1bCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_1aCE">Ackermann.sub_1aCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub_int">Ackermann.sub_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1a">Ackermann.sub1a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1b">Ackermann.sub1b</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sub1Nfin">Ackermann.sub1Nfin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psiCE">Ackermann.succ_psiCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_prCE">Ackermann.succ_prCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi2">Ackermann.succ_psi2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_psi1">Ackermann.succ_psi1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_is_add_one">Ackermann.succ_is_add_one</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_inj">Ackermann.succ_inj</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_p1">Ackermann.succ_p1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.succ_pred">Ackermann.succ_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec_antisym">Alt.Eps0_prec.prec_antisym</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec_dec">Alt.Eps0_prec.prec_dec</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.eq_symbol_dec">Alt.Eps0_sig.eq_symbol_dec</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_inc_rpo_0">Alt.lt_inc_rpo_0</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_subterm2">Alt.lt_subterm2</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.lt_subterm1">Alt.lt_subterm1</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term_mono">Alt.nat_2_term_mono</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_lt_cons">Alt.nat_lt_cons</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nf_Acc">Alt.nf_Acc</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nf_Wf">Alt.nf_Wf</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.rpo_trans">Alt.rpo_trans</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R_inc_rpo">Alt.R_inc_rpo</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R1">Alt.R1</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R2">Alt.R2</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R3">Alt.R3</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.R4">Alt.R4</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size3">Alt.T1_size3</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size2">Alt.T1_size2</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size1">Alt.T1_size1</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.eq_variable_dec">Alt.Vars.eq_variable_dec</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded_rpo">Alt.well_founded_rpo</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="additions.AM.html#AM_power_Ok">AM_power_Ok</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andE1">andE1</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andE1">andE1</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andE1">andE1</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andE2">andE2</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andE2">andE2</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andE2">andE2</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andI">andI</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andI">andI</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andI">andI</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#andRelPR">andRelPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#andSym">andSym</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#andSym">andSym</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#andSym">andSym</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#approx_ok">approx_ok</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_plusR">ap_plusR</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_plus">ap_plus</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#ap_cases">ap_cases</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_plus_AP">AP_plus_AP</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_to_phi0">AP_to_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_phi0">AP_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_o_segment">AP_o_segment</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed">AP_closed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_sup">AP_sup</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_unbounded">AP_unbounded</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_unbounded_0">AP_unbounded_0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_mult_fin_r_closed">AP_mult_fin_r_closed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_mult_Sn_closed">AP_mult_Sn_closed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_plus_closed">AP_plus_closed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_finite_eq_one">AP_finite_eq_one</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_omega">AP_omega</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_one">AP_one</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#ap_ref">ap_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Axm">Axm</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Axm">Axm</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Axm">Axm</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_scheme_length1">axp_scheme_length1</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_correct">axp_correct</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_full">A_full</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_Cr">A_Cr</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A_closed">A_closed</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A_denum">A_denum</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A1_A2">A1_A2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#A2_A1">A2_A1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="lemma_B"></a><h2>B </h2>
+<a href="additions.FirstSteps.html#Bad.correct_exp_too_strong">Bad.correct_exp_too_strong</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad.F3_correct">Bad.F3_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Examples.html#Bad.height_lt_size">Bad.height_lt_size</a> [in <a href="hydras.Hydra.Hydra_Examples.html">hydras.Hydra.Hydra_Examples</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.mul_not_associative">Bad.mul_not_associative</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.one_not_neutral">Bad.one_not_neutral</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad2.Fcompose_correct">Bad2.Fcompose_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad4.F23_ok">Bad4.F23_ok</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#battle_length_std_Hardy">battle_length_std_Hardy</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#battle_length_std">battle_length_std</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#battle_example">battle_example</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Bbeta_denum">Bbeta_denum</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaEquiv">betaEquiv</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaRepresentable">betaRepresentable</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaThm2">betaThm2</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaThm3">betaThm3</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaThm4">betaThm4</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaThm5">betaThm5</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#beta_def">beta_def</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#Bigstep">Bigstep</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#big_to_small">big_to_small</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#big_to_small">big_to_small</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#big_to_small">big_to_small</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="additions.Pow_variant.html#binary_power_mult_ok">binary_power_mult_ok</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#binary_power_mult_ok">binary_power_mult_ok</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_chain_length">binary_chain_length</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_generator_not_optimal">binary_generator_not_optimal</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_generator_correct">binary_generator_correct</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_correct">binary_correct</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#bin_pow_scheme_length1">bin_pow_scheme_length1</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.fib.html#bitsP">bitsP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#bits_cat">bits_cat</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#bool_eq_iff">bool_eq_iff</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide_eq_false">bool_decide_eq_false</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide_eq_true">bool_decide_eq_true</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputationMonotone">boundComputationMonotone</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.MoreDecidable.html#boundedCheck">boundedCheck</a> [in <a href="hydras.Prelude.MoreDecidable.html">hydras.Prelude.MoreDecidable</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#boundedLT">boundedLT</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch1">boundedSearch1</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch2">boundedSearch2</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#boundedSearch3">boundedSearch3</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#boundedSearch4">boundedSearch4</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundMakeTrace">boundMakeTrace</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp">boundSubFormulaHelp</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp1">boundSubFormulaHelp1</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubFormulaHelp2">boundSubFormulaHelp2</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubTerm">boundSubTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundSubTerms">boundSubTerms</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_iota_from'">but_last_iota_from'</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_app">but_last_app</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_shift'">but_last_shift'</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_interval">but_last_interval</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last_iota_from">but_last_iota_from</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<br/><br/><a id="lemma_C"></a><h2>C </h2>
+<a href="hydras.Epsilon0.F_alpha.html#call_lt_wf">call_lt_wf</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonSSn">CanonSSn</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_rel_rounds">canonS_rel_rounds</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota">canonS_iota</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_i">canonS_iota_i</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_final">canonS_iota_final</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_6">canonS_iota_6</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_5">canonS_iota_5</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#canonS_iota_4">canonS_iota_4</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_ocons_succE">canonS_ocons_succE</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_zero_inv">canonS_zero_inv</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_phi0_succE">canonS_phi0_succE</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_cons_not_zero">canonS_cons_not_zero</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_lt">canonS_lt</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canonS_LE">canonS_LE</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_phi0_Succ">CanonS_phi0_Succ</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_lt">CanonS_lt</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#CanonS_phi0_lim">CanonS_phi0_lim</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_LE">canonS_LE</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_mono">canonS_limit_mono</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_lub">canonS_limit_lub</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_limit_strong">canonS_limit_strong</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_cons_not_zero">canonS_cons_not_zero</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_LT">canonS_LT</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_zero_inv">canonS_zero_inv</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_cons_succ_eqn2">canonS_cons_succ_eqn2</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_phi0_succ_eqn">canonS_phi0_succ_eqn</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_succ">canonS_succ</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_lim2">canonS_lim2</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonS_lim1">canonS_lim1</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#CanonS_Phi0_Succ_eqn">CanonS_Phi0_Succ_eqn</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#CanonS_plus_1">CanonS_plus_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_of">canon_limit_of</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_mono">canon_limit_mono</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_limit_strong">canon_limit_strong</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim3">canon_lim3</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim2">canon_lim2</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_tail">canon_tail</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lim1">canon_lim1</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_SSn_zero">canon_SSn_zero</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_lt">canon_lt</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon_succ">canon_succ</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#canon_not_null">canon_not_null</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_of_limit_not_null">Canon_of_limit_not_null</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_lt">Canon_lt</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_Omega">Canon_Omega</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon_Succ">Canon_Succ</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_limit_mono">canon_limit_mono</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_limit_strong">canon_limit_strong</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lt">canon_lt</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_LT">canon_LT</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_SSn_zero">canon_SSn_zero</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_succ">canon_succ</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim3">canon_lim3</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim2">canon_lim2</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_lim1">canon_lim1</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_tail">canon_tail</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon_zero">canon_zero</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_first_step_lim">Canon_plus_first_step_lim</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_first_step">Canon_plus_first_step</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_mono1">Canon_mono1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus_inv">Canon_plus_inv</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon0_lt">canon0_lt</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon0_phi0_succE">canon0_phi0_succE</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_LT">canon0_LT</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_cons_succ_eqn2">canon0_cons_succ_eqn2</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_phi0_succ_eqn">canon0_phi0_succ_eqn</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_succ">canon0_succ</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon0_lim2">canon0_lim2</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.addC_CE">CantorOrdinal.addC_CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl3">CantorOrdinal.add_simpl3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_normal">CantorOrdinal.add_normal</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le4">CantorOrdinal.add_le4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le3">CantorOrdinal.add_le3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_inj">CantorOrdinal.add_inj</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le2">CantorOrdinal.add_le2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_le1">CantorOrdinal.add_le1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl2">CantorOrdinal.add_simpl2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_simpl1">CantorOrdinal.add_simpl1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_to_cons">CantorOrdinal.add_to_cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_fin_omega">CantorOrdinal.add_fin_omega</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add_int">CantorOrdinal.add_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.add1Nfin">CantorOrdinal.add1Nfin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr4">CantorOrdinal.ap_pr4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr3">CantorOrdinal.ap_pr3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr2CE">CantorOrdinal.ap_pr2CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr2">CantorOrdinal.ap_pr2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr1">CantorOrdinal.ap_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ap_pr0">CantorOrdinal.ap_pr0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantorCE2">CantorOrdinal.cantorCE2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantorCE3">CantorOrdinal.cantorCE3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_CE1">CantorOrdinal.cantor_CE1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_unique">CantorOrdinal.cantor_unique</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cantor_exists">CantorOrdinal.cantor_exists</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.div_by_omega_pr">CantorOrdinal.div_by_omega_pr</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_eq1">CantorOrdinal.expF_eq1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_eq0">CantorOrdinal.expF_eq0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_1n">CantorOrdinal.expF_1n</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_mul">CantorOrdinal.expF_mul</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expF_add">CantorOrdinal.expF_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_add">CantorOrdinal.expO_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_eq1">CantorOrdinal.expO_eq1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_eq0">CantorOrdinal.expO_eq0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_1n">CantorOrdinal.expO_1n</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_n0">CantorOrdinal.expO_n0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expO_mul1">CantorOrdinal.expO_mul1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx_nat">CantorOrdinal.expx_nat</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx_pnat">CantorOrdinal.expx_pnat</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx0">CantorOrdinal.expx0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.expx1CE">CantorOrdinal.expx1CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE3">CantorOrdinal.limit_CE3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE2">CantorOrdinal.limit_CE2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_CE1">CantorOrdinal.limit_CE1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_unique2">CantorOrdinal.limit_unique2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit3">CantorOrdinal.limit3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.minus_le">CantorOrdinal.minus_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.mul_omega2_pr1">CantorOrdinal.mul_omega2_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_succ_CE">CantorOrdinal.pred_succ_CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_succ">CantorOrdinal.pred_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_lt">CantorOrdinal.pred_lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.pred_le">CantorOrdinal.pred_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_add">CantorOrdinal.split_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_limit1">CantorOrdinal.split_limit1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_le">CantorOrdinal.split_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_pred">CantorOrdinal.split_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_succ">CantorOrdinal.split_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_finite">CantorOrdinal.split_finite</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_is_succ">CantorOrdinal.split_is_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.split_limit">CantorOrdinal.split_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1rCE">CantorOrdinal.sub_pr1rCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_nzCE">CantorOrdinal.sub_nzCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_nz">CantorOrdinal.sub_nz</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1r">CantorOrdinal.sub_pr1r</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1CE">CantorOrdinal.sub_pr1CE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr1">CantorOrdinal.sub_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_pr">CantorOrdinal.sub_pr</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_le1">CantorOrdinal.sub_le1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_int">CantorOrdinal.sub_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_1bCE">CantorOrdinal.sub_1bCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub_1aCE">CantorOrdinal.sub_1aCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1a">CantorOrdinal.sub1a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1b">CantorOrdinal.sub1b</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sub1Nfin">CantorOrdinal.sub1Nfin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_is_add_one">CantorOrdinal.succ_is_add_one</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_injCE">CantorOrdinal.succ_injCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_inj">CantorOrdinal.succ_inj</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_p1">CantorOrdinal.succ_p1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_predCE">CantorOrdinal.succ_predCE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_pred">CantorOrdinal.succ_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.succ_lt">CantorOrdinal.succ_lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_limit">CantorOrdinal.sup_Oalpha_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_succ">CantorOrdinal.sup_Oalpha_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_Oalpha_zero">CantorOrdinal.sup_Oalpha_zero</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.sup_unique">CantorOrdinal.sup_unique</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.tail_lt_cons">CantorOrdinal.tail_lt_cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_pred">CantorOrdinal.toNF_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_succ">CantorOrdinal.toNF_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_ex2">CantorOrdinal.toNF_ex2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_ex1">CantorOrdinal.toNF_ex1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_mon">CantorOrdinal.toNF_mon</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_nf">CantorOrdinal.toNF_nf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.toNF_nz">CantorOrdinal.toNF_nz</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addA">CantorOrdinal.T1addA</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addn0">CantorOrdinal.T1addn0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1addS">CantorOrdinal.T1addS</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add_eq0">CantorOrdinal.T1add_eq0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1add0n">CantorOrdinal.T1add0n</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ap_phi0">CantorOrdinal.T1ap_phi0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eqE">CantorOrdinal.T1eqE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1eq_le">CantorOrdinal.T1eq_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_mul2l">CantorOrdinal.T1lt_mul2l</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_add2r">CantorOrdinal.T1lt_add2r</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_add2l">CantorOrdinal.T1lt_add2l</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_cons_le">CantorOrdinal.T1lt_cons_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_le_trans">CantorOrdinal.T1lt_le_trans</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_trans">CantorOrdinal.T1lt_trans</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_anti">CantorOrdinal.T1lt_anti</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_trichotomy">CantorOrdinal.T1lt_trichotomy</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_neAle">CantorOrdinal.T1lt_neAle</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_ne'">CantorOrdinal.T1lt_ne'</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt_ne">CantorOrdinal.T1lt_ne</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1lt0n">CantorOrdinal.T1lt0n</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1muln1">CantorOrdinal.T1muln1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#cons_standard_pathS">cons_standard_pathS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#cons_lt_epsilon0">cons_lt_epsilon0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#cons_rw">cons_rw</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#contrad">contrad</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#contrad">contrad</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#contradiction">contradiction</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#contradiction">contradiction</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#contradiction">contradiction</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Contradiction">Contradiction</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="additions.AM.html#correct_app">correct_app</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Cor12">Cor12</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12">Cor12</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_E0">Cor12_E0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_3">Cor12_3</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_2">Cor12_2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Cor12_1">Cor12_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#countable_members">countable_members</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#countable_not_Unbounded">countable_not_Unbounded</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#countable_segment_proper">countable_segment_proper</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_image">countable_image</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_bij_funR">countable_bij_funR</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_bij_fun">countable_bij_fun</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#countable_surj">countable_surj</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#counter_ex">counter_ex</a> [in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInjHelp">cPairInjHelp</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInj1">cPairInj1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairInj2">cPairInj2</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLemma1">cPairLemma1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe1">cPairLe1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe1A">cPairLe1A</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe2">cPairLe2</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe2A">cPairLe2A</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLe3">cPairLe3</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLt1">cPairLt1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairLt2">cPairLt2</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections">cPairProjections</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjectionsHelp">cPairProjectionsHelp</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections1">cPairProjections1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairProjections2">cPairProjections2</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#cp1">cp1</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#cp1">cp1</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#cp1">cp1</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#cp2">cp2</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#cp2">cp2</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#cp2">cp2</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_1_iff">Cr_1_iff</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_incl">Cr_incl</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_lt">Cr_lt</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos_iff">Cr_pos_iff</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos_inv">Cr_pos_inv</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero_AP">Cr_zero_AP</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero_inv">Cr_zero_inv</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_pos">Cr_pos</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_zero">Cr_zero</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_inv">Cr_inv</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_equation">Cr_equation</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Cr_extensional">Cr_extensional</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#cst_correct">cst_correct</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#cst0_correct">cst0_correct</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj">cTripleProj</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj1">cTripleProj1</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj2">cTripleProj2</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleProj3">cTripleProj3</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="additions.Addition_Chains.html#C87_ok'">C87_ok'</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<br/><br/><a id="lemma_D"></a><h2>D </h2>
+<a href="Goedel.rosserPA.html#decideAxioms">decideAxioms</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#decideAxioms">decideAxioms</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#decide_hltE">decide_hltE</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp">decomp</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#decompose">decompose</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decompos2">decompos2</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp1">decomp1</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#decomp2">decomp2</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#decr_seq">decr_seq</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.Deduction.html#DeductionTheorem">DeductionTheorem</a> [in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#den_U">den_U</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthForall">depthForall</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthImp1">depthImp1</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthImp2">depthImp2</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#depthNot">depthNot</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_gt">dicho_gt</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_lt">dicho_lt</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_gt">dicho_aux_gt</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_lt">dicho_aux_lt</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xIXI">dicho_aux_le_xIXI</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xIXO">dicho_aux_le_xIXO</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xOXI">dicho_aux_le_xOXI</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_le_xOXO">dicho_aux_le_xOXO</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xIxI">dicho_aux_xIxI</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xIxO">dicho_aux_xIxO</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xOxI">dicho_aux_xOxI</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="additions.Dichotomy.html#dicho_aux_xOxO">dicho_aux_xOxO</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="gaia_hydras.GPrelude.html#diffP">diffP</a> [in <a href="gaia_hydras.GPrelude.html">gaia_hydras.GPrelude</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.nf_Acc">Direct_proof.nf_Acc</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_implies_Acc_strong">Direct_proof.Acc_implies_Acc_strong</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_strong_stronger">Direct_proof.Acc_strong_stronger</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.acc_impl">Direct_proof.acc_impl</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.wf_LT">Direct_proof.wf_LT</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.Acc_zero">Direct_proof.Acc_zero</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div_eucl_unique">div_eucl_unique</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div_not_qlt">div_not_qlt</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.Strategies.html#div_gamma_pos">div_gamma_pos</a> [in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_even_plus">div2_even_plus</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_incr">div2_incr</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_of_Even">div2_of_Even</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#div2_double_is_id">div2_double_is_id</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_iterate">dominates_iterate</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_from_le">dominates_from_le</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_trans_strong">dominates_trans_strong</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_trans">dominates_trans</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#dominates_from_trans">dominates_from_trans</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR">dom_AckNotPR</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#Done">Done</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#doubleIsPR">doubleIsPR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#doubleS_law">doubleS_law</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_is_even">double_is_even</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_inj">double_inj</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_plus">double_plus</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#double_S">double_S</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_3">DS_iota_3</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_2">DS_iota_2</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota_1">DS_iota_1</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<br/><br/><a id="lemma_E"></a><h2>E </h2>
+<a href="hydras.Schutte.Correctness_E0.html#embedding">embedding</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#E0_g2h_h2gK">E0_g2h_h2gK</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_h2g_g2hK">E0_h2g_g2hK</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_h2g_nf">E0_h2g_nf</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_lt_ge">E0_lt_ge</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_lt_eq_lt">E0_lt_eq_lt</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_not_Lt_zero">E0_not_Lt_zero</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_Lt_Succ_inv">E0_Lt_Succ_inv</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_Lt_irrefl">E0_Lt_irrefl</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_pred_Lt">E0_pred_Lt</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_pred_of_Succ">E0_pred_of_Succ</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_dec">E0_eq_dec</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_iff">E0_eq_iff</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_eq_intro">E0_eq_intro</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="lemma_F"></a><h2>F </h2>
+<a href="gaia_hydras.onType.html#F">F</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact_alt_ok">fact_alt_ok</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#fact_correct">fact_correct</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#Fake_thm">Fake_thm</a> [in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="additions.fib.html#fastexp3P">fastexp3P</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#fbij">fbij</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_correct">Fcompose_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_correct_nat">Fcompose_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_correct">Fexp2_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_nat_correct">Fexp2_nat_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#FF">FF</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="additions.AM.html#FFK_correct">FFK_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_correct">FFK_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_correct_nat">FFK_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.fib.html#fibmP">fibmP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibtP">fibtP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibt_aux">fibt_aux</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibZ2P">fibZ2P</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#fibZ3P">fibZ3P</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Fib2.html#fib_mul2_OK">fib_mul2_OK</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_mul2_OK_0">fib_mul2_OK_0</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_SSn">fib_SSn</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#fib_ind">fib_ind</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_alt_Ok">fib_alt_Ok</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_OK0">fib_OK0</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.rpo.more_list.html#find_not_mem">find_not_mem</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#finite_ltR">finite_ltR</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#finite_lt">finite_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Finite_ref">Finite_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_not_limit">finite_not_limit</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_lt_omega">finite_lt_omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_inj">finite_inj</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_mono">finite_mono</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#finite_lt_inv">finite_lt_inv</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#finite_is_finite">finite_is_finite</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#finite_plus_infinite">finite_plus_infinite</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_lt_wf">finord_lt_wf</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_compare_correct">finord_compare_correct</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#FinS_Succ_eq">FinS_Succ_eq</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#FinS_eq">FinS_eq</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#fin_lt_omega">fin_lt_omega</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="Goedel.goedel1.html#FirstIncompletenessA">FirstIncompletenessA</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#FirstIncompletenessA">FirstIncompletenessA</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel1.html#FirstIncompletenessB">FirstIncompletenessB</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.fixPoint.html#FixPointLNN">FixPointLNN</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#FixPointLNT">FixPointLNT</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#FixR_F_eq">FixR_F_eq</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="additions.AM.html#FK_correct">FK_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FK_correct">FK_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#flatten">flatten</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_valid_2">flatten_valid_2</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_valid">flatten_valid</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#flatten_aux_valid">flatten_aux_valid</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallE">forallE</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallE">forallE</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallE">forallE</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallI">forallI</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallI">forallI</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallI">forallI</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#forallSimp">forallSimp</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#forallSimp">forallSimp</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#forallSimp">forallSimp</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#forall_weak">forall_weak</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_forall">Forall_forall</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_and">Forall_and</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall_inv">Forall_inv</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_iff">Forall2R_iff</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_cons">Forall2R_cons</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_indR">Forall2_indR</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_RR">Forall2_RR</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_R">Forall2_R</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Forall2_inv">Forall2_inv</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_ind2">Formula_depth_ind2</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_forall">Formula_depth_rec2_forall</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_not">Formula_depth_rec2_not</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2_imp">Formula_depth_rec2_imp</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec2rec_nice">Formula_depth_rec2rec_nice</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula_depth_rec_indep">Formula_depth_rec_indep</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#formula_eqdec">formula_eqdec</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddExists1">freeVarAddExists1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddExists2">freeVarAddExists2</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddForalls1">freeVarAddForalls1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarAddForalls2">freeVarAddForalls2</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosed">freeVarClosed</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosedList1">freeVarClosedList1</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarClosedList2">freeVarClosedList2</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPf">freeVarCodeSysPf</a> [in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPrf">freeVarCodeSysPrf</a> [in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#freeVarCodeSysPrfN">freeVarCodeSysPrfN</a> [in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.goedel1.html#freeVarG">freeVarG</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpFormula">freeVarInterpFormula</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpRel">freeVarInterpRel</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarInterpTerm">freeVarInterpTerm</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarListFormulaApp">freeVarListFormulaApp</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarLT">freeVarLT</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarMap_ext">freeVarMap_ext</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarMap1">freeVarMap1</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.model.html#freeVarNNHelp">freeVarNNHelp</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarPlus">freeVarPlus</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarPlus">freeVarPlus</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarPrimRecPiFormulaHelp1">freeVarPrimRecPiFormulaHelp1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#freeVarPrimRecSigmaFormulaHelp1">freeVarPrimRecSigmaFormulaHelp1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.rosserPA.html#freeVarRepT'">freeVarRepT'</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllFormula1">freeVarSubAllFormula1</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllFormula2">freeVarSubAllFormula2</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerms1">freeVarSubAllTerms1</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerms2">freeVarSubAllTerms2</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerm1">freeVarSubAllTerm1</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#freeVarSubAllTerm2">freeVarSubAllTerm2</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula1">freeVarSubFormula1</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula2">freeVarSubFormula2</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula3">freeVarSubFormula3</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubFormula4">freeVarSubFormula4</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms1">freeVarSubTerms1</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms2">freeVarSubTerms2</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms3">freeVarSubTerms3</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerms4">freeVarSubTerms4</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm1">freeVarSubTerm1</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm2">freeVarSubTerm2</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm3">freeVarSubTerm3</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#freeVarSubTerm4">freeVarSubTerm4</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarSucc">freeVarSucc</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarSucc">freeVarSucc</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTApply">freeVarTApply</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarTimes">freeVarTimes</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarTimes">freeVarTimes</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#freeVarZero">freeVarZero</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#freeVarZero">freeVarZero</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#fstar_S">fstar_S</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Fstar_S">Fstar_S</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#FS_rw">FS_rw</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#fun_le_trans">fun_le_trans</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection_is_ZL_bijection">fun_bijection_is_ZL_bijection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection_codomain">fun_bijection_codomain</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_not_PR">F_alpha_not_PR</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_not_PR_E0">F_alpha_not_PR_E0</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_restricted_mono_l">F_restricted_mono_l</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_limE">F_limE</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_succE">F_succE</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_0_eq">F_alpha_0_eq</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_mono_l">F_mono_l</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_zeroE">F_zeroE</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_positive">F_alpha_positive</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_Succ_le">F_alpha_Succ_le</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_dom">F_alpha_dom</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_mono">F_alpha_mono</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#F_alpha_gt">F_alpha_gt</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_succ_eqn">f_succ_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_lim_eqn">f_lim_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_zero_eqn">f_zero_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_iterate">f_star_iterate</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_Succ">f_star_Succ</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_eq2">f_eq2</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#f_star_zero_eqn">f_star_zero_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_mono_l">F_mono_l</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_mono_l_0">F_mono_l_0</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_restricted_mono_l">F_restricted_mono_l</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_dom">F_alpha_dom</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_Succ_le">F_alpha_Succ_le</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_positive">F_alpha_positive</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_gt">F_alpha_gt</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_mono">F_alpha_mono</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_One_Zero_ge">F_One_Zero_ge</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_One_Zero_dom">F_One_Zero_dom</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_0_eq">F_alpha_0_eq</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_succ_eqn">F_succ_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_lim_eqn">F_lim_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_zero_eqn">F_zero_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_iterate">F_star_iterate</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_Succ">F_star_Succ</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_eq2">F_eq2</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_star_zero_eqn">F_star_zero_eqn</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#f_ok_inv">f_ok_inv</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR_inv">F_alpha_notPR_inv</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_PR_inv">F_alpha_PR_inv</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_not_PR">F_alpha_not_PR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_omega_not_PR">F_omega_not_PR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_vs_Ack">F_vs_Ack</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_iterate_rw">F_iterate_rw</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#F_alpha_Sn">F_alpha_Sn</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#F_3_eqn">F_3_eqn</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#f_minoration">f_minoration</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#f_sup_commutes">f_sup_commutes</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#F_not_lim">F_not_lim</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F0">F0</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#F1">F1</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F1">F1</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#F1">F1</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="additions.AM.html#F1_correct">F1_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_correct">F1_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_neutral_r">F1_neutral_r</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_neutral_l">F1_neutral_l</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F1_simpl">F1_simpl</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F2">F2</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#F2">F2</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F2">F2</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#F2">F2</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F2C_correct">F2C_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#F2q_correct">F2q_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#F2q_correct_1">F2q_correct_1</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#F2q_correct_0">F2q_correct_0</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F2_correct">F2_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#F3">F3</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="additions.AM.html#F3_correct">F3_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F3_correct">F3_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F5">F5</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F6">F6</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F7">F7</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F8">F8</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F9">F9</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F9_correct">F9_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="lemma_G"></a><h2>G </h2>
+<a href="hydras.Epsilon0.Paths.html#gamma_positive">gamma_positive</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_transitive">Gamma0_prec.prec_transitive</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_antisym">Gamma0_prec.prec_antisym</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.prec_dec">Gamma0_prec.prec_dec</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.eq_symbol_dec">Gamma0_sig.eq_symbol_dec</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le4">Gamma0.add_le4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le3">Gamma0.add_le3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_inj">Gamma0.add_inj</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le2">Gamma0.add_le2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_le1">Gamma0.add_le1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_to_cons">Gamma0.add_to_cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_fin_omega">Gamma0.add_fin_omega</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add_int">Gamma0.add_int</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.add1Nfin">Gamma0.add1Nfin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr4">Gamma0.ap_pr4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr3">Gamma0.ap_pr3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr2">Gamma0.ap_pr2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr1">Gamma0.ap_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_limit">Gamma0.ap_limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.ap_pr0">Gamma0.ap_pr0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.le_succ_succE">Gamma0.le_succ_succE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.le_succ_succ">Gamma0.le_succ_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.limit_pr">Gamma0.limit_pr</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.limit_pr1">Gamma0.limit_pr1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_3">Gamma0.lt_succ_le_3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_2">Gamma0.lt_succ_le_2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_le_1">Gamma0.lt_succ_le_1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_succE">Gamma0.lt_succ_succE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_succ_succ">Gamma0.lt_succ_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.lt_tail">Gamma0.lt_tail</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.minus_le">Gamma0.minus_le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.minus_lt">Gamma0.minus_lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_phi">Gamma0.nf_phi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_add">Gamma0.nf_add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_sub">Gamma0.nf_sub</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_pred">Gamma0.nf_pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_succ">Gamma0.nf_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.nf_psi">Gamma0.nf_psi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_minus_one">Gamma0.omega_minus_one</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_least_inf2">Gamma0.omega_least_inf2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_least_inf1">Gamma0.omega_least_inf1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.omega_lt_epsilon0">Gamma0.omega_lt_epsilon0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv2">Gamma0.phi_inv2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inv0">Gamma0.phi_inv0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_eqE">Gamma0.phi_eqE</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.phi_inj1">Gamma0.phi_inj1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2leP">Gamma0.T2leP</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_add2r">Gamma0.T2le_add2r</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_add2l">Gamma0.T2le_add2l</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_psi_b">Gamma0.T2le_psi_b</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_psi1">Gamma0.T2le_psi1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le_trans">Gamma0.T2le_trans</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2le0n">Gamma0.T2le0n</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltn0">Gamma0.T2ltn0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltP">Gamma0.T2ltP</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltW">Gamma0.T2ltW</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.GCanon.html#g2h_canon">g2h_canon</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_E0zero">g2h_E0zero</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_E0_succ">g2h_E0_succ</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#g2h_plusE">g2h_plusE</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_multE">g2h_multE</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#g2h_phi0">g2h_phi0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="hydras.Hydra.Hydra_Lemmas.html#hcons_mult_app">hcons_mult_app</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="additions.fib.html#headbits">headbits</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#head_LT_cons">head_LT_cons</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#head_lt">head_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Epsilon0.F_alpha.html#HF0">HF0</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
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+<a href="hydras.Ackermann.LNT.html#iffRefl">iffRefl</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffRefl">iffRefl</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffSym">iffSym</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffSym">iffSym</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffSym">iffSym</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#iffTrans">iffTrans</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#iffTrans">iffTrans</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#iffTrans">iffTrans</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.Schutte.PartialFun.html#image_as_Im">image_as_Im</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#image_B_g_seg">image_B_g_seg</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impE">impE</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impE">impE</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impE">impE</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#impForall">impForall</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impI">impI</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impI">impI</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impI">impI</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.Ackermann.LNN.html#impRefl">impRefl</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impRefl">impRefl</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impRefl">impRefl</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#impTrans">impTrans</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#impTrans">impTrans</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#impTrans">impTrans</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#incl_inv">incl_inv</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="Goedel.rosser.html#Incompleteness">Incompleteness</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Inc_U_A">Inc_U_A</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
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+<a href="hydras.Schutte.Critical.html#induct_step">induct_step</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_ord">Inh_ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#interval_sorted_ge">interval_sorted_ge</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_unroll">interval_unroll</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_app">interval_app</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_singleton">interval_singleton</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_not_empty_iff">interval_not_empty_iff</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_not_empty">interval_not_empty</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval_length">interval_length</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#interval_def">interval_def</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#inv_Pi">inv_Pi</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_fun_bij">inv_fun_bij</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_composeR">inv_composeR</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_compose">inv_compose</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.rpo.closure.html#inv_trans">inv_trans</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_2_1">In_betaFormula_subst_2_1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_2">In_betaFormula_subst_2</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula">In_betaFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1">In_betaFormula_subst_1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1_2">In_betaFormula_subst_1_2</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#In_betaFormula_subst_1_2_0">In_betaFormula_subst_1_2_0</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#In_sorted_ge_inv">In_sorted_ge_inv</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_remove">in_remove</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_map_in">in_map_in</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#in_in_map">in_in_map</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarListFormulaE">In_freeVarListFormulaE</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarListFormula">In_freeVarListFormula</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
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+<a href="hydras.MoreAck.FolExamples.html#In_add2">In_add2</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
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+<a href="hydras.Prelude.MoreVectors.html#In_cases">In_cases</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
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+<a href="hydras.Hydra.O2H.html#iota_phi0">iota_phi0</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
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+<a href="hydras.Hydra.O2H.html#iota_iotas">iota_iotas</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
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+<a href="hydras.Prelude.MoreLists.html#iota_from_sorted_ge">iota_from_sorted_ge</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_unroll">iota_from_unroll</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_plus">iota_from_plus</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from_app">iota_from_app</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#iota_adapt">iota_adapt</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#iota_ind">iota_ind</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
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+<a href="hydras.OrdinalNotations.ON_Finite.html#iota_compare_commute">iota_compare_commute</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Schutte.AP.html#is_limit_phi0">is_limit_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#is_in_seq_not_Acc">is_in_seq_not_Acc</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit_sup_members">is_limit_sup_members</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit_omega">is_limit_omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_ab">is_limit_ab</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_intro">is_limit_intro</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_cons_inv">is_limit_cons_inv</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_limit_not_succ">is_limit_not_succ</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#is_finite_finite">is_finite_finite</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge_diag'">iterate_ge_diag'</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_gt_diag'">iterate_gt_diag'</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge_from">iterate_ge_from</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ext2">iterate_ext2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_dom_prop">iterate_dom_prop</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono_1">iterate_mono_1</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge''">iterate_ge''</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge'">iterate_ge'</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_Sge">iterate_Sge</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ge">iterate_ge</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono">iterate_mono</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_mono2">iterate_mono2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le_np_le">iterate_le_np_le</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le_n_Sn">iterate_le_n_Sn</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_lt_from">iterate_lt_from</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_lt">iterate_lt</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_le">iterate_le</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_ext">iterate_ext</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_rw">iterate_rw</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_S_eqn2">iterate_S_eqn2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_S_eqn">iterate_S_eqn</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat3">iterate_compat3</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat2">iterate_compat2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_compat">iterate_compat</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate_comm">iterate_comm</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#iterate_nat_rec">iterate_nat_rec</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.Iterate_compat.html#iterate_extEqual">iterate_extEqual</a> [in <a href="hydras.MoreAck.Iterate_compat.html">hydras.MoreAck.Iterate_compat</a>]<br/>
+<a href="hydras.MoreAck.Iterate_compat.html#iterate_compat">iterate_compat</a> [in <a href="hydras.MoreAck.Iterate_compat.html">hydras.MoreAck.Iterate_compat</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono2">iterate2_mono2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono3">iterate2_mono3</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono_weak">iterate2_mono_weak</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate2_mono">iterate2_mono</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="additions.fib.html#iter_combine">iter_combine</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#iter_comm">iter_comm</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#iter_mul">iter_mul</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="gaia_hydras.onType.html#I_i_wf">I_i_wf</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<br/><br/><a id="lemma_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#Kchain_correct_conv">Kchain_correct_conv</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFF_correct">KFF_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF_correct">KFF_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF_correct_nat">KFF_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFK_correct">KFK_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_correct">KFK_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_proper">KFK_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK_correct_nat">KFK_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_5_iii">KP_5_iii</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_arrowS_zero">KP_arrowS_zero</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_beta_eq">ksi_plus_beta_eq</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_beta">ksi_plus_beta</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_seq_n'">ksi_plus_seq_n'</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#ksi_plus_seq_n">ksi_plus_seq_n</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.GPaths.html#KS_thm_2_4">KS_thm_2_4</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_E0">KS_thm_2_4_E0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4">KS_thm_2_4</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma5">KS_thm_2_4_lemma5</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma4">KS_thm_2_4_lemma4</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma3">KS_thm_2_4_lemma3</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma3_0">KS_thm_2_4_lemma3_0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma2">KS_thm_2_4_lemma2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma1'">KS_thm_2_4_lemma1'</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KS_thm_2_4_lemma1">KS_thm_2_4_lemma1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F_correct">K2F_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F_correct_nat">K2F_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k3_1_proper">k3_1_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k3_1_correct">k3_1_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k7_3_correct">k7_3_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="lemma_L"></a><h2>L </h2>
+<a href="hydras.Epsilon0.F_omega.html#L">L</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L">L</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#L">L</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_n_R">largeb_n_R</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_n">largeb_n</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_finite">largeb_finite</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Largeb_Sb">Largeb_Sb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb_Sb">largeb_Sb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeSb_b">LargeSb_b</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeSb_b">largeSb_b</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeS_iff">largeS_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_iff">large_omega_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_2">large_omega_2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_omega_1">large_omega_1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_n_iff">large_n_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large_iff">large_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Least_zero">Least_zero</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.AP.html#least_AP">least_AP</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Least_is_0">Least_is_0</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least_not_Limit">Least_not_Limit</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least_not_Succ">Least_not_Succ</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_of_eq">least_member_of_eq</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_ex_unique">least_member_ex_unique</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_unicity">least_member_unicity</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_glb">least_member_glb</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#least_member_lower_bound">least_member_lower_bound</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_5_2">Lemma_4_5_2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_5_1">Lemma_4_5_1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_2">Lemma_4_4_2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_4_0">Lemma_4_4_0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_4_1">Lemma_4_4_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3">Lemma_4_3</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_0">Lemma_4_3_0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Lemma2_6_1">Lemma2_6_1</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma2_6_1_E0">Lemma2_6_1_E0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma2_6_1">Lemma2_6_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma4_4">Lemma4_4</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5">Lemma5</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_7">Lemma5_7</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_2">Lemma5_2</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_1">Lemma5_1</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_04">Lemma5_04</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_03">Lemma5_03</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_02">Lemma5_02</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_01">Lemma5_01</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Lemma5_0">Lemma5_0</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#lengthTerms">lengthTerms</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
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+<a href="hydras.OrdinalNotations.ON_Omega2.html#lt_succ">lt_succ</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
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+<a href="hydras.Hydra.O2H.html#LT_to_round_plus">LT_to_round_plus</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Lt_n_F_Zero_n">Lt_n_F_Zero_n</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_eq_lt">lt_eq_lt</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_incl_le">lt_incl_le</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_le_trans">lt_le_trans</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_not_ge">lt_not_ge</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_not_gt">lt_not_gt</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#lt_wf">lt_wf</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="gaia_hydras.GPaths.html#LT_to_standard_path">LT_to_standard_path</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#LT_path_to">LT_path_to</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
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+<a href="hydras.Prelude.DecPreOrder.html#lt_le_trans">lt_le_trans</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_not_equiv">lt_not_equiv</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
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+<a href="hydras.Schutte.AP.html#lt_phi0">lt_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#lt_beta_exists">lt_beta_exists</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#LT_ref">LT_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#lt_ref">lt_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_square_eqn1">L_omega_square_eqn1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_eqn">L_omega_mult_eqn</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_ok">L_omega_mult_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_mult_Sle">L_omega_mult_Sle</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_ok">L_omega_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_smono">L_omega_smono</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_omega_Sle">L_omega_Sle</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_fin_ok">L_fin_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_S_succ_rw">L_S_succ_rw</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_fin_smono">L_fin_smono</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_finS_S_le">L_finS_S_le</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_lim_ok">L_lim_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_ok">L_succ_ok</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_Sle">L_succ_Sle</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_succ_mono">L_succ_mono</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_unicity">L_spec_unicity</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_pos_inv">L_pos_inv</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_zero_inv">L_zero_inv</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_compat">L_spec_compat</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec_inv2">L_spec_inv2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_correct">L_correct</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok">L_ok</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok_lim">L_ok_lim</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok_succ">L_ok_succ</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ok0">L_ok0</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_succ_ok">L_succ_ok</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ge_S">L_ge_S</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_ge_id">L_ge_id</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_omega">L_omega</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_finite">L_finite</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_lim_eqn">L_lim_eqn</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_succ_eqn">L_succ_eqn</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_eq2">L_eq2</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_zero_eqn">L_zero_eqn</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#l_std_ok">l_std_ok</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_0_3">L_0_3</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_2_3">L_2_3</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L_0_2">L_0_2</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="gaia_hydras.GHydra.html#l_std_ok">l_std_ok</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#l_stdE">l_stdE</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#L_fin_ok">L_fin_ok</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_correct">L_correct</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_pos_inv">L_pos_inv</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_spec1">L_spec1</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_spec0">L_spec0</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_ge_S">L_ge_S</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_ge_id">L_ge_id</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_omega">L_omega</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_finite">L_finite</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_limE">L_limE</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_succE">L_succE</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_eq2">L_eq2</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="gaia_hydras.GL_alpha.html#L_zeroE">L_zeroE</a> [in <a href="gaia_hydras.GL_alpha.html">gaia_hydras.GL_alpha</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L0">L0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="additions.AM.html#L0">L0</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#L0">L0</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L0_95">L0_95</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#L00">L00</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#L02">L02</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L03">L03</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L04">L04</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L04'">L04'</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L05">L05</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#L06">L06</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L1">L1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#L1">L1</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L1">L1</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#L1">L1</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#L1">L1</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#L1">L1</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#L1'">L1'</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L10">L10</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L10">L10</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L11">L11</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L12">L12</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L13">L13</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L14">L14</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L2">L2</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#L2">L2</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L2">L2</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#L2">L2</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#L2">L2</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#L2">L2</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L2">L2</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#L2">L2</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#l2hsK">l2hsK</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L2_95_S">L2_95_S</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L2_95">L2_95</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#L2_6_2">L2_6_2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L22">L22</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L3">L3</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L3">L3</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#L3">L3</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L3">L3</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#L3a">L3a</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#L3_u">L3_u</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L4">L4</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L4">L4</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#L4">L4</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L4">L4</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#L4">L4</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L46">L46</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L5">L5</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L5">L5</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#L5">L5</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L6">L6</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#L6">L6</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L7">L7</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L8">L8</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="additions.Addition_Chains.html#L87''">L87''</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#L9">L9</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L94">L94</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#L95">L95</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<br/><br/><a id="lemma_M"></a><h2>M </h2>
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+<a href="hydras.MoreAck.AckNotPR.html#majorProjection">majorProjection</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#majorPR1">majorPR1</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#majorPR2">majorPR2</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#majorPR2_strict">majorPR2_strict</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#majorSucc">majorSucc</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#majorZero">majorZero</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceCorrect">makeTraceCorrect</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceForallNice">makeTraceForallNice</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceImpNice">makeTraceImpNice</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTraceNotNice">makeTraceNotNice</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTrace1">makeTrace1</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#makeTrace2">makeTrace2</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.Euclidean_Chains.html#make_chain_correct">make_chain_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.acc_build">Make.acc_build</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.acc_lex_drop_proof">Make.acc_lex_drop_proof</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.ac_syntactic">Make.ac_syntactic</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.ac_syntactic_aux">Make.ac_syntactic_aux</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_comm">Make.add_comm</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_12">Make.add_12</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_2">Make.add_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_1">Make.add_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.add_prf">Make.add_prf</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_eq_set">Make.cardinal_eq_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union">Make.cardinal_union</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_inter_12">Make.cardinal_union_inter_12</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_2">Make.cardinal_union_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_union_1">Make.cardinal_union_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.cardinal_subset">Make.cardinal_subset</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.cons_permut_in">Make.cons_permut_in</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_multiset_extension_step_app2">Make.context_multiset_extension_step_app2</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_trans_clos_multiset_extension_step_app1">Make.context_trans_clos_multiset_extension_step_app1</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.context_multiset_extension_step_app1">Make.context_multiset_extension_step_app1</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_app2">Make.context_list_permut_app2</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_app1">Make.context_list_permut_app1</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.context_list_permut_cons">Make.context_list_permut_cons</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.DecVar.eq_A_dec">Make.DecVar.eq_A_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson">Make.dickson</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux3">Make.dickson_aux3</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux2">Make.dickson_aux2</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.dickson_aux1">Make.dickson_aux1</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.term.html#Make.empty_subst_is_id_list">Make.empty_subst_is_id_list</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.empty_subst_is_id">Make.empty_subst_is_id</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.eq_term_dec">Make.eq_term_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_list_permut_support">Make.eq_set_list_permut_support</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_trans">Make.eq_set_trans</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_sym">Make.eq_set_sym</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_refl">Make.eq_set_refl</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.eq_set_dec">Make.eq_set_dec</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_union">Make.filter_union</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_2">Make.filter_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_2_list">Make.filter_2_list</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_1">Make.filter_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.filter_1_list">Make.filter_1_list</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.included_remove_red">Make.included_remove_red</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.included_filter_aux">Make.included_filter_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_12">Make.inter_12</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_12_aux">Make.inter_12_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_2">Make.inter_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_2_aux">Make.inter_2_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_1">Make.inter_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.inter_1_aux">Make.inter_1_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.in_permut_in">Make.in_permut_in</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.in_mult_S">Make.in_mult_S</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.in_sn_sn">Make.in_sn_sn</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.is_a_pos_exists_subtem">Make.is_a_pos_exists_subtem</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex1">Make.lex1</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex1_bis">Make.lex1_bis</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex2">Make.lex2</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.lex3">Make.lex3</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_acc">Make.list_permut_acc</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_multiset_extension_step_2">Make.list_permut_multiset_extension_step_2</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.list_permut_multiset_extension_step_1">Make.list_permut_multiset_extension_step_1</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_dec">Make.list_permut_dec</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length_2">Make.list_permut_length_2</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length_1">Make.list_permut_length_1</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_map">Make.list_permut_map</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_size">Make.list_permut_size</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_length">Make.list_permut_length</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_remove_hd">Make.list_permut_remove_hd</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_app_app">Make.list_permut_app_app</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_add_cons_inside">Make.list_permut_add_cons_inside</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_add_inside">Make.list_permut_add_inside</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_nil">Make.list_permut_nil</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_trans">Make.list_permut_trans</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_sym">Make.list_permut_sym</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.list_permut_refl">Make.list_permut_refl</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.list_permut_map_acc">Make.list_permut_map_acc</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.mem_dec">Make.mem_dec</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.multiplicity_app">Make.multiplicity_app</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.multiset_closure">Make.multiset_closure</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.out_mult_O">Make.out_mult_O</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.o_size3_trans">Make.o_size3_trans</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.projection_list_of_SN_terms">Make.projection_list_of_SN_terms</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.remove_red_included">Make.remove_red_included</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.remove_context_list_permut_app2">Make.remove_context_list_permut_app2</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Make.remove_context_list_permut_cons">Make.remove_context_list_permut_cons</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_list_replace_at_pos_in_subterm">Make.replace_at_pos_list_replace_at_pos_in_subterm</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_is_replace_at_pos2">Make.replace_at_pos_is_replace_at_pos2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_is_replace_at_pos1">Make.replace_at_pos_is_replace_at_pos1</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos_unfold">Make.replace_at_pos_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_add_context">Make.rpo_add_context</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_subst">Make.rpo_subst</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_trans_clos">Make.rpo_mul_trans_clos</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_rest_same_length">Make.rpo_lex_rest_same_length</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_trans">Make.rpo_trans</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_closure">Make.rpo_closure</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_subterm">Make.rpo_subterm</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_same_length">Make.rpo_lex_same_length</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_subterm_at_pos">Make.size_subterm_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_direct_subterm">Make.size_direct_subterm</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_ge_one">Make.size_ge_one</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size_unfold">Make.size_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_cardinal_not_eq_not_eq_set">Make.subset_cardinal_not_eq_not_eq_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_subset_union">Make.subset_subset_union</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat">Make.subset_compat</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat_2">Make.subset_compat_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_compat_1">Make.subset_compat_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_filter">Make.subset_filter</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_inter_2">Make.subset_inter_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_inter_1">Make.subset_inter_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_union_2">Make.subset_union_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_union_1">Make.subset_union_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset_dec">Make.subset_dec</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp">Make.subst_comp_is_subst_comp</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp_aux2">Make.subst_comp_is_subst_comp_aux2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp_is_subst_comp_aux1">Make.subst_comp_is_subst_comp_aux1</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos">Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.two_cases">Make.two_cases</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.two_cases_rpo">Make.two_cases_rpo</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_eq_set">Make.union_compat_eq_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_subset_2">Make.union_compat_subset_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_compat_subset_1">Make.union_compat_subset_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_assoc">Make.union_assoc</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_comm">Make.union_comm</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_empty_2">Make.union_empty_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_empty_1">Make.union_empty_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_12">Make.union_12</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_12_aux">Make.union_12_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_2">Make.union_2</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_2_aux">Make.union_2_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_1">Make.union_1</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union_1_aux">Make.union_1_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_apply_subst">Make.well_formed_apply_subst</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_fold">Make.well_formed_fold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed_unfold">Make.well_formed_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_rpo">Make.wf_rpo</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_rpo_term">Make.wf_rpo_term</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_mul_rest">Make.wf_on_mul_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_lex_rest">Make.wf_on_lex_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_on_rest">Make.wf_on_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size3">Make.wf_size3</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size2">Make.wf_size2</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.wf_size">Make.wf_size</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_permut">Make.without_red_permut</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_not_common">Make.without_red_remove_not_common</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_not_common_aux">Make.without_red_remove_not_common_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_add_without_red">Make.without_red_add_without_red</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_singleton">Make.without_red_singleton</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_nil">Make.without_red_nil</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_add">Make.without_red_add</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove">Make.without_red_remove</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_remove_red">Make.without_red_remove_red</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.without_red_filter_aux">Make.without_red_filter_aux</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#mapListLemma">mapListLemma</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.rpo.more_list.html#map_app">map_app</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#map_map">map_map</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxApp">maxApp</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma">maxLemma</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma2">maxLemma2</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxLemma3">maxLemma3</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxSubTerm">maxSubTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxSubTerms">maxSubTerms</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#maxVarFreeVar">maxVarFreeVar</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#max_nf">max_nf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_regL">max_le_regL</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_regR">max_le_regR</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#max_le_plus">max_le_plus</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_refl">max_refl</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_ge_b">max_ge_b</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_ge_a">max_ge_a</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_comm">max_comm</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_dec">max_dec</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_tl">max_v_tl</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_ge">max_v_ge</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_lub">max_v_lub</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#max_v_2">max_v_2</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html#membersOk">membersOk</a> [in <a href="hydras.solutions_exercises.OnCodeList.html">hydras.solutions_exercises.OnCodeList</a>]<br/>
+<a href="hydras.solutions_exercises.OnCodeList.html#membersOk'">membersOk'</a> [in <a href="hydras.solutions_exercises.OnCodeList.html">hydras.solutions_exercises.OnCodeList</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#members_omega">members_omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#members_eq">members_eq</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#members_proper">members_proper</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#mem_head_mult_inv">mem_head_mult_inv</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_sort_correct">merge_sort_correct</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_permutation">merge_permutation</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_LocallySorted">merge_LocallySorted</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_Forall">merge_Forall</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_equation">merge_equation</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#merge_rect">merge_rect</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="Goedel.PRrepresentable.html#minimize1">minimize1</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration">minoration</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_3">minoration_3</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_2">minoration_2</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_1">minoration_1</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#minoration_0">minoration_0</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minPR_correct">minPR_correct</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_le">minus_le</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_a_a">minus_a_a</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#minus_lt">minus_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#minus_semi_assoc">minus_semi_assoc</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Schutte.Addition.html#minus_exists">minus_exists</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_refl">min_refl</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_le_b">min_le_b</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_le_ad">min_le_ad</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_dec">min_dec</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_comm">min_comm</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_max_iff">min_max_iff</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#min_alt_correct">min_alt_correct</a> [in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlargeS_iff">mlargeS_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_omega">mlarge_omega</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_FS">mlarge_FS</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_iff">mlarge_iff</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_unshift">mlarge_unshift</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#mlarge_unicity">mlarge_unicity</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#mlarge_unicity">mlarge_unicity</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#ml_1">ml_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#ml_psi">ml_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.model.html#ModelConsistent">ModelConsistent</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_weak">mono_weak</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_injective">mono_injective</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#mono_le">mono_le</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mono_F_Zero">mono_F_Zero</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq_weak2">mono_seq_weak2</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq2">mono_seq2</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#mono_seq">mono_seq</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#MPSys">MPSys</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_eqn3">ms_eqn3</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_nf">ms_nf</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#ms_eqn2">ms_eqn2</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#multE4">multE4</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#multE5">multE5</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#multLemma1">multLemma1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#multLemma2">multLemma2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_plus_distr_l">mult_plus_distr_l</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_mono">mult_mono</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_nf">mult_nf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_Sn_add">mult_Sn_add</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_0_a">mult_0_a</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_nf_fin">mult_nf_fin</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_a_1">mult_a_1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_1_a">mult_1_a</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_a_0">mult_a_0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_compat">mult_compat</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#mult_fin_omega">mult_fin_omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#mult_Sn_dist">mult_Sn_dist</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#mult_lt_lt">mult_lt_lt</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mult_ref">mult_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mult_ref0">mult_ref0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="additions.Monoid_instances.html#mult_mod_associative">mult_mod_associative</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mult_plus_distr_l">mult_plus_distr_l</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mult_fin_rw">mult_fin_rw</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn_mono3">mult_Sn_mono3</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn_mono2">mult_Sn_mono2</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r_mono">mult_fin_r_mono</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r_one">mult_fin_r_one</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#mult_correct">mult_correct</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#mult_eqn1">mult_eqn1</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu_beta_h">mu_beta_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#mu_positive">mu_positive</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="gaia_hydras.GHydra.html#mVariant">mVariant</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="additions.fib.html#my_pow_m2lmul">my_pow_m2lmul</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#my_pow_m3lmul">my_pow_m3lmul</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#my_powP">my_powP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_lt">m_lt</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_variant_LT">m_variant_LT</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#m_ge">m_ge</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_dominates_Ack">m_dominates_Ack</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_dominates_Ack_from_3">m_dominates_Ack_from_3</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_ge_Ack">m_ge_Ack</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_ge_F_omega">m_ge_F_omega</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#m_eqn">m_eqn</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#m_ms">m_ms</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#m_nf">m_nf</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_lt">m_lt</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_decrease">m_decrease</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#m_ge">m_ge</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="additions.Monoid_instances.html#m_neq_0">m_neq_0</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="gaia_hydras.GHydra.html#m_nf">m_nf</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="gaia_hydras.GHydra.html#m_ref">m_ref</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#m_strict_mono">m_strict_mono</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#m_ge_generic">m_ge_generic</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#m_ge_0">m_ge_0</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_ge">m_ge</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_lt">m_lt</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#m_big_h_not_null">m_big_h_not_null</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#m_def">m_def</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#m_ge">m_ge</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#m_lt">m_lt</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="additions.fib.html#m2lfibP">m2lfibP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m2lmulP">m2lmulP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_eq_intros">M2_eq_intros</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Naive.html#M2.Id2_neutral">M2.Id2_neutral</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.M2_mult_assoc">M2.M2_mult_assoc</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#M2.power_of_plus">M2.power_of_plus</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.fib.html#m3lfibP">m3lfibP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#m3lmulP">m3lmulP</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<br/><br/><a id="lemma_N"></a><h2>N </h2>
+<a href="hydras.Ackermann.LNN.html#nAnd">nAnd</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nAnd">nAnd</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nAnd">nAnd</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natLE">natLE</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natLT">natLT</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natNE">natNE</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natPlus">natPlus</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#natTimes">natTimes</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#nat_double_or_s_double">nat_double_or_s_double</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="additions.Compatibility.html#nat_power_ok">nat_power_ok</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#nat_eqb_false">nat_eqb_false</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_2_term_mono">nat_2_term_mono</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_lt_psi">nat_lt_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nat_lt_cons">nat_lt_cons</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.rpo.more_list.html#nb_occ_app">nb_occ_app</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#nested_Ack_bound">nested_Ack_bound</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Fib2.html#neutral_r">neutral_r</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Fib2.html#neutral_l">neutral_l</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#newVar1">newVar1</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#newVar2">newVar2</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nExist">nExist</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nExist">nExist</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nExist">nExist</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="additions.Fib2.html#next_fib">next_fib</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nForall">nForall</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nForall">nForall</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nForall">nForall</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_LT_right">nf_LT_right</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_exp_F">nf_exp_F</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_LT_iff">nf_LT_iff</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega_omega">nf_omega_omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_phi0">nf_phi0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega">nf_omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_Wf">nf_Wf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_tail_lt">nf_tail_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_LT">nf_cons_LT</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_omega_tower">nf_omega_tower</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_coeff_irrelevance">nf_coeff_irrelevance</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_intro'">nf_intro'</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_intro">nf_intro</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_tail_lt_nf">nf_tail_lt_nf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_of_finite">nf_of_finite</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_fin">nf_fin</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_FS">nf_FS</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_b_cons_eq">nf_b_cons_eq</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_iff">nf_cons_iff</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_cons_inv">nf_cons_inv</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv3">nf_inv3</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv2">nf_inv2</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#nf_inv1">nf_inv1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_epsilon">nf_epsilon</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_epsilon0">nf_epsilon0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_omega">nf_omega</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#nf_fin">nf_fin</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Schutte.CNF.html#nf_bounded">nf_bounded</a> [in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_Acc">NF_Acc</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_inv2">NF_inv2</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF_inv1">NF_inv1</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#nf_ref">nf_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#nf_canon">nf_canon</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#nf_proof_unicity">nf_proof_unicity</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#nf_m">nf_m</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_intro">nf_intro</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_coeff_irrelevance">nf_coeff_irrelevance</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_Wf">nf_Wf</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c2">nf_c2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c1">nf_c1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b2">nf_b2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b1">nf_b1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a2">nf_a2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a1">nf_a1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_subterm">nf_subterm</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_fin_inv">nf_fin_inv</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_inv_tail">nf_inv_tail</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_c">nf_c</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_b">nf_b</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#nf_a">nf_a</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.fol.html#nilTerms">nilTerms</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nImp">nImp</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nImp">nImp</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nImp">nImp</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nnE">nnE</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nnE">nnE</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nnE">nnE</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nnI">nnI</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nnI">nnI</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nnI">nnI</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.NNtheory.html#nnPlusNotNeeded">nnPlusNotNeeded</a> [in <a href="hydras.Ackermann.NNtheory.html">hydras.Ackermann.NNtheory</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn1">nn1</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn2">nn2</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN2PA.html#NN2PA">NN2PA</a> [in <a href="hydras.Ackermann.NN2PA.html">hydras.Ackermann.NN2PA</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn3">nn3</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn4">nn4</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn5">nn5</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn6">nn6</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn7">nn7</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN72PA">NN72PA</a> [in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn8">nn8</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN82PA">NN82PA</a> [in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.NN.html#nn9">nn9</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#NN92PA">NN92PA</a> [in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#noMiddle">noMiddle</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#noMiddle">noMiddle</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#noMiddle">noMiddle</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.rpo.more_list.html#none_nb_occ_O">none_nb_occ_O</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Non_denum">Non_denum</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#nOr">nOr</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#nOr">nOr</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#nOr">nOr</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Schutte.AP.html#normal_phi0">normal_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Addition.html#normal_plus_alpha">normal_plus_alpha</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="Goedel.PRrepresentable.html#notBoundedForall">notBoundedForall</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.wConsistent.html#notCon2wNotCon">notCon2wNotCon</a> [in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#notInFreeVarSys">notInFreeVarSys</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notRelPR">notRelPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_LT_zero">not_LT_zero</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_zero_lt">not_zero_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_zero_gt_0">not_zero_gt_0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#not_lt_zero">not_lt_zero</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#not_double_is_s_double">not_double_is_s_double</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#not_le_gt">not_le_gt</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#not_le_ge">not_le_ge</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Schutte.CNF.html#not_AP_inv2">not_AP_inv2</a> [in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.CNF.html#not_AP_inv_0">not_AP_inv_0</a> [in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#not_acc">not_acc</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_countable_unbounded">not_countable_unbounded</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Not_Unbounded_countable">Not_Unbounded_countable</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Not_Unbounded_bounded">Not_Unbounded_bounded</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_succ_limit">not_is_succ_limit</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_limit_succ">not_is_limit_succ</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_limit_zero">not_is_limit_zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_is_succ_zero">not_is_succ_zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_lt_zero">not_lt_zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#not_gt_le">not_gt_le</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#not_lt_zero">not_lt_zero</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#not_decreasing">not_decreasing</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_decreasing">not_decreasing</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_decreasing_aux">not_decreasing_aux</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#not_acc">not_acc</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="additions.Compatibility.html#Npos_power_compat">Npos_power_compat</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.More_on_positive.html#Npos_gt_0">Npos_gt_0</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#Npos_diff_zero">Npos_diff_zero</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.rpo.more_list.html#nth_error_ok_in">nth_error_ok_in</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#nth_error_map">nth_error_map</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="additions.Compatibility.html#N_size_gt">N_size_gt</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#N_pow_compat">N_pow_compat</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#N_pow_power">N_pow_power</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_commute">N_bpow_commute</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_ok_R">N_bpow_ok_R</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow_ok">N_bpow_ok</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#N_bpow_commute">N_bpow_commute</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#N_bpow_ok_R">N_bpow_ok_R</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#N_bpow_ok">N_bpow_ok</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.More_on_positive.html#N_le_mul_pos">N_le_mul_pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_q0">N_pos_div_eucl_q0</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_rest">N_pos_div_eucl_rest</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_div_eucl_divides">N_pos_div_eucl_divides</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_pos_N2pos">N_pos_N2pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_le_1_pos">N_le_1_pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N_0_le_n">N_0_le_n</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#N_simpl">N_simpl</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="additions.More_on_positive.html#N2pos_lt_switch2">N2pos_lt_switch2</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N2pos_lt_switch">N2pos_lt_switch</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#N2pos_pos">N2pos_pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<br/><br/><a id="lemma_O"></a><h2>O </h2>
+<a href="hydras.Schutte.Ordering_Functions.html#of_image">of_image</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#of_beta'">of_beta'</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Addition.html#of_u">of_u</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#of_g">of_g</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.Euclidean_Chains.html#OK87">OK87</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#omeganf">omeganf</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
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+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_rw">o_finite_mult_rw</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult_nf">o_finite_mult_nf</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#o_finite_mult_S_rw">o_finite_mult_S_rw</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota">o2iota</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota_1">o2iota_1</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#o2iota_0">o2iota_0</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<br/><br/><a id="lemma_P"></a><h2>P </h2>
+<a href="hydras.Ackermann.NN2PA.html#PAboundedLT">PAboundedLT</a> [in <a href="hydras.Ackermann.NN2PA.html">hydras.Ackermann.NN2PA</a>]<br/>
+<a href="hydras.Ackermann.PAconsistent.html#PAconsistent">PAconsistent</a> [in <a href="hydras.Ackermann.PAconsistent.html">hydras.Ackermann.PAconsistent</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PAdec">PAdec</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="Goedel.rosserPA.html#PAIncomplete">PAIncomplete</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#paPlusSym">paPlusSym</a> [in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_for_free">param_correctness_for_free</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness">param_correctness</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_nat">param_correctness_nat</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#param_correctness_aux">param_correctness_aux</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.O2H.html#path_to_round_plus">path_to_round_plus</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#path_toS_round_plus">path_toS_round_plus</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#path_toS_trace">path_toS_trace</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_LT">path_to_LT</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_iff1">path_to_iff1</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to_inv1">path_to_inv1</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#path_to_omega_mult">path_to_omega_mult</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_standard_equiv">path_to_standard_equiv</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_standardS_equiv">path_toS_standardS_equiv</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_S_standard_pathS">path_to_S_standard_pathS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_S_iota_from">path_to_S_iota_from</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_LT">path_to_LT</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_LT">path_toS_LT</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_gnaw">path_to_gnaw</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_mult">path_to_mult</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_mult">path_toS_mult</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_tail">path_to_tail</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_tail">path_toS_tail</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_not_In_zero">path_to_not_In_zero</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_zero">path_to_zero</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero">path_toS_zero</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_inv">path_toS_zero_inv</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_one">path_toS_zero_one</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_zero_but_last">path_toS_zero_but_last</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_appR">path_to_appR</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_appR">path_toS_appR</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_decompose">path_to_decompose</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_decompose">path_toS_decompose</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_app">path_to_app</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_app">path_toS_app</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_gnawS">path_toS_gnawS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_acc_from">path_acc_from</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_nf">path_toS_nf</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_path_toS_iff">path_to_path_toS_iff</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_path_toS">path_to_path_toS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_path_to">path_toS_path_to</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_interval_inv_le">path_to_interval_inv_le</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_not_nil">path_to_not_nil</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.PAtheory.html#paZeroOrSucc">paZeroOrSucc</a> [in <a href="hydras.Ackermann.PAtheory.html">hydras.Ackermann.PAtheory</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA_Succ">PA_Succ</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA_Zero">PA_Zero</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa1">pa1</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa2">pa2</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa3">pa3</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa4">pa4</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa5">pa5</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#pa6">pa6</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PC_Zero">PC_Zero</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PD_Zero">PD_Zero</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PF87">PF87</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_RR">phi_mono_RR</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_psi">phi_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_6">phi_to_psi_6</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_5">phi_to_psi_5</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_4">phi_to_psi_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_3">phi_to_psi_3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_2">phi_to_psi_2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi_1">phi_to_psi_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_inj_r">phi_inj_r</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_weak_r">phi_mono_weak_r</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_mono_r">phi_mono_r</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi_plus_fin">phi_of_psi_plus_fin</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_alpha_zero_gt_alpha">phi_alpha_zero_gt_alpha</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_principalR">phi_principalR</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_spec1">phi_spec1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_le_ge">phi_le_ge</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_le">phi_le</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_fix">phi_fix</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_any_cons">phi_of_any_cons</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_nf">phi_nf</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases">phi_cases</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases'">phi_cases'</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_cases_aux">phi_cases_aux</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi_succ">phi_of_psi_succ</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_alpha_zero">phi_alpha_zero</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_principal">phi_principal</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi">phi_to_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_of_psi">phi_of_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_eq">phi0_eq</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_eq_bad">phi0_eq_bad</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_plus_mult">phi0_plus_mult</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono">phi0_mono</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono_strict_LT">phi0_mono_strict_LT</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_mono_strict">phi0_mono_strict</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_ltR">phi0_ltR</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0_lt">phi0_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_plus_lt_phi0R">phi0_mult_plus_lt_phi0R</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_plus_lt_phi0">phi0_mult_plus_lt_phi0</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#phi0_mult_lt_phi0">phi0_mult_lt_phi0</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Critical.html#phi0_well_named">phi0_well_named</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_lt_epsilon0_R">phi0_lt_epsilon0_R</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_lt_epsilon0">phi0_lt_epsilon0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_of_limit">phi0_of_limit</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_sup">phi0_sup</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_alpha_phi0_beta">phi0_alpha_phi0_beta</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_positive">phi0_positive</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_inj">phi0_inj</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_R_weak">phi0_mono_R_weak</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_R">phi0_mono_R</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono_weak">phi0_mono_weak</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_mono">phi0_mono</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_zero">phi0_zero</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_elim">phi0_elim</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0_ordering">phi0_ordering</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#phi0_ref">phi0_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#phi0_mono">phi0_mono</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Phi0_def">Phi0_def</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus_cons_cons_eqn">plus_cons_cons_eqn</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#plus_zero">plus_zero</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#plus_alpha_mult_phi0">plus_alpha_mult_phi0</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
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+<a href="hydras.Epsilon0.E0.html#Plus_rw">Plus_rw</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#plus_alpha_0">plus_alpha_0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_bi">plus_mono_bi</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
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+<a href="hydras.Schutte.Addition.html#plus_assoc'">plus_assoc'</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc3">plus_assoc3</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc2">plus_assoc2</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc1">plus_assoc1</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_FF">plus_FF</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_limit">plus_limit</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_reg_r">plus_reg_r</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_r_weak">plus_mono_r_weak</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_of_succ">plus_of_succ</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_mono_r">plus_mono_r</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_elim">plus_elim</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_ordering">plus_ordering</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#plus_correct">plus_correct</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.More_on_positive.html#positive_4step_ind">positive_4step_ind</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.AM.html#Pos_to_nat_diff_0">Pos_to_nat_diff_0</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow_compat">Pos_pow_compat</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow_power">Pos_pow_power</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_ok">Pos_iter_ok</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_op_ok">Pos_iter_op_ok</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Pos_iter_op_ok_0">Pos_iter_op_ok_0</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow_ok_R">Pos_bpow_ok_R</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow_ok">Pos_bpow_ok</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_of_bpow">Pos_bpow_of_bpow</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_of_plus">Pos_bpow_of_plus</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_ok_R">Pos_bpow_ok_R</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_ok">Pos_bpow_ok</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
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+<a href="additions.More_on_positive.html#pos_div_eucl_quotient_pos">pos_div_eucl_quotient_pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_gt_3">pos_gt_3</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_lt_wf">pos_lt_wf</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos_lt_mul">pos_lt_mul</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
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+<a href="additions.Compatibility.html#Pos2Nat_morph">Pos2Nat_morph</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
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+<a href="additions.More_on_positive.html#Pos2Nat_le_1_p">Pos2Nat_le_1_p</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.More_on_positive.html#pos2N_inj_add">pos2N_inj_add</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
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+<a href="hydras.Prelude.More_Arith.html#power_of_1">power_of_1</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
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+<a href="additions.Pow_variant.html#power_of_square">power_of_square</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_of_power">power_of_power</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_commute_with_x">power_commute_with_x</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_commute">power_commute</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="additions.Pow_variant.html#power_eq3">power_eq3</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#power_eq2">power_eq2</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
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+<a href="additions.Pow.html#power_eq3">power_eq3</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_eq2">power_eq2</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#power_eq1">power_eq1</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_is_a_refinement">power_R_is_a_refinement</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_1">power_R_1</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R_Mult">power_R_Mult</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3Min">pow3Min</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
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+<a href="additions.Euclidean_Chains.html#PO2">PO2</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO3">PO3</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO4">PO4</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO6">PO6</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO8">PO8</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#PO9">PO9</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
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+<a href="hydras.rpo.more_list.html#prop_map_without_repetition">prop_map_without_repetition</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Gamma0.T2.html#psi_eq">psi_eq</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_principal">psi_principal</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_lt_epsilon0">psi_lt_epsilon0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_lt_head">psi_lt_head</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#psi_le_cons">psi_le_cons</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PZero">PZero</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PZero">PZero</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#P_3">P_3</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#P_alpha">P_alpha</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#P_alpha_0">P_alpha_0</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#P_well_founded_induction_type">P_well_founded_induction_type</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#P0">P0</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<br/><br/><a id="lemma_Q"></a><h2>Q </h2>
+<a href="hydras.Epsilon0.F_alpha.html#QA0">QA0</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#QD0">QD0</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<br/><br/><a id="lemma_R"></a><h2>R </h2>
+<a href="hydras.Epsilon0.F_alpha.html#RAlim">RAlim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RAlim">RAlim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RB">RB</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RB">RB</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RBlim">RBlim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RBlim">RBlim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RC">RC</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RC">RC</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RClim">RClim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RClim">RClim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RD">RD</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RD">RD</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RDlim">RDlim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RDlim">RDlim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RE">RE</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RE">RE</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#reachable_S">reachable_S</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#rebindExist">rebindExist</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#rebindForall">rebindForall</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddExists">reduceAddExists</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddExistsOneWay">reduceAddExistsOneWay</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#reduceAddForalls">reduceAddForalls</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceAnd">reduceAnd</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceCloseList">reduceCloseList</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceExist">reduceExist</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceForall">reduceForall</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceIff">reduceIff</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceIfThenElse">reduceIfThenElse</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceImp">reduceImp</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceNot">reduceNot</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceOr">reduceOr</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#reduceSub">reduceSub</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#reduceSubAll">reduceSubAll</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#reduce_lt_plus">reduce_lt_plus</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.more_list.html#reduce_assoc_list">reduce_assoc_list</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#reduce1stCompose">reduce1stCompose</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#reduce2ndCompose">reduce2ndCompose</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines1_R">refines1_R</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines2_R">refines2_R</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RElim">RElim</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RElim">RElim</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#remove_first_sumand">remove_first_sumand</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#remove_heads_r_free">remove_heads_r_free</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#remove_heads_r">remove_heads_r</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem0">Rem0</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem01">Rem01</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem02">Rem02</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem03">Rem03</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem04">Rem04</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem05">Rem05</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem06">Rem06</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem08">Rem08</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem09">Rem09</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem1">rem1</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem1">Rem1</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem1">Rem1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem10">Rem10</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem11">Rem11</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem2">rem2</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem2">Rem2</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Rem2">Rem2</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem3">rem3</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Rem3">Rem3</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Rem3">Rem3</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Rem4">Rem4</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem6">rem6</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem61">rem61</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#rem62">rem62</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#replaceCompose2">replaceCompose2</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermMonotone">ReplaceFormulaTermMonotone</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermSub">ReplaceFormulaTermSub</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermMonotone">ReplaceTermsTermMonotone</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermSub">ReplaceTermsTermSub</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermMonotone">ReplaceTermTermMonotone</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTermMonotone">ReplaceTermTermsTermMonotone</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermSub">ReplaceTermTermSub</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableAlternate">RepresentableAlternate</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalfHelp">RepresentableHalfHelp</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf1Alternate">RepresentableHalf1Alternate</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf2Alternate">RepresentableHalf2Alternate</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable_ext">Representable_ext</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable2Expressible">Representable2Expressible</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#restrict">restrict</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restriction_fwd">restriction_fwd</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser'sIncompleteness">Rosser'sIncompleteness</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#Rosser'sIncompleteness">Rosser'sIncompleteness</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_free_equiv2">rounds_free_equiv2</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_free_equiv1">rounds_free_equiv1</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#rounds_trans">rounds_trans</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#round_decr">round_decr</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_n_inv">round_n_inv</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_n_remove_h">round_n_remove_h</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#round_plus_trans">round_plus_trans</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#round_S">round_S</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_star_intro">round_star_intro</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#RP">RP</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#RP">RP</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_7_1">rpo_7_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_6_4">rpo_6_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_6_1">rpo_6_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_4">rpo_5_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_1">rpo_5_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_3">rpo_5_3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_5_2">rpo_5_2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_4">rpo_4_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_1">rpo_4_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_3">rpo_4_3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_4_2">rpo_4_2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_4">rpo_3_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_1">rpo_3_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_3">rpo_3_3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_3_2">rpo_3_2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_4">rpo_2_4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_1">rpo_2_1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_3">rpo_2_3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_2_2">rpo_2_2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#rpo_trans">rpo_trans</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.First_toggle.html#Rwf">Rwf</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee_domain">R_nat_elaguee_domain</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee_fun">R_nat_elaguee_fun</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv_inj">R_inv_inj</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv_surj">R_inv_surj</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R_pred_Sn">R_pred_Sn</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R_inc_rpo">R_inc_rpo</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R0">R0</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#R0">R0</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="gaia_hydras.onType.html#R0">R0</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R0">R0</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R00">R00</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R00">R00</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R01">R01</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#R02">R02</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#R1">R1</a> [in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#R1">R1</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#R1">R1</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#R1">R1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#R1">R1</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#R1">R1</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#R1">R1</a> [in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<a href="gaia_hydras.onType.html#R1">R1</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#R1">R1</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#R1">R1</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#R1">R1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#R1">R1</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_mem_head">R1_mem_head</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_R2">R1_R2</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_hcons_mult">R1_hcons_mult</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#R1_hcons">R1_hcons</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R1_decr">R1_decr</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#R1_decr_0">R1_decr_0</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_remove_r_i">R1_remove_r_i</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_app">R1_app</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R1_iff">R1_iff</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#R12">R12</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R13">R13</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R14">R14</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R15">R15</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#R16">R16</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Epsilon0.Paths.html#R18">R18</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="hydras.Hydra.Battle_length.html#R2">R2</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
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+<a href="hydras.Schutte.Ordering_Functions.html#R2">R2</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
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+<a href="hydras.MoreAck.Ack.html#R5">R5</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
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+<br/><br/><a id="lemma_S"></a><h2>S </h2>
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+<a href="hydras.Gamma0.Gamma0.html#T2_size4">T2_size4</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size3">T2_size3</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size2">T2_size2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size1">T2_size1</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="lemma_U"></a><h2>U </h2>
+<a href="hydras.Schutte.Schutte_basics.html#unbounded">unbounded</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Unbounded_Cr">Unbounded_Cr</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Unbounded_not_countable">Unbounded_not_countable</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Addition.html#Unbounded_ge">Unbounded_ge</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#unique_decomposition">unique_decomposition</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#unique_decomposition">unique_decomposition</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_pred">unshift_pred</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_shift">unshift_shift</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_interval_pred">unshift_interval_pred</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_interval">unshift_interval</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_iota_from">unshift_iota_from</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_not_nil">unshift_not_nil</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_app">unshift_app</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift_but_last">unshift_but_last</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<br/><br/><a id="lemma_V"></a><h2>V </h2>
+<a href="hydras.Prelude.WfVariant.html#Variant_termination">Variant_termination</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Variant_lt_standard">Variant_lt_standard</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Variant_LT_standard">Variant_LT_standard</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Variant_lt_free">Variant_lt_free</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#Variant_LT_free_0">Variant_LT_free_0</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#variant_mono_free">variant_mono_free</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Vars.eq_variable_dec">Vars.eq_variable_dec</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Vid_eq">Vid_eq</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<br/><br/><a id="lemma_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wConsistent.html#wCon2Con">wCon2Con</a> [in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedFormulaCorrect1">wellFormedFormulaCorrect1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedFormulaCorrect2">wellFormedFormulaCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermCorrect1">wellFormedTermCorrect1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermCorrect2">wellFormedTermCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsCorrect1">wellFormedTermsCorrect1</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsCorrect2">wellFormedTermsCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTermsCorrect2">wellFormedTermTermsCorrect2</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.rpo.more_list.html#well_founded_length">well_founded_length</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded_rpo">well_founded_rpo</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_correct">Wfsum.f_correct</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_eqn">Wfsum.f_eqn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec_simp">Wfsum.f_spec_simp</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.lt_dec">Wfsum.lt_dec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.psum_exten">Wfsum.psum_exten</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Wf_transparent.html#wf_inverse_image_transparent">wf_inverse_image_transparent</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="additions.Wf_transparent.html#wf_incl_transparent">wf_incl_transparent</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#wf_measure">wf_measure</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="gaia_hydras.onType.html#wf_ltn">wf_ltn</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.rpo.closure.html#wf_trans">wf_trans</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_correct">Wf_ex3.f_correct</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_eqn">Wf_ex3.f_eqn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.odd_dec">Wf_ex3.odd_dec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0c">Wf_ex3.f0c</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0b">Wf_ex3.f0b</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f0a">Wf_ex3.f0a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp2">Wf_ex3.f_spec_simp2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp1">Wf_ex3.f_spec_simp1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec_simp">Wf_ex3.f_spec_simp</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte_wf">Wf_ex3.lte_wf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_correct">Wf_ex.f_correct</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_eqn">Wf_ex.f_eqn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f0">Wf_ex.f0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec_simp">Wf_ex.f_spec_simp</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#wf_lexico">wf_lexico</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.rpo.rpo.html#wf_lex">wf_lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<br/><br/><a id="lemma_X"></a><h2>X </h2>
+<a href="hydras.Ackermann.cPair.html#xPairDef">xPairDef</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<br/><br/><a id="lemma_Z"></a><h2>Z </h2>
+<a href="Goedel.PRrepresentable.html#zeroRepresentable">zeroRepresentable</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#zero_le">zero_le</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#zero_nf">zero_nf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#zero_lt">zero_lt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#zero_lt">zero_lt</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#zero_le">zero_le</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.AP.html#zero_lt_beta">zero_lt_beta</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#zero_ref">zero_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero_lt_omega">zero_lt_omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero_lt_succ">zero_lt_succ</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero_or_positive">zero_or_positive</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero_or_greater">zero_or_greater</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero_le">zero_le</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#zero_not_lim">zero_not_lim</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#zero_lt_succ">zero_lt_succ</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#zero_plus_alpha">zero_plus_alpha</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#ZLS_dec">ZLS_dec</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="additions.fib.html#ZtoRD">ZtoRD</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#ZtoRM">ZtoRM</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Compatibility.html#Z_pow_compat">Z_pow_compat</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Compatibility.html#Z_pow_compat_pos">Z_pow_compat_pos</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Naive.html#Z.power_of_plus">Z.power_of_plus</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#Z.power_S">Z.power_S</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><hr/>
+<h1>Constructor Index</h1>
+<a id="constructor_A"></a><h2>A </h2>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Eq">Ackermann.CompareT3Eq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Gt">Ackermann.CompareT3Gt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.CompareT3Lt">Ackermann.CompareT3Lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.cons">Ackermann.cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GeqNotGtn">Ackermann.T3GeqNotGtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GeqNotLtn">Ackermann.T3GeqNotLtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3GtnNotLeq">Ackermann.T3GtnNotLeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3LtnNotGeq">Ackermann.T3LtnNotGeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.zero">Ackermann.zero</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="additions.Trace_exercise.html#Add">Add</a> [in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.Mul">Alt.Eps0_prec.Mul</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.Lex">Alt.Eps0_prec.Lex</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.Free">Alt.Eps0_sig.Free</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.C">Alt.Eps0_sig.C</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.AC">Alt.Eps0_sig.AC</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_cons">Alt.Eps0_sig.ord_cons</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.ord_zero">Alt.Eps0_sig.ord_zero</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_S">Alt.Eps0_sig.nat_S</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.nat_0">Alt.Eps0_sig.nat_0</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alt.hnode">Alt.hnode</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#apply">apply</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap_intro">ap_intro</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#ap_intro">ap_intro</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#assoc">assoc</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.fol.html#atomic">atomic</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AXM">AXM</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.Addition_Chains.html#A_node">A_node</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<br/><br/><a id="constructor_B"></a><h2>B </h2>
+<a href="additions.Euclidean_Chains.html#Bad3.Fchain_proper_bad_prf">Bad3.Fchain_proper_bad_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Build_OF">Build_OF</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="constructor_C"></a><h2>C </h2>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Eq">CantorOrdinal.CompareT1Eq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Gt">CantorOrdinal.CompareT1Gt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.CompareT1Lt">CantorOrdinal.CompareT1Lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.cons">CantorOrdinal.cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GeqNotGtn">CantorOrdinal.T1GeqNotGtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GeqNotLtn">CantorOrdinal.T1GeqNotLtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1GtnNotLeq">CantorOrdinal.T1GtnNotLeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1LtnNotGeq">CantorOrdinal.T1LtnNotGeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.zero">CantorOrdinal.zero</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare">compare</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#composeFunc">composeFunc</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#cons">cons</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#cons_succ">cons_succ</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#CP">CP</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<br/><br/><a id="constructor_D"></a><h2>D </h2>
+<a href="hydras.Prelude.STDPP_compat.html#decide">decide</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide_rel">decide_rel</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="constructor_E"></a><h2>E </h2>
+<a href="hydras.Ackermann.fol.html#equal">equal</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="additions.Monoid_def.html#equiv">equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ1">EQ1</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ2">EQ2</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ3">EQ3</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ4">EQ4</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#EQ5">EQ5</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<br/><br/><a id="constructor_F"></a><h2>F </h2>
+<a href="hydras.Ackermann.folProof.html#FA1">FA1</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#FA2">FA2</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#FA3">FA3</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fchain_proper_prf">Fchain_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Gamma0.T2.html#finite_succ">finite_succ</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper_prf">Fkont_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#FO">FO</a> [in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Ackermann.fol.html#forallH">forallH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_last">Forall2R_last</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R_nil">Forall2R_nil</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#FS">FS</a> [in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bij_i">fun_bij_i</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_inj_i">fun_inj_i</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<br/><br/><a id="constructor_G"></a><h2>G </h2>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.Mul">Gamma0_prec.Mul</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.Lex">Gamma0_prec.Lex</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.Free">Gamma0_sig.Free</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.C">Gamma0_sig.C</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.AC">Gamma0_sig.AC</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_cons">Gamma0_sig.ord_cons</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_psi">Gamma0_sig.ord_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.ord_zero">Gamma0_sig.ord_zero</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_S">Gamma0_sig.nat_S</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.nat_0">Gamma0_sig.nat_0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Eq">Gamma0.CompareT2Eq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Gt">Gamma0.CompareT2Gt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.CompareT2Lt">Gamma0.CompareT2Lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.cons">Gamma0.cons</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GeqNotGtn">Gamma0.T2GeqNotGtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GeqNotLtn">Gamma0.T2GeqNotLtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2GtnNotLeq">Gamma0.T2GtnNotLeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2LtnNotGeq">Gamma0.T2LtnNotGeq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.zero">Gamma0.zero</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons">gcons</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons_lt_e0">gcons_lt_e0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gcons_nf">gcons_nf</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#GEN">GEN</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.Euclidean_Chains.html#gen_K">gen_K</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#gen_F">gen_F</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.mkg0">G0.mkg0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="constructor_H"></a><h2>H </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#hcons">hcons</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.O2H.html#hh_2">hh_2</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#hh_1">hh_1</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hnil">hnil</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="constructor_I"></a><h2>I </h2>
+<a href="hydras.Ackermann.fol.html#impH">impH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#IMP1">IMP1</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#IMP2">IMP2</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InHWit">InHWit</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="additions.Trace_exercise.html#Init">Init</a> [in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit_cons">is_limit_cons</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit_0">is_limit_0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr_rel">is_pr_rel</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr">is_pr</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="constructor_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#Kchain_proper_prf">Kchain_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="constructor_L"></a><h2>L </h2>
+<a href="hydras.Ackermann.fol.html#language">language</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#lex_2">lex_2</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#lex_1">lex_1</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex1">lex1</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex1">lex1</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex2">lex2</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex2">lex2</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex3">lex3</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lex4">lex4</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#limit_step">limit_step</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_F_cons">limit_plus_F_cons</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_F_0">limit_plus_F_0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.ap_intro">LO.ap_intro</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT_right">LT_right</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT_middle">LT_middle</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT_left">LT_left</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_c">lt_a_phi0_b_c</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b_z">lt_a_phi0_b_z</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_right">lt_right</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_middle">lt_middle</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_left">lt_left</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_7">lt_7</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_6">lt_6</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_5">lt_5</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_4">lt_4</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_3">lt_3</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_2">lt_2</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_1">lt_1</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LT_">LT_</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec1">L_spec1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec0">L_spec0</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<br/><br/><a id="constructor_M"></a><h2>M </h2>
+<a href="hydras.rpo.rpo.html#Make.Eq">Make.Eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul_rest_step">Make.List_mul_rest_step</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul_rest">Make.List_mul_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_eq_rest">Make.List_eq_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_gt_rest">Make.List_gt_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_mul">Make.List_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_eq">Make.List_eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.List_gt">Make.List_gt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Lt">Make.Lt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.mk_set">Make.mk_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.mk_sn">Make.mk_sn</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.dickson.html#Make.rmv_case">Make.rmv_case</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Subterm">Make.Subterm</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Term">Make.Term</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_eq_mul">Make.Top_eq_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_eq_lex">Make.Top_eq_lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.Top_gt">Make.Top_gt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Var">Make.Var</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked0">marked0</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked1">marked1</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#mkE0">mkE0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#mkord">mkord</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mkP">mkP</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#mkP">mkP</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#mkQ">mkQ</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#mks">mks</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Ackermann.model.html#model">model</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#MP">MP</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.AM.html#MUL">MUL</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#Mult">Mult</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#mult_op">mult_op</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Addition_Chains.html#Mul_node">Mul_node</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.FirstSteps.html#M2.mat">M2.mat</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.mat">M2.mat</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="constructor_N"></a><h2>N </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#node">node</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#notH">notH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<br/><br/><a id="constructor_O"></a><h2>O </h2>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Ok">Ok</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Mixin">ONDef.Mixin</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Pack">ONDef.Pack</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="additions.Addition_Chains.html#One_node">One_node</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.mkord">OO.mkord</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<br/><br/><a id="constructor_P"></a><h2>P </h2>
+<a href="hydras.Epsilon0.Paths.html#path_toS_cons">path_toS_cons</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_toS_1">path_toS_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_cons">path_to_cons</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to_1">path_to_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Plus_">Plus_</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_omega">PP_omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_exp">PP_exp</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_mult">PP_mult</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_add">PP_add</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#PP_fin">PP_fin</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRcons">PRcons</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.Lex">Precedence.Lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.Mul">Precedence.Mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#primRecFunc">primRecFunc</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRnil">PRnil</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#projFunc">projFunc</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.AM.html#PUSH">PUSH</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<br/><br/><a id="constructor_R"></a><h2>R </h2>
+<a href="hydras.Schutte.GRelations.html#rel_surj_i">rel_surj_i</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_inj_i">rel_inj_i</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_bij_i">rel_bij_i</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inj_i">rel_inj_i</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Remaining">Remaining</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="additions.AM.html#result_equiv_success">result_equiv_success</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#result_equiv_fail">result_equiv_fail</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Addition_Chains.html#Return">Return</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds_n">rounds_n</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds_1">rounds_1</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Eq">RPO.Eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_mul">RPO.List_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_eq">RPO.List_eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.List_gt">RPO.List_gt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Lt">RPO.Lt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Subterm">RPO.Subterm</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_eq_mul">RPO.Top_eq_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_eq_lex">RPO.Top_eq_lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.Top_gt">RPO.Top_gt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R1_intro">R1_intro</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2_intro_2">R2_intro_2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2_intro">R2_intro</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="constructor_S"></a><h2>S </h2>
+<a href="hydras.rpo.term.html#Signature.AC">Signature.AC</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.C">Signature.C</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.Free">Signature.Free</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Gamma0.T2.html#single_nf">single_nf</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_cons">sorted_ge_cons</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_one">sorted_ge_one</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge_nil">sorted_ge_nil</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.AM.html#SQR">SQR</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#stack_equivn">stack_equivn</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#stack_equiv0">stack_equiv0</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#stdS_S">stdS_S</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#stdS_1">stdS_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#std_S">std_S</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#std_1">std_1</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps_S">steps_S</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps1">steps1</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step1">step1</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step2">step2</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#step3">step3</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_trans">subterm_trans</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_c">subterm_c</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_b">subterm_b</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm_a">subterm_a</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#succFunc">succFunc</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.T2.html#succ_finite">succ_finite</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#succ_step">succ_step</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Succ_">Succ_</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="additions.AM.html#SWAP">SWAP</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.mk_set">S.mk_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S0_rest">S0_rest</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S0_first">S0_first</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S1_next">S1_next</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S1_first">S1_first</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S2_next">S2_next</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S2_first">S2_first</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="constructor_T"></a><h2>T </h2>
+<a href="hydras.Ackermann.fol.html#Tcons">Tcons</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Term">Term.Term</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Var">Term.Var</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Times_">Times_</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Tnil">Tnil</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#Too_far">Too_far</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#totalness">totalness</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.a_">Toy.a_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.A_">Toy.A_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.b_">Toy.b_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.B_">Toy.B_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.C_">Toy.C_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.f_">Toy.f_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.g_">Toy.g_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.h_">Toy.h_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.P_">Toy.P_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Q_">Toy.Q_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.R_">Toy.R_</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#trace_toS">trace_toS</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#trace_to1">trace_to1</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.closure.html#t_trans">t_trans</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.rpo.closure.html#t_step">t_step</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<br/><br/><a id="constructor_V"></a><h2>V </h2>
+<a href="hydras.Ackermann.fol.html#var">var</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<br/><br/><a id="constructor_Z"></a><h2>Z </h2>
+<a href="hydras.Epsilon0.T1.html#zero">zero</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#zero">zero</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#zeroFunc">zeroFunc</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.T2.html#zero_lt_e0">zero_lt_e0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#zero_nf">zero_nf</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#zero_finite">zero_finite</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#Zero_">Zero_</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<br/><br/><hr/>
+<h1>Axiom Index</h1>
+<a id="axiom_A"></a><h2>A </h2>
+<a href="hydras.Schutte.Schutte_basics.html#AX1">AX1</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#AX2">AX2</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#AX3">AX3</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="axiom_F"></a><h2>F </h2>
+<a href="hydras.Prelude.LibHyps_Experiments.html#f">f</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<br/><br/><a id="axiom_G"></a><h2>G </h2>
+<a href="hydras.Prelude.LibHyps_Experiments.html#g">g</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<br/><br/><a id="axiom_H"></a><h2>H </h2>
+<a href="hydras.Prelude.LibHyps_Experiments.html#h">h</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<br/><br/><a id="axiom_I"></a><h2>I </h2>
+<a href="hydras.Schutte.Schutte_basics.html#inh_Ord">inh_Ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="axiom_L"></a><h2>L </h2>
+<a href="hydras.Schutte.Schutte_basics.html#lt">lt</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="axiom_O"></a><h2>O </h2>
+<a href="hydras.Schutte.Schutte_basics.html#Ord">Ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="axiom_P"></a><h2>P </h2>
+<a href="hydras.Prelude.LibHyps_Experiments.html#P">P</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.A">Precedence.A</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec">Precedence.prec</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_transitive">Precedence.prec_transitive</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_antisym">Precedence.prec_antisym</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.prec_dec">Precedence.prec_dec</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.status">Precedence.status</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<br/><br/><a id="axiom_R"></a><h2>R </h2>
+<a href="hydras.rpo.rpo.html#RPO.rpo_add_context">RPO.rpo_add_context</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_subst">RPO.rpo_subst</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_trans">RPO.rpo_trans</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_closure">RPO.rpo_closure</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.wf_rpo">RPO.wf_rpo</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<br/><br/><a id="axiom_S"></a><h2>S </h2>
+<a href="hydras.rpo.term.html#Signature.arity">Signature.arity</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.eq_symbol_dec">Signature.eq_symbol_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.symb">Signature.symb</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.decidable_set.html#S.A">S.A</a> [in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.cardinal_subset">S.cardinal_subset</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.decidable_set.html#S.eq_A_dec">S.eq_A_dec</a> [in <a href="hydras.rpo.decidable_set.html">hydras.rpo.decidable_set</a>]<br/>
+<br/><br/><a id="axiom_T"></a><h2>T </h2>
+<a href="hydras.rpo.term.html#Term.empty_subst_is_id_list">Term.empty_subst_is_id_list</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.empty_subst_is_id">Term.empty_subst_is_id</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.eq_term_dec">Term.eq_term_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.is_a_pos_exists_subtem">Term.is_a_pos_exists_subtem</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos_list_replace_at_pos_in_subterm">Term.replace_at_pos_list_replace_at_pos_in_subterm</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos_is_replace_at_pos2">Term.replace_at_pos_is_replace_at_pos2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos_is_replace_at_pos1">Term.replace_at_pos_is_replace_at_pos1</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.replace_at_pos_unfold">Term.replace_at_pos_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_subterm_at_pos">Term.size_subterm_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_direct_subterm">Term.size_direct_subterm</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_ge_one">Term.size_ge_one</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.size_unfold">Term.size_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subst_comp_is_subst_comp">Term.subst_comp_is_subst_comp</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subst_comp_is_subst_comp_aux1">Term.subst_comp_is_subst_comp_aux1</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos">Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec8">Term.term_rec8</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec7">Term.term_rec7</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec4">Term.term_rec4</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec3">Term.term_rec3</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term_rec2">Term.term_rec2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_apply_subst">Term.well_formed_apply_subst</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_fold">Term.well_formed_fold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.well_formed_unfold">Term.well_formed_unfold</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<br/><br/><a id="axiom_V"></a><h2>V </h2>
+<a href="hydras.rpo.term.html#Variables.eq_variable_dec">Variables.eq_variable_dec</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Variables.var">Variables.var</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<br/><br/><hr/>
+<h1>Inductive Index</h1>
+<a id="inductive_A"></a><h2>A </h2>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.compare_T3">Ackermann.compare_T3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3">Ackermann.T3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3leq_xor_gtn">Ackermann.T3leq_xor_gtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ltn_xor_geq">Ackermann.T3ltn_xor_geq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.status_type">Alt.Eps0_prec.status_type</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.arity_type">Alt.Eps0_sig.arity_type</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb0">Alt.Eps0_sig.symb0</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alt.Hydra">Alt.Hydra</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.empty_set">Alt.Vars.empty_set</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#answer">answer</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ap">ap</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#ap">ap</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Assoc">Assoc</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="inductive_B"></a><h2>B </h2>
+<a href="additions.Euclidean_Chains.html#Bad3.Fchain_proper">Bad3.Fchain_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="inductive_C"></a><h2>C </h2>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.compare_T1">CantorOrdinal.compare_T1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1">CantorOrdinal.T1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1leq_xor_gtn">CantorOrdinal.T1leq_xor_gtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.T1ltn_xor_geq">CantorOrdinal.T1ltn_xor_geq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Compare">Compare</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="additions.Addition_Chains.html#computation">computation</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<br/><br/><a id="inductive_D"></a><h2>D </h2>
+<a href="hydras.Prelude.STDPP_compat.html#Decision">Decision</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="inductive_E"></a><h2>E </h2>
+<a href="additions.Monoid_def.html#Equiv">Equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<br/><br/><a id="inductive_F"></a><h2>F </h2>
+<a href="additions.Euclidean_Chains.html#Fchain_proper">Fchain_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper">Fkont_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2R">Forall2R</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Formula">Formula</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Fuel.html#fuel">fuel</a> [in <a href="hydras.Prelude.Fuel.html">hydras.Prelude.Fuel</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_bijection">fun_bijection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#fun_injection">fun_injection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<br/><br/><a id="inductive_G"></a><h2>G </h2>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_prec.status_type">Gamma0_prec.status_type</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.arity_type">Gamma0_sig.arity_type</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Gamma0_sig.symb0">Gamma0_sig.symb0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.compare_T2">Gamma0.compare_T2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2">Gamma0.T2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2leq_xor_gtn">Gamma0.T2leq_xor_gtn</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2ltn_xor_geq">Gamma0.T2ltn_xor_geq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="inductive_H"></a><h2>H </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydra">Hydra</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydrae">Hydrae</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="inductive_I"></a><h2>I </h2>
+<a href="additions.Trace_exercise.html#info">info</a> [in <a href="additions.Trace_exercise.html">additions.Trace_exercise</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InH">InH</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="additions.AM.html#instr">instr</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPR">isPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRrel">isPRrel</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_limit">is_limit</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_successor">is_successor</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#is_finite">is_finite</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<br/><br/><a id="inductive_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#Kchain_proper">Kchain_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="inductive_L"></a><h2>L </h2>
+<a href="hydras.Prelude.Simple_LexProd.html#lexico">lexico</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#lexpower">lexpower</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod">lex_prod</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#limit_plus_fin">limit_plus_fin</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNNRelation">LNNRelation</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTFunction">LNTFunction</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#LNTRelation">LNTRelation</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.ap">LO.ap</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt">lt</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT_cases">LT_cases</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_a_phi0_b">lt_a_phi0_b</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_cases">lt_cases</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#lt_epsilon0">lt_epsilon0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#L_spec">L_spec</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<br/><br/><a id="inductive_M"></a><h2>M </h2>
+<a href="hydras.rpo.dickson.html#Make.multiset_extension_step">Make.multiset_extension_step</a> [in <a href="hydras.rpo.dickson.html">hydras.rpo.dickson</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo">Make.rpo</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_rest_step">Make.rpo_mul_rest_step</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul_rest">Make.rpo_mul_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex_rest">Make.rpo_lex_rest</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_mul">Make.rpo_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_lex">Make.rpo_lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_eq">Make.rpo_eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term">Make.term</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#marked">marked</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Hydra.O2H.html#mem_head">mem_head</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="additions.Addition_Chains.html#Monoid_Exp">Monoid_Exp</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_def.html#Mult_op">Mult_op</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<br/><br/><a id="inductive_N"></a><h2>N </h2>
+<a href="hydras.Gamma0.T2.html#nf">nf</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<br/><br/><a id="inductive_O"></a><h2>O </h2>
+<a href="hydras.Hydra.BigBattle.html#one_step">one_step</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<br/><br/><a id="inductive_P"></a><h2>P </h2>
+<a href="hydras.Epsilon0.Paths.html#path_toS">path_toS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path_to">path_to</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#ppT1">ppT1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.rpo.rpo.html#Precedence.status_type">Precedence.status_type</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Prf">Prf</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRec">PrimRec</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PrimRecs">PrimRecs</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="inductive_R"></a><h2>R </h2>
+<a href="hydras.Prelude.STDPP_compat.html#RelDecision">RelDecision</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_surjection">rel_surjection</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#rel_injection">rel_injection</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_bijection">rel_bijection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_injection">rel_injection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="additions.AM.html#result_equiv">result_equiv</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#rounds">rounds</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo">RPO.rpo</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_mul">RPO.rpo_mul</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_lex">RPO.rpo_lex</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#RPO.rpo_eq">RPO.rpo_eq</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R1">R1</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#R2">R2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="inductive_S"></a><h2>S </h2>
+<a href="additions.Euclidean_Chains.html#signature">signature</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.rpo.term.html#Signature.arity_type">Signature.arity_type</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#sorted_ge">sorted_ge</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.AM.html#stack_equiv">stack_equiv</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_to">standard_path_to</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path_toS">standard_path_toS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#step">step</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#steps">steps</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#subterm">subterm</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S0">S0</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S1">S1</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#S2">S2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="inductive_T"></a><h2>T </h2>
+<a href="hydras.Ackermann.fol.html#Term">Term</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Terms">Terms</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.rpo.term.html#Term.term">Term.term</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Total">Total</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Fun">Toy.Fun</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Rel">Toy.Rel</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#trace_to">trace_to</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.rpo.closure.html#trans_clos">trans_clos</a> [in <a href="hydras.rpo.closure.html">hydras.rpo.closure</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1">T1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#T2">T2</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<br/><br/><a id="inductive_V"></a><h2>V </h2>
+<a href="hydras.Gamma0.Gamma0.html#Vars.empty_set">Vars.empty_set</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><hr/>
+<h1>Projection Index</h1>
+<a id="projection_A"></a><h2>A </h2>
+<a href="hydras.Ackermann.fol.html#arityF">arityF</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#arityR">arityR</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#assoc">assoc</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="projection_B"></a><h2>B </h2>
+<a href="additions.Euclidean_Chains.html#Bad3.Fchain_proper_bad_prf">Bad3.Fchain_proper_bad_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#battle_ok">battle_ok</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#battle_rel">battle_rel</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="projection_C"></a><h2>C </h2>
+<a href="gaia_hydras.T1Bridge.html#cnf">cnf</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf">cnf</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#cnf_ok">cnf_ok</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeFInj">codeFInj</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#codeRInj">codeRInj</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#comparable_comp_spec">comparable_comp_spec</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#comparable_sto">comparable_sto</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#compare">compare</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="additions.Monoid_instances.html#c00">c00</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#c01">c01</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#c10">c10</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#c11">c11</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><a id="projection_D"></a><h2>D </h2>
+<a href="hydras.Prelude.STDPP_compat.html#decide">decide</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#decide_rel">decide_rel</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#decr">decr</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<br/><br/><a id="projection_E"></a><h2>E </h2>
+<a href="additions.Monoid_def.html#Eone_right">Eone_right</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eone_left">Eone_left</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_comm">Eop_comm</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_assoc">Eop_assoc</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eop_proper">Eop_proper</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#equiv">equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Eq_equiv">Eq_equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<br/><br/><a id="projection_F"></a><h2>F </h2>
+<a href="additions.Euclidean_Chains.html#Fchain_proper_prf">Fchain_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fkont_proper_prf">Fkont_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Ackermann.model.html#func">func</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Functions">Functions</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<br/><br/><a id="projection_G"></a><h2>G </h2>
+<a href="additions.Strategies.html#gamma_gt">gamma_gt</a> [in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="additions.Strategies.html#gamma_lt">gamma_lt</a> [in <a href="additions.Strategies.html">additions.Strategies</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.vnf">G0.vnf</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.vnf_ok">G0.vnf_ok</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="projection_I"></a><h2>I </h2>
+<a href="hydras.Schutte.MoreEpsilonIota.html#InHWit">InHWit</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_inv2">iso_inv2</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_inv1">iso_inv1</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#iso_compare">iso_compare</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr_rel">is_pr_rel</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#is_pr">is_pr</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="projection_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#Kchain_proper_prf">Kchain_proper_prf</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="projection_L"></a><h2>L </h2>
+<a href="hydras.Schutte.Well_Orders.html#Lt_irreflexive">Lt_irreflexive</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Lt_trans">Lt_trans</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<br/><br/><a id="projection_M"></a><h2>M </h2>
+<a href="hydras.rpo.list_set.html#Make.is_a_set">Make.is_a_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.sn">Make.sn</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.support">Make.support</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.tt">Make.tt</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="additions.Monoid_def.html#mult_op">mult_op</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#m_bounded">m_bounded</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c00">M2.c00</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c00">M2.c00</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c01">M2.c01</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c01">M2.c01</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c10">M2.c10</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c10">M2.c10</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.c11">M2.c11</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Naive.html#M2.c11">M2.c11</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="projection_N"></a><h2>N </h2>
+<a href="hydras.Hydra.BigBattle.html#nh">nh</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#n1">n1</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#n2">n2</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<br/><br/><a id="projection_O"></a><h2>O </h2>
+<a href="hydras.Schutte.Ordering_Functions.html#OF_mono">OF_mono</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#OF_onto">OF_onto</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#OF_total">OF_total</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#OF_segment">OF_segment</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.type">ONDef.type</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="additions.Monoid_def.html#one_right">one_right</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#one_left">one_left</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#On_compare_spec">On_compare_spec</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_correct_onto">ON_correct_onto</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_correct_inj">ON_correct_inj</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_wf">ON_wf</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_comp">ON_comp</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.data">OO.data</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.data_ok">OO.data_ok</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="additions.Monoid_def.html#op_comm">op_comm</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#op_assoc">op_assoc</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<br/><br/><a id="projection_P"></a><h2>P </h2>
+<a href="hydras.Epsilon0.F_alpha.html#PA">PA</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PA">PA</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PB">PB</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PB">PB</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PC">PC</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PC">PC</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PD">PD</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PD">PD</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#PE">PE</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#PE">PE</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<br/><br/><a id="projection_Q"></a><h2>Q </h2>
+<a href="hydras.Epsilon0.F_alpha.html#QA">QA</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#QD">QD</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#QE">QE</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<br/><br/><a id="projection_R"></a><h2>R </h2>
+<a href="hydras.Ackermann.model.html#rel">rel</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Relations">Relations</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#round">round</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<br/><br/><a id="projection_S"></a><h2>S </h2>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.dec_dec">Semibundled.dec_dec</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec_total">Semibundled.total_dec_total</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec_pre_order">Semibundled.total_dec_pre_order</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Semibundled.total_dec">Semibundled.total_dec</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_onto">SubON_onto</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_incl">SubON_incl</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_compare">SubON_compare</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.is_a_set">S.is_a_set</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.list_set.html#S.support">S.support</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<br/><br/><a id="projection_T"></a><h2>T </h2>
+<a href="hydras.Prelude.DecPreOrder.html#totalness">totalness</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#total_pre_order_total">total_pre_order_total</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#total_pre_order_pre">total_pre_order_pre</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<br/><br/><a id="projection_U"></a><h2>U </h2>
+<a href="hydras.Ackermann.model.html#U">U</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<br/><br/><a id="projection_V"></a><h2>V </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#variant_decr">variant_decr</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="projection_W"></a><h2>W </h2>
+<a href="hydras.Schutte.Well_Orders.html#well_order">well_order</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Prelude.WfVariant.html#wf">wf</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<br/><br/><hr/>
+<h1>Section Index</h1>
+<a id="section_A"></a><h2>A </h2>
+<a href="hydras.Epsilon0.Hprime.html#Abstract_Properties">Abstract_Properties</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.rel_to_fun">AB_given.rel_to_fun</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given.f_given">AB_given.f_given</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#AB_given">AB_given</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties.Induc_step">Ack_Properties.Induc_step</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack_Properties">Ack_Properties</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.Compatibility.html#Adaptation">Adaptation</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.well_founded">Alt.well_founded</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_closed">AP_closed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded.ksi_fixed">AP_Unbounded.ksi_fixed</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP_Unbounded">AP_Unbounded</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#Arith_lemmas">Arith_lemmas</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A_def">A_def</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#A_given">A_given</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<br/><br/><a id="section_B"></a><h2>B </h2>
+<a href="hydras.MoreAck.BadSubst.html#BadExample">BadExample</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.CounterExample">Bad.CounterExample</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#bare_syntax">bare_syntax</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#battle_length_notPR">battle_length_notPR</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="hydras.Hydra.Battle_length.html#Battle_length">Battle_length</a> [in <a href="hydras.Hydra.Battle_length.html">hydras.Hydra.Battle_length</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_power_proof">binary_power_proof</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#Bounded">Bounded</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_by_induction">building_ordering_function_by_induction</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1.beta_fixed">building_ordering_function_1.beta_fixed</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#building_ordering_function_1">building_ordering_function_1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="section_C"></a><h2>C </h2>
+<a href="hydras.Epsilon0.Canon.html#Canon_examples">Canon_examples</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.AddLocalNotation">CantorOrdinal.AddLocalNotation</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#Check_Proof">Check_Proof</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#close">close</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#Code_Free_Vars">Code_Free_Vars</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#Code_Substitute_Term">Code_Substitute_Term</a> [in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Code_Substitute_Formula">Code_Substitute_Formula</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.codeTermFormDef">Code_Term_Formula_Proof.codeTermFormDef</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof.LcodeDef">Code_Term_Formula_Proof.LcodeDef</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.code.html#Code_Term_Formula_Proof">Code_Term_Formula_Proof</a> [in <a href="hydras.Ackermann.code.html">hydras.Ackermann.code</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#code_nat_list">code_nat_list</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf.LNN">code_SysPrf.LNN</a> [in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="Goedel.codeSysPrf.html#code_SysPrf">code_SysPrf</a> [in <a href="Goedel.codeSysPrf.html">Goedel.codeSysPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#Code_PA">Code_PA</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Comparable">Comparable</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#compose2Examples">compose2Examples</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Composition">Composition</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.AM.html#CompositionProofs">CompositionProofs</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.App">CompositionProofs.App</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FFK">CompositionProofs.FFK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.FK">CompositionProofs.FK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#CompositionProofs.KFK">CompositionProofs.KFK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Constant_to_standard_Proof">Constant_to_standard_Proof</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Correctness">Correctness</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable">Countable</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_bijection">Countable.Countable_bijection</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Countable_seq_range">Countable.Countable_seq_range</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#Countable.Definitions">Countable.Definitions</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Counter_Example">Counter_Example</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_Order">CPair_Order</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_projections">CPair_projections</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#CPair_Injectivity">CPair_Injectivity</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.solutions_exercises.schutte_cnf_counter_example.html#Cter_example">Cter_example</a> [in <a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">hydras.solutions_exercises.schutte_cnf_counter_example</a>]<br/>
+<br/><br/><a id="section_D"></a><h2>D </h2>
+<a href="hydras.Ackermann.Deduction.html#Deduction_Theorem">Deduction_Theorem</a> [in <a href="hydras.Ackermann.Deduction.html">hydras.Ackermann.Deduction</a>]<br/>
+<a href="additions.FirstSteps.html#Definitions">Definitions</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Definitions">Definitions</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Definitions">Definitions</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Defs">Defs</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#Defs">Defs</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#Defs">Defs</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#depth_rec_demo">depth_rec_demo</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof.First_attempt">Direct_proof.well_foundedness_proof.First_attempt</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Direct_proof.well_foundedness_proof">Direct_proof.well_foundedness_proof</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#dom_AckNotPR">dom_AckNotPR</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_6">DS_iota.Proof_case_6</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_5">DS_iota.Proof_case_5</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota.Proof_case_4">DS_iota.Proof_case_4</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Hydra.O2H.html#DS_iota">DS_iota</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<br/><br/><a id="section_E"></a><h2>E </h2>
+<a href="hydras.Ackermann.folLogic3.html#Equality_Logic_Rules">Equality_Logic_Rules</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case3">Equations_for_addition.case3</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case2">Equations_for_addition.case2</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition.case1">Equations_for_addition.case1</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Equations_for_addition">Equations_for_addition</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="additions.Compatibility.html#Equivalence">Equivalence</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#essai">essai</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#evalList">evalList</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Examples">Examples</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Examples">Examples</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Exs">Exs</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="section_F"></a><h2>F </h2>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Depth_Induction">First_Order_Logic.Formula_Depth_Induction</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic.Formula_Decidability">First_Order_Logic.Formula_Decidability</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#First_Order_Logic">First_Order_Logic</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#Fix">Fix</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Substitution_Properties">Fol_Properties.Substitution.Substitution_Properties</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution.Extensions">Fol_Properties.Substitution.Extensions</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Substitution">Fol_Properties.Substitution</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties.Free_Variables">Fol_Properties.Free_Variables</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Fol_Properties">Fol_Properties</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#Forall2_right_induction">Forall2_right_induction</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Free_Variables">Free_Variables</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Free_Variables">Free_Variables</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_lt">F_monotony_l.case_lt</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l.case_eq">F_monotony_l.case_eq</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_monotony_l">F_monotony_l</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR.case_lt">F_alpha_notPR.case_lt</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_alpha_notPR">F_alpha_notPR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_omega_notPR">F_omega_notPR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<br/><br/><a id="section_G"></a><h2>G </h2>
+<a href="additions.AM.html#Gamma">Gamma</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Gamma">Gamma</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#Gamma.All_OK">Gamma.All_OK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage.to_nat_elagage">General_Relations.elagage.to_nat_elagage</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.elagage">General_Relations.elagage</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.surjection2injection">General_Relations.surjection2injection</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.injection2surjection">General_Relations.injection2surjection</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations.Definitions">General_Relations.Definitions</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#General_Relations">General_Relations</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic">Generic</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging">Generic.Merging</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness">Generic.Merging.Correctness</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Merging.Correctness.merge_sort">Generic.Merging.Correctness.merge_sort</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#Generic.Splitting">Generic.Splitting</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="Goedel.goedel1.html#Goedel's_1st_Incompleteness">Goedel's_1st_Incompleteness</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness.Goedel1PA">Goedel's_2nd_Incompleteness.Goedel1PA</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="Goedel.goedel2.html#Goedel's_2nd_Incompleteness">Goedel's_2nd_Incompleteness</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<br/><br/><a id="section_H"></a><h2>H </h2>
+<a href="hydras.Prelude.First_toggle.html#Hypos">Hypos</a> [in <a href="hydras.Prelude.First_toggle.html">hydras.Prelude.First_toggle</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#H'_F">H'_F</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_omega_cube_3">H'_omega_cube_3</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_cons">H'_cons</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<br/><br/><a id="section_I"></a><h2>I </h2>
+<a href="hydras.Ackermann.primRec.html#Ignore_Params">Ignore_Params</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.GHydra.html#ImpossibilityProof">ImpossibilityProof</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Free.html#Impossibility_Proof">Impossibility_Proof</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Free.html">hydras.Hydra.Epsilon0_Needed_Free</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Impossibility_Proof">Impossibility_Proof</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Std.html#Impossibility_Proof">Impossibility_Proof</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Std.html">hydras.Hydra.Epsilon0_Needed_Std</a>]<br/>
+<a href="hydras.Hydra.Omega2_Small.html#Impossibility_Proof">Impossibility_Proof</a> [in <a href="hydras.Hydra.Omega2_Small.html">hydras.Hydra.Omega2_Small</a>]<br/>
+<a href="hydras.Hydra.Omega_Small.html#Impossibility_Proof">Impossibility_Proof</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#Impossibility1">Impossibility1</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Inclusion_ij">Inclusion_ij</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#inductive_step">inductive_step</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="additions.Wf_transparent.html#Inverse_Image_transp">Inverse_Image_transp</a> [in <a href="additions.Wf_transparent.html">additions.Wf_transparent</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inversion_of_bijection">inversion_of_bijection</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<br/><br/><a id="section_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#KFK_proof">KFK_proof</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#KP">KP</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<br/><br/><a id="section_L"></a><h2>L </h2>
+<a href="hydras.Gamma0.Gamma0.html#lemmas_on_length">lemmas_on_length</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Lemma_4_4_Proof">Lemma_4_4_Proof</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Lemma_4_3_Proof">Lemma_4_3_Proof</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Ackermann.fol.html#LExamples">LExamples</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lexprod">lexprod</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#lim">lim</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Ackermann.ListExt.html#List_Remove">List_Remove</a> [in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<a href="Goedel.fixPoint.html#LNN_FixPoint">LNN_FixPoint</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="Goedel.fixPoint.html#LNT_FixPoint">LNT_FixPoint</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Logic">Logic</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Logic">Logic</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Other_Rules">Logic_Rules.Other_Rules</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules.Not_Rules">Logic_Rules.Not_Rules</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Ackermann.folLogic.html#Logic_Rules">Logic_Rules</a> [in <a href="hydras.Ackermann.folLogic.html">hydras.Ackermann.folLogic</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#lt_incl_rpo">lt_incl_rpo</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#L_correct_proof">L_correct_proof</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<br/><br/><a id="section_M"></a><h2>M </h2>
+<a href="hydras.rpo.term.html#Make.DoubleRecursion">Make.DoubleRecursion</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.Recursion">Make.Recursion</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory.Consistent_Theory">Model_Theory.Consistent_Theory</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#Model_Theory">Model_Theory</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType">MoreOrderType</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#MoreOrderType.Succ_no_limit">MoreOrderType.Succ_no_limit</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Ackermann.folLogic2.html#More_Logic_Rules">More_Logic_Rules</a> [in <a href="hydras.Ackermann.folLogic2.html">hydras.Ackermann.folLogic2</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.Power_of_op">M_given.Power_of_op</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given.About_power">M_given.About_power</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow_variant.html#M_given">M_given</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#M_given.Power_of_op">M_given.Power_of_op</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow.html#M_given">M_given</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_def">M2_def</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Naive.html#M2.Definitions">M2.Definitions</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.FirstSteps.html#M2.M2_Definitions">M2.M2_Definitions</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<br/><br/><a id="section_N"></a><h2>N </h2>
+<a href="additions.Monoid_instances.html#Nmodulo">Nmodulo</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.NN.html#NN">NN</a> [in <a href="hydras.Ackermann.NN.html">hydras.Ackermann.NN</a>]<br/>
+<a href="additions.Addition_Chains.html#non_optimality_proof">non_optimality_proof</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#NotIncl">NotIncl</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="additions.Naive.html#N_mod.m_fixed">N_mod.m_fixed</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="section_O"></a><h2>O </h2>
+<a href="hydras.OrdinalNotations.ON_O.html#OA_given">OA_given</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.Lemmas">ONDef.Exports.Lemmas</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Packing">ONDef.Packing</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONEpsilon0Def">ONEpsilon0Def</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onFiniteDef">onFiniteDef</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaDef">onOmegaDef</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Gamma0.T2.html#on_length">on_length</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S2">On_alpha.S2</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha.S1">On_alpha.S1</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#On_alpha">On_alpha</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#On_Iota">On_Iota</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity">ordering_function_unicity</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_function_unicity_1">ordering_function_unicity_1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="section_P"></a><h2>P </h2>
+<a href="hydras.Ackermann.PA.html#PA">PA</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi_to_psi">phi_to_psi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#phi0_mult">phi0_mult</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Primitive_Recursive_Representable">Primitive_Recursive_Representable</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Primitive_recursion">Primitive_recursion</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH">ProofH</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#ProofH.Example">ProofH.Example</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#ProofOfLexwf">ProofOfLexwf</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S2">Proofs_of_unicity.Proofs.S2</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs.S1">Proofs_of_unicity.Proofs.S1</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity.Proofs">Proofs_of_unicity.Proofs</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.predSuccUnicity.html#Proofs_of_unicity">Proofs_of_unicity</a> [in <a href="hydras.solutions_exercises.predSuccUnicity.html">hydras.solutions_exercises.predSuccUnicity</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs.S1">Proofs_of_lt_succ_le.Proofs.S1</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le.Proofs">Proofs_of_lt_succ_le.Proofs</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.solutions_exercises.lt_succ_le.html#Proofs_of_lt_succ_le">Proofs_of_lt_succ_le</a> [in <a href="hydras.solutions_exercises.lt_succ_le.html">hydras.solutions_exercises.lt_succ_le</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_dist">Proof_of_dist</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf.Induction">Proof_of_mult_nf.Induction</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#Proof_of_mult_nf">Proof_of_mult_nf</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#Proof_of_MinIsPR">Proof_of_MinIsPR</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_lt1">Proof_of_oplus_lt1</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_assoc">Proof_of_oplus_assoc</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_oplus_comm">Proof_of_oplus_comm</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#Proof_of_plus_nf">Proof_of_plus_nf</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR.S_step">Proof_of_Ackn_PR.S_step</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#Proof_of_Ackn_PR">Proof_of_Ackn_PR</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Proof_of_4_5_2">Proof_of_4_5_2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.closedness">Proof_of_Lemma5.Alpha_positive.closedness</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive.Proof_unbounded">Proof_of_Lemma5.Alpha_positive.Proof_unbounded</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5.Alpha_positive">Proof_of_Lemma5.Alpha_positive</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Schutte.Critical.html#Proof_of_Lemma5">Proof_of_Lemma5</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Proof_of_nested_Ack_bound">Proof_of_nested_Ack_bound</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Proof_of_mult_ref">Proof_of_mult_ref</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#Proof_of_FibIsPR">Proof_of_FibIsPR</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#Proof_of_MinIsPR">Proof_of_MinIsPR</a> [in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Limit_case">Proof_of_H'_mono_l.Limit_case</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l.Succ_case">Proof_of_H'_mono_l.Succ_case</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_H'_mono_l">Proof_of_H'_mono_l</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_limit">Proof_of_Abstract_Properties.The_induction.alpha_limit</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction.alpha_Succ">Proof_of_Abstract_Properties.The_induction.alpha_Succ</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties.The_induction">Proof_of_Abstract_Properties.The_induction</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Proof_of_Abstract_Properties">Proof_of_Abstract_Properties</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Schutte.Addition.html#proof_of_associativity">proof_of_associativity</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf.seq_intro">Proof_of_lt_not_wf.seq_intro</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#Proof_of_lt_not_wf">Proof_of_lt_not_wf</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties">Properties</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha.The_induction">Properties_of_f_alpha.The_induction</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties_of_f_alpha">Properties_of_f_alpha</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_limit">Properties.The_induction.alpha_limit</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction.alpha_Succ">Properties.The_induction.alpha_Succ</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#Properties.The_induction">Properties.The_induction</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<br/><br/><a id="section_R"></a><h2>R </h2>
+<a href="additions.Addition_Chains.html#Refinement_proof">Refinement_proof</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Ackermann.folReplace.html#Replacement">Replacement</a> [in <a href="hydras.Ackermann.folReplace.html">hydras.Ackermann.folReplace</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableExpressible">RepresentableExpressible</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restricted_recursion">restricted_recursion</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restricted_recursion">restricted_recursion</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.rosserPA.html#Rosser's_Incompleteness">Rosser's_Incompleteness</a> [in <a href="Goedel.rosserPA.html">Goedel.rosserPA</a>]<br/>
+<a href="Goedel.rosser.html#Rosser's_Incompleteness">Rosser's_Incompleteness</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#R_given">R_given</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<br/><br/><a id="section_S"></a><h2>S </h2>
+<a href="additions.AM.html#Semantics">Semantics</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Sequences">Sequences</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR">sqrtIsPR</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#sqrtIsPR.Proof_isqrt">sqrtIsPR.Proof_isqrt</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#step">step</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#Strong_Recursion">Strong_Recursion</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars">SubAllVars</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SubAllVars.subAllCloseFrom">SubAllVars.subAllCloseFrom</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_properties">SubON_properties</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.subProp.html#Substitution_Properties">Substitution_Properties</a> [in <a href="hydras.Ackermann.subProp.html">hydras.Ackermann.subProp</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#succ">succ</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.solutions_exercises.is_F_monotonous.html#S1">S1</a> [in <a href="hydras.solutions_exercises.is_F_monotonous.html">hydras.solutions_exercises.is_F_monotonous</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1">S1</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#S1">S1</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit">S1.Limit</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Limit.S3">S1.Limit.S3</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor">S1.Successor</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#S1.Successor.S2">S1.Successor.S2</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#S2">S2</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html#S256">S256</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><a id="section_T"></a><h2>T </h2>
+<a href="hydras.rpo.term.html#Term.DoubleRecursion">Term.DoubleRecursion</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Term.Recursion">Term.Recursion</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context.About_WO">the_context.About_WO</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#the_context">the_context</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso.U_fixed">Th13_5.verso.U_fixed</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.verso">Th13_5.verso</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto.M_fixed">Th13_5.recto.M_fixed</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5.recto">Th13_5.recto</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#Th13_5">Th13_5</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Drinkers_theorem">Toy.Drinkers_theorem</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.OnSubstF">Toy.OnSubstF</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.PeirceProof">Toy.PeirceProof</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.PrimedSymbols">Toy.PrimedSymbols</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.ProofOfEx3">Toy.ProofOfEx3</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.Ackermann.LNN2LNT.html#Translate_Proof">Translate_Proof</a> [in <a href="hydras.Ackermann.LNN2LNT.html">hydras.Ackermann.LNN2LNT</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#trans_proof">trans_proof</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="section_V"></a><h2>V </h2>
+<a href="hydras.Prelude.WfVariant.html#Variants">Variants</a> [in <a href="hydras.Prelude.WfVariant.html">hydras.Prelude.WfVariant</a>]<br/>
+<br/><br/><a id="section_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term.Well_Formed_Formula">Well_Formed_Term.Well_Formed_Formula</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#Well_Formed_Term">Well_Formed_Term</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#well_founded">well_founded</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="additions.fib.html#with_matrices">with_matrices</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<br/><br/><hr/>
+<h1>Instance Index</h1>
+<a id="instance_A"></a><h2>A </h2>
+<a href="hydras.MoreAck.AckNotPR.html#Ackn_IsPR">Ackn_IsPR</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.addIsPR">Alt.addIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.addIsPR'">Alt.addIsPR'</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.const0_NIsPR">Alt.const0_NIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.pi2_5IsPR">Alt.pi2_5IsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.predIsPR">Alt.predIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.succIsPR">Alt.succIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.zeroIsPR">Alt.zeroIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="instance_B"></a><h2>B </h2>
+<a href="additions.Euclidean_Chains.html#Bad3.Fcompose_proper">Bad3.Fcompose_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.BinaryStrat.html#Binary_strat">Binary_strat</a> [in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boddIsPR">boddIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Monoid_def.html#bool_and_binop">bool_and_binop</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputationIsPR">boundComputationIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundSearchIsPR">boundSearchIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="gaia_hydras.GHydra.html#bVar">bVar</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<br/><br/><a id="instance_C"></a><h2>C </h2>
+<a href="hydras.Ackermann.cPair.html#callIsPR">callIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfAXMIsPR">checkPrfAXMIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfCPIsPR">checkPrfCPIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ1IsPR">checkPrfEQ1IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ2IsPR">checkPrfEQ2IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ3IsPR">checkPrfEQ3IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ4IsPR">checkPrfEQ4IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfEQ5IsPR">checkPrfEQ5IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA1IsPR">checkPrfFA1IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA2IsPR">checkPrfFA2IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfFA3IsPR">checkPrfFA3IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfGENIsPR">checkPrfGENIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfHelpIsPR">checkPrfHelpIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP1IsPR">checkPrfIMP1IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIMP2IsPR">checkPrfIMP2IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfIsPR">checkPrfIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#checkPrfMPIsPR">checkPrfMPIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#checkTraceIsPR">checkTraceIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeAndIsPR">codeAndIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeAppIsPR">codeAppIsPR</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNNRIsPR">codeArityLNNRIsPR</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTFIsPR">codeArityLNTFIsPR</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.Languages.html#codeArityLNTRIsPR">codeArityLNTRIsPR</a> [in <a href="hydras.Ackermann.Languages.html">hydras.Ackermann.Languages</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeAxmEqHelpIsPR">codeAxmEqHelpIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseIsPR">codeCloseIsPR</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeCloseListIsPR">codeCloseListIsPR</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarFormulaIsPR">codeFreeVarFormulaIsPR</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarListFormulaIsPR">codeFreeVarListFormulaIsPR</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermsIsPR">codeFreeVarTermsIsPR</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.codeFreeVar.html#codeFreeVarTermTermsIsPR">codeFreeVarTermTermsIsPR</a> [in <a href="hydras.Ackermann.codeFreeVar.html">hydras.Ackermann.codeFreeVar</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeIffIsPR">codeIffIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html#codeImpIsPR">codeImpIsPR</a> [in <a href="hydras.Ackermann.prLogic.html">hydras.Ackermann.prLogic</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeInductionSchemaIsPR">codeInductionSchemaIsPR</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeInIsPR">codeInIsPR</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeLengthIsPR">codeLengthIsPR</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeListRemoveIsPR">codeListRemoveIsPR</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#codeNatToTermIsPR">codeNatToTermIsPR</a> [in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeNewVarIsPR">codeNewVarIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeList.html#codeNoDupIsPR">codeNoDupIsPR</a> [in <a href="hydras.Ackermann.codeList.html">hydras.Ackermann.codeList</a>]<br/>
+<a href="hydras.Ackermann.prLogic.html#codeNotIsPR">codeNotIsPR</a> [in <a href="hydras.Ackermann.prLogic.html">hydras.Ackermann.prLogic</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#codeNthIsPR">codeNthIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars1IsPR">codeNVars1IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeNVars2IsPR">codeNVars2IsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codeOpenIsPR">codeOpenIsPR</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.checkPrf.html#codeOrIsPR">codeOrIsPR</a> [in <a href="hydras.Ackermann.checkPrf.html">hydras.Ackermann.checkPrf</a>]<br/>
+<a href="hydras.Ackermann.codePA.html#codePAIsPR">codePAIsPR</a> [in <a href="hydras.Ackermann.codePA.html">hydras.Ackermann.codePA</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#codeSubFormulaIsPR">codeSubFormulaIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermIsPR">codeSubTermIsPR</a> [in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.Ackermann.codeSubTerm.html#codeSubTermTermsIsPR">codeSubTermTermsIsPR</a> [in <a href="hydras.Ackermann.codeSubTerm.html">hydras.Ackermann.codeSubTerm</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#comp">comp</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#compare_T1">compare_T1</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#compare_omega2">compare_omega2</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#compare_O">compare_O</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#compare_nat_nat">compare_nat_nat</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#compare_mult">compare_mult</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#compare_plus">compare_plus</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#compare_fin">compare_fin</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#compare_E0">compare_E0</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#compare_nat">compare_nat</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#compare_T2">compare_T2</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#comparison_eq_dec">comparison_eq_dec</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_NIsPR">compose1_NIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_3IsPR">compose1_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_2IsPR">compose1_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose1_1IsPR">compose1_1IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2IsPR">compose2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_4IsPR">compose2_4IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_3IsPR">compose2_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_2IsPR">compose2_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose2_1IsPR">compose2_1IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_3IsPR">compose3_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_2IsPR">compose3_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose3_1IsPR">compose3_1IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose4_3IsPR">compose4_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#compose4_2IsPR">compose4_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Euclidean_Chains.html#comp_equiv_equivalence">comp_equiv_equivalence</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Comp_equiv">Comp_equiv</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Cons">Cons</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#const0_NIsPR">const0_NIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#const1_NIsPR">const1_NIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairIsPR">cPairIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi1IsPR">cPairPi1IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cPairPi2IsPR">cPairPi2IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTripleIsPR">cTripleIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi1IsPR">cTriplePi1IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi2IsPR">cTriplePi2IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#cTriplePi3IsPR">cTriplePi3IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<br/><br/><a id="instance_D"></a><h2>D </h2>
+<a href="additions.Monoid_def.html#Demo.nat_mult_op">Demo.nat_mult_op</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Dichotomy.html#Dicho_strat">Dicho_strat</a> [in <a href="additions.Dichotomy.html">additions.Dichotomy</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#div2IsPR">div2IsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#doubleIsPR">doubleIsPR</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<br/><br/><a id="instance_E"></a><h2>E </h2>
+<a href="additions.Pow_variant.html#Eop_proper">Eop_proper</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#Eop_proper">Eop_proper</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#Epsilon0">Epsilon0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Epsilon0">Epsilon0</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#epsilon0_dec">epsilon0_dec</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#epsilon0_pre_order">epsilon0_pre_order</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#Epsilon0_correct">Epsilon0_correct</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#eqIsPR">eqIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#equiv_equiv">equiv_equiv</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#equiv_equiv">equiv_equiv</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Trans">Equiv_Trans</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Sym">Equiv_Sym</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Refl">Equiv_Refl</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv_Equiv">Equiv_Equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#eq_equiv">eq_equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#evalStrongRecIsPR">evalStrongRecIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="additions.AM.html#exec_Proper">exec_Proper</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#expIsPR">expIsPR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#exp_alt_PR">exp_alt_PR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0add">E0add</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0compare">E0compare</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0fin">E0fin</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0finS">E0finS</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0mul">E0mul</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0one">E0one</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0zero">E0zero</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_comp">E0_comp</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0_sto">E0_sto</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_comp">E0_comp</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_sto">E0_sto</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_phi0">E0_phi0</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_succ">E0_succ</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0_omega">E0_omega</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="instance_F"></a><h2>F </h2>
+<a href="hydras.solutions_exercises.MorePRExamples.html#factIsPR">factIsPR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#fact_altIsPR">fact_altIsPR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fcompose_proper">Fcompose_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_proper">Fexp2_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Fexp2_nat_proper">Fexp2_nat_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FFK_proper">FFK_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fibIsPR">fibIsPR</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fibPRIsPR">fibPRIsPR</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#fib_altIsPR">fib_altIsPR</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter001IsPR">filter001IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter0011IsPR">filter0011IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter01IsPR">filter01IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter010IsPR">filter010IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter011IsPR">filter011IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter10IsPR">filter10IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter100IsPR">filter100IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1000IsPR">filter1000IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter101IsPR">filter101IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1011IsPR">filter1011IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter110IsPR">filter110IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#filter1100IsPR">filter1100IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#FinOrd">FinOrd</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_ON">finord_ON</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_comp">finord_comp</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_sto">finord_sto</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#finord_compare">finord_compare</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#FinOrd_Omega">FinOrd_Omega</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="additions.Euclidean_Chains.html#FK_proper">FK_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#free">free</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_n_PR">F_n_PR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#F_0_isPR">F_0_isPR</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#F_incl_ij">F_incl_ij</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F1_proper">F1_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F2_proper">F2_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#F3_proper">F3_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="instance_G"></a><h2>G </h2>
+<a href="hydras.Ackermann.primRec.html#gtIsPR">gtIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.compare_G0">G0.compare_G0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Finite">G0.Finite</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Gamma0">G0.Gamma0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Gamma0_comp">G0.Gamma0_comp</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt_sto">G0.lt_sto</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Omega">G0.Omega</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Phi">G0.Phi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Plus">G0.Plus</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.zero">G0.zero</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="instance_H"></a><h2>H </h2>
+<a href="hydras.Hydra.Omega_Small.html#height_var">height_var</a> [in <a href="hydras.Hydra.Omega_Small.html">hydras.Hydra.Omega_Small</a>]<br/>
+<a href="gaia_hydras.GHydra.html#hVar">hVar</a> [in <a href="gaia_hydras.GHydra.html">gaia_hydras.GHydra</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#HVariant">HVariant</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<a href="hydras.Hydra.Hydra_Termination.html#HVariant_0">HVariant_0</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<br/><br/><a id="instance_I"></a><h2>I </h2>
+<a href="hydras.Ackermann.primRec.html#idIsPR">idIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#If2IsPR">If2IsPR</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ignoreParamsIsPR">ignoreParamsIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Incl">Incl</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#indIsPR">indIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ind1ParamIsPR">ind1ParamIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ind2ParamIsPR">ind2ParamIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#inhNat">inhNat</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_Ord_Ord">Inh_Ord_Ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#Inh_OSets">Inh_OSets</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#InH_Ord">InH_Ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.Monoid_instances.html#Int63mult">Int63mult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#int63_mult_op">int63_mult_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Inverse_fun">Inverse_fun</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRextEqual">isPRextEqual</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#isPRTrans">isPRTrans</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#iSPR_Ack_Sn">iSPR_Ack_Sn</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#issqrtIsPR">issqrtIsPR</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#iterateIsPR">iterateIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="instance_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#KFF_proper">KFF_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F_proper">K2F_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k7_3_proper">k7_3_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<br/><br/><a id="instance_L"></a><h2>L </h2>
+<a href="hydras.Ackermann.codeNatToTerm.html#LcodeLNN">LcodeLNN</a> [in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.codeNatToTerm.html#LcodeLNT">LcodeLNT</a> [in <a href="hydras.Ackermann.codeNatToTerm.html">hydras.Ackermann.codeNatToTerm</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#leIsPR">leIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#leq_trans">leq_trans</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#lex_prod_dec">lex_prod_dec</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Lex_nat_wf">Lex_nat_wf</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le_trans">Le_trans</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#List_length">List_length</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.compare_oo">LO.compare_oo</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lo_comparable">LO.lo_comparable</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.lo_strorder">LO.lo_strorder</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.plus_assoc">LO.plus_assoc</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ltIsPR">ltIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LT_St">LT_St</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#lt_dec">lt_dec</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#lt_transitive">lt_transitive</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#lt_strorder">lt_strorder</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#lt_strorder">lt_strorder</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html#L_3_plus_omega">L_3_plus_omega</a> [in <a href="hydras.OrdinalNotations.Example_3PlusOmega.html">hydras.OrdinalNotations.Example_3PlusOmega</a>]<br/>
+<br/><br/><a id="instance_M"></a><h2>M </h2>
+<a href="hydras.Ackermann.primRec.html#maxIsPR">maxIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#max_assoc">max_assoc</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minIsPR">minIsPR</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#minIsPR">minIsPR</a> [in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR2.html#minPR_PR">minPR_PR</a> [in <a href="hydras.solutions_exercises.MinPR2.html">hydras.solutions_exercises.MinPR2</a>]<br/>
+<a href="hydras.solutions_exercises.MinPR.html#minPR_PR">minPR_PR</a> [in <a href="hydras.solutions_exercises.MinPR.html">hydras.solutions_exercises.MinPR</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#minusIndIsPR">minusIndIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#minusIsPR">minusIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#min_assoc">min_assoc</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Hydra.Hydra_Theorems.html#mIsPR">mIsPR</a> [in <a href="hydras.Hydra.Hydra_Theorems.html">hydras.Hydra.Hydra_Theorems</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_Equiv">mod_Equiv</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_op">mod_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#mod_equiv">mod_equiv</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_def.html#Monoid_EMonoid">Monoid_EMonoid</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#multIndIsPR">multIndIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#multIsPR">multIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#mult_comp">mult_comp</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="additions.Monoid_instances.html#mult_mod_proper">mult_mod_proper</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Mult_i">Mult_i</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.Fib2.html#Mul2">Mul2</a> [in <a href="additions.Fib2.html">additions.Fib2</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_Monoid">M2_Monoid</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#M2_op">M2_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><a id="instance_N"></a><h2>N </h2>
+<a href="additions.Monoid_instances.html#Natmult">Natmult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Natplus">Natplus</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Nat_le_TO">Nat_le_TO</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Nat_le_dec">Nat_le_dec</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Monoid_instances.html#nat_plus_op">nat_plus_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#nat_mult_op">nat_mult_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#neqIsPR">neqIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Monoid_instances.html#Nmod_Monoid">Nmod_Monoid</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#NMult">NMult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#notZeroIsPR">notZeroIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Monoid_instances.html#NPlus">NPlus</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#N_plus_op">N_plus_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#N_mult_op">N_mult_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><a id="instance_O"></a><h2>O </h2>
+<a href="hydras.Epsilon0.F_alpha.html#Olt">Olt</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Epsilon0.L_alpha.html#Olt">Olt</a> [in <a href="hydras.Epsilon0.L_alpha.html">hydras.Epsilon0.L_alpha</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Olt">Olt</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#Omega">Omega</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Omega_plus_Omega">Omega_plus_Omega</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Omega_2">Omega_2</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Omega_term">Omega_term</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#omega_ok">omega_ok</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#Omega_comp">Omega_comp</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Omega2">Omega2</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#ON_O">ON_O</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#ON_O_comp">ON_O_comp</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#ON_mult">ON_mult</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#ON_plus">ON_plus</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.compare_OO">OO.compare_OO</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.embed">OO.embed</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Fin">OO.Fin</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.mult">OO.mult</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.ON_OO">OO.ON_OO</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.OO_comp">OO.OO_comp</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.oo_str">OO.oo_str</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.phi0">OO.phi0</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.plus">OO.plus</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Succ">OO.Succ</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.Zero">OO.Zero</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO._Omega">OO._Omega</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Oplus">Oplus</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Oplus_assoc">Oplus_assoc</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="instance_P"></a><h2>P </h2>
+<a href="hydras.Ackermann.primRec.html#pi1_4IsPR">pi1_4IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi1_3IsPR">pi1_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi1_2IsPR">pi1_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_4IsPR">pi2_4IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_3IsPR">pi2_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi2_2IsPR">pi2_2IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi3_4IsPR">pi3_4IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi3_3IsPR">pi3_3IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#pi4_4IsPR">pi4_4IsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#plusIndIsPR">plusIndIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#plusIsPR">plusIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#plus_assoc">plus_assoc</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#plus_comp">plus_comp</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#plus_assoc">plus_assoc</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus_assoc">plus_assoc</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.Monoid_instances.html#PMult">PMult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow_proper">Pos_bpow_proper</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Pow_variant.html#power_proper">power_proper</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#power_proper">power_proper</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3IsPR">pow3IsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PO_lex_prod">PO_lex_prod</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#predIsPR">predIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PrO_Included">PrO_Included</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Monoid_instances.html#P_mult_op">P_mult_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#P_dec">P_dec</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<br/><br/><a id="instance_R"></a><h2>R </h2>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#RelDecision_Inverse_fun">RelDecision_Inverse_fun</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTermIsPR">ReplaceFormulaTermIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTermIsPR">ReplaceTermsTermIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermIsPR">ReplaceTermTermIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTermIsPR">ReplaceTermTermsTermIsPR</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="additions.AM.html#result_equiv_equiv">result_equiv_equiv</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Return_proper">Return_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R01">R01</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R02">R02</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.Epsilon0.F_omega.html#R03">R03</a> [in <a href="hydras.Epsilon0.F_omega.html">hydras.Epsilon0.F_omega</a>]<br/>
+<a href="hydras.MoreAck.AckNotPR.html#R2">R2</a> [in <a href="hydras.MoreAck.AckNotPR.html">hydras.MoreAck.AckNotPR</a>]<br/>
+<br/><br/><a id="instance_S"></a><h2>S </h2>
+<a href="hydras.Ackermann.cPair.html#searchXYIsPR">searchXYIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="additions.AM.html#Stack_equiv_equiv">Stack_equiv_equiv</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.AM.html#Stack_equiv_refl">Stack_equiv_refl</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#standard">standard</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#sto">sto</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#sto">sto</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Strict_lex">Strict_lex</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="additions.Monoid_def.html#string_op">string_op</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="Goedel.fixPoint.html#subStarIsPR">subStarIsPR</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#succIsPR">succIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#sumToNIsPR">sumToNIsPR</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#swapIsPR">swapIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#switchIsPR">switchIsPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="instance_T"></a><h2>T </h2>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Total_Inverse_fun">Total_Inverse_fun</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder.html#Total_Reflexive">Total_Reflexive</a> [in <a href="hydras.Prelude.DecPreOrder.html">hydras.Prelude.DecPreOrder</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2IsPR">tower2IsPR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.solutions_exercises.MorePRExamples.html#tower2_alt_PR">tower2_alt_PR</a> [in <a href="hydras.solutions_exercises.MorePRExamples.html">hydras.solutions_exercises.MorePRExamples</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#To_lex_prod">To_lex_prod</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.Simple_LexProd.html#Trans_lex">Trans_lex</a> [in <a href="hydras.Prelude.Simple_LexProd.html">hydras.Prelude.Simple_LexProd</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Two">Two</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="additions.BinaryStrat.html#Two_strat">Two_strat</a> [in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1addAssoc">T1addAssoc</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1compare">T1compare</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#t1_strorder">t1_strorder</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<br/><br/><a id="instance_V"></a><h2>V </h2>
+<a href="hydras.Hydra.Hydra_Termination.html#var">var</a> [in <a href="hydras.Hydra.Hydra_Termination.html">hydras.Hydra.Hydra_Termination</a>]<br/>
+<br/><br/><a id="instance_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedFormulaIsPR">wellFormedFormulaIsPR</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermIsPR">wellFormedTermIsPR</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermsIsPR">wellFormedTermsIsPR</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTermsIsPR">wellFormedTermTermsIsPR</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#WF">WF</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#WF">WF</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#WO_ord">WO_ord</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#WO_nat">WO_nat</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<br/><br/><a id="instance_Z"></a><h2>Z </h2>
+<a href="additions.Monoid_instances.html#ZMult">ZMult</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Monoid_instances.html#ZMult_Abelian">ZMult_Abelian</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Z_le_TO">Z_le_TO</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Z_le_dec">Z_le_dec</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="additions.Monoid_instances.html#Z_mult_op">Z_mult_op</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<br/><br/><hr/>
+<h1>Abbreviation Index</h1>
+<a id="abbreviation_A"></a><h2>A </h2>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.Tf">Ackermann.Tf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="abbreviation_B"></a><h2>B </h2>
+<a href="hydras.Hydra.Hydra_Definitions.html#battle">battle</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="abbreviation_C"></a><h2>C </h2>
+<a href="gaia_hydras.GCanon.html#canonS">canonS</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canonSSn">canonSSn</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.LT">CantorOrdinal.LT</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.Tf">CantorOrdinal.Tf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<br/><br/><a id="abbreviation_E"></a><h2>E </h2>
+<a href="hydras.Gamma0.T2.html#epsilon0">epsilon0</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#EqDecision">EqDecision</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#exH'">exH'</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0succb">E0succb</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="abbreviation_F"></a><h2>F </h2>
+<a href="hydras.Schutte.Schutte_basics.html#F">F</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#fin">fin</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Fin">Fin</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.fol.html#FolNotations.exH'">FolNotations.exH'</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#Formula">Formula</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Formula">Formula</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Formula">Formula</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Formula">Formula</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#Formulas">Formulas</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Formulas">Formulas</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Formulas">Formulas</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Formulas">Formulas</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarFormula">freeVarFormula</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTerm">freeVarTerm</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#freeVarTerms">freeVarTerms</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#FS">FS</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#FS">FS</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Epsilon0.F_alpha.html#F_alpha_ge_S">F_alpha_ge_S</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<br/><br/><a id="abbreviation_G"></a><h2>G </h2>
+<a href="hydras.Gamma0.Gamma0.html#G0.omega">G0.omega</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.phi">G0.phi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.phi0">G0.phi0</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="abbreviation_H"></a><h2>H </h2>
+<a href="gaia_hydras.GPaths.html#hbounded_transitionS">hbounded_transitionS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GCanon.html#hcanon">hcanon</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hCanon_plus">hCanon_plus</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hcnf">hcnf</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hcons">hcons</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hconst_path">hconst_path</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hconst_pathS">hconst_pathS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#head">head</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0">hE0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0fin">hE0fin</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0is_succ">hE0is_succ</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0le">hE0le</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0limit">hE0limit</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0lt">hE0lt</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0omega">hE0omega</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0phi0">hE0phi0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hE0zero">hE0zero</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#hF_">hF_</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hgnaw">hgnaw</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hgnawS">hgnawS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hlarge">hlarge</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hlargeS">hlargeS</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hL_spec">hL_spec</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hmlarge">hmlarge</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#hmlargeS">hmlargeS</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#hoplus">hoplus</a> [in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpath">hpath</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpathS">hpathS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hpath_to">hpath_to</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#hstandard_gnaw">hstandard_gnaw</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#htransition">htransition</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#hT1">hT1</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hT2">hT2</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#hyd">hyd</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd1">hyd1</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd2">hyd2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd3">hyd3</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#hyd4">hyd4</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#hzero">hzero</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#h1">h1</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#h2">h2</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#h3">h3</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<br/><br/><a id="abbreviation_I"></a><h2>I </h2>
+<a href="hydras.Ackermann.ListExt.html#In_list_remove2">In_list_remove2</a> [in <a href="hydras.Ackermann.ListExt.html">hydras.Ackermann.ListExt</a>]<br/>
+<br/><br/><a id="abbreviation_L"></a><h2>L </h2>
+<a href="hydras.Epsilon0.T1.html#le">le</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#le">le</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#LE">LE</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#limitb">limitb</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Limitb">Limitb</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.caar">LispAbbreviations.caar</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.caddr">LispAbbreviations.caddr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cadr">LispAbbreviations.cadr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.car">LispAbbreviations.car</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cddddr">LispAbbreviations.cddddr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cdddr">LispAbbreviations.cdddr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cddr">LispAbbreviations.cddr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.Ackermann.cPair.html#LispAbbreviations.cdr">LispAbbreviations.cdr</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.cons">LO.cons</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.FS">LO.FS</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.omega">LO.omega</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#LO.phi0">LO.phi0</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#LT">LT</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#LT">LT</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_irrefl">lt_irrefl</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#lt_trans">lt_trans</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<br/><br/><a id="abbreviation_M"></a><h2>M </h2>
+<a href="hydras.Epsilon0.T1.html#mult">mult</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Mult">Mult</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="abbreviation_N"></a><h2>N </h2>
+<a href="hydras.solutions_exercises.MultisetWf.html#NF">NF</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#NNnotations.S_">NNnotations.S_</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#n2t">n2t</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<br/><br/><a id="abbreviation_O"></a><h2>O </h2>
+<a href="hydras.Epsilon0.T1.html#omega">omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#omega">omega</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#omega">omega</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#omega">omega</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega">omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.ON">ONDef.Exports.ON</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.on">ONDef.Exports.on</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.ONMixin">ONDef.Exports.ONMixin</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#one">one</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#one">one</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.OmegaOmega.html#OO.omega">OO.omega</a> [in <a href="hydras.OrdinalNotations.OmegaOmega.html">hydras.OrdinalNotations.OmegaOmega</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#ordinal">ordinal</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<br/><br/><a id="abbreviation_P"></a><h2>P </h2>
+<a href="gaia_hydras.GPaths.html#path">path</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#phi0">phi0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Schutte.AP.html#phi0">phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#plus">plus</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Plus">Plus</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#power">power</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PReval">PReval</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRevalN">PRevalN</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_3">PRNotations.pi1_3</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_2">PRNotations.pi1_2</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi1_1">PRNotations.pi1_1</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi2_3">PRNotations.pi2_3</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi2_2">PRNotations.pi2_2</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.pi3_3">PRNotations.pi3_3</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.PRcomp">PRNotations.PRcomp</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#PRNotations.PRrec">PRNotations.PRrec</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<br/><br/><a id="abbreviation_R"></a><h2>R </h2>
+<a href="Goedel.goedel1.html#reflection">reflection</a> [in <a href="Goedel.goedel1.html">Goedel.goedel1</a>]<br/>
+<a href="Goedel.PRrepresentable.html#rem">rem</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<br/><br/><a id="abbreviation_S"></a><h2>S </h2>
+<a href="hydras.Ackermann.folProp.html#substF2">substF2</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF3">substF3</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF4">substF4</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF5">substF5</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF6">substF6</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF7">substF7</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF8">substF8</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF9">substF9</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormula">substituteFormula</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteTerm">substituteTerm</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteTerms">substituteTerms</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Succ">Succ</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succb">succb</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#SysPrf">SysPrf</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#System">System</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#System">System</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#System">System</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#System">System</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#S_">S_</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<br/><br/><a id="abbreviation_T"></a><h2>T </h2>
+<a href="hydras.Ackermann.subAll.html#Term">Term</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Term">Term</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Term">Term</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Term">Term</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#Terms">Terms</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Terms">Terms</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#Terms">Terms</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#Terms">Terms</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.A">Toy.A</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.a">Toy.a</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.B">Toy.B</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.b">Toy.b</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.C">Toy.C</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.f">Toy.f</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.g">Toy.g</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.h">Toy.h</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.P">Toy.P</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.Q">Toy.Q</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Toy.R">Toy.R</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#T1">T1</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#T1omega">T1omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#T2">T2</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<br/><br/><a id="abbreviation_U"></a><h2>U </h2>
+<a href="hydras.Ackermann.folProof.html#undecidable">undecidable</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="Goedel.PRrepresentable.html#ed2bf37be4fac66de48eeab40fec4772">Usual.β</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<br/><br/><a id="abbreviation_V"></a><h2>V </h2>
+<a href="hydras.Prelude.MoreVectors.html#vfourth">vfourth</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vfst">vfst</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vsnd">vsnd</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vthird">vthird</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<br/><br/><a id="abbreviation_Z"></a><h2>Z </h2>
+<a href="hydras.Epsilon0.E0.html#Zero">Zero</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="abbreviation__"></a><h2>_ </h2>
+<a href="hydras.Epsilon0.E0.html#_Omega">_Omega</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="abbreviation_*"></a><h2>other </h2>
+<a href="Goedel.PRrepresentable.html#dc5233cb1d950ecad15b1e9b2514f665">β</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#cb118be89dab27ceb8f4a738e65da758">βR</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#45bf03a575f6e81359314e906fb2bff3">ω</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<br/><br/><hr/>
+<h1>Definition Index</h1>
+<a id="definition_A"></a><h2>A </h2>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#A">A</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="gaia_hydras.GPaths.html#acc_from">acc_from</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#acc_from">acc_from</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack">Ack</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Ack">Ack</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.all_zero">Ackermann.all_zero</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.epsilon0">Ackermann.epsilon0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit_of">Ackermann.limit_of</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.limit12_hyp">Ackermann.limit12_hyp</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi">Ackermann.lt_psi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_rec">Ackermann.lt_rec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.lt_psi_rec">Ackermann.lt_psi_rec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.normal">Ackermann.normal</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.omega">Ackermann.omega</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.one">Ackermann.one</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi">Ackermann.phi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phia">Ackermann.phia</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec">Ackermann.phi_rec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi_rec_psi">Ackermann.phi_rec_psi</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi0">Ackermann.phi0</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi1">Ackermann.phi1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi10">Ackermann.phi10</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi11">Ackermann.phi11</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12a">Ackermann.phi12a</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b2">Ackermann.phi12b2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b3">Ackermann.phi12b3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b4">Ackermann.phi12b4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi12b5">Ackermann.phi12b5</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi3">Ackermann.phi3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi4">Ackermann.phi4</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi5">Ackermann.phi5</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi6">Ackermann.phi6</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi7">Ackermann.phi7</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi8">Ackermann.phi8</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.phi9">Ackermann.phi9</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.psi_succ">Ackermann.psi_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.size">Ackermann.size</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.sup_of">Ackermann.sup_of</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T1_T3">Ackermann.T1_T3</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3add">Ackermann.T3add</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3ap">Ackermann.T3ap</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3bad">Ackermann.T3bad</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3eq">Ackermann.T3eq</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3finite">Ackermann.T3finite</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3is_succ">Ackermann.T3is_succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3le">Ackermann.T3le</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3limit">Ackermann.T3limit</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lt">Ackermann.T3lt</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3lta">Ackermann.T3lta</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3nat">Ackermann.T3nat</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3nf">Ackermann.T3nf</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3pred">Ackermann.T3pred</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3split">Ackermann.T3split</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3sub">Ackermann.T3sub</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3succ">Ackermann.T3succ</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3_eqType">Ackermann.T3_eqType</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Ackermann.T3_eqMixin">Ackermann.T3_eqMixin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="Goedel.PRrepresentable.html#addExists">addExists</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#addForalls">addForalls</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_r">add_r</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_head_r_plus">add_head_r_plus</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#add_head_r">add_head_r</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Alive">Alive</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha_0_eq">alpha_0_eq</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#alpha_0">alpha_0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#alpha1">alpha1</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt.Ack">Alt.Ack</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.add'">Alt.add'</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="additions.Demo.html#Alt.double">Alt.double</a> [in <a href="additions.Demo.html">additions.Demo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.status">Alt.Eps0_prec.status</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.prec">Alt.Eps0_prec.prec</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_prec.A">Alt.Eps0_prec.A</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.arity">Alt.Eps0_sig.arity</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Eps0_sig.symb">Alt.Eps0_sig.symb</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.nat_2_term">Alt.nat_2_term</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor">Alt.transfinite_recursor</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.transfinite_recursor_lt">Alt.transfinite_recursor_lt</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_size">Alt.T1_size</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.T1_2_term">Alt.T1_2_term</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.Epsilon0.Epsilon0rpo.html#Alt.Vars.var">Alt.Vars.var</a> [in <a href="hydras.Epsilon0.Epsilon0rpo.html">hydras.Epsilon0.Epsilon0rpo</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Alt.xpred">Alt.xpred</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.Ack">Alt2.Ack</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.test1">Alt2.test1</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt2.test2">Alt2.test2</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="hydras.MoreAck.Ack.html#Alt3.ack">Alt3.ack</a> [in <a href="hydras.MoreAck.Ack.html">hydras.MoreAck.Ack</a>]<br/>
+<a href="additions.AM.html#AM_power">AM_power</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="hydras.Ackermann.fol.html#andH">andH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#andRel">andRel</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="additions.Addition_Chains.html#any_chain_parametric">any_chain_parametric</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#ap">ap</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.AP.html#AP">AP</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#approx">approx</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AxmEq4">AxmEq4</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#AxmEq5">AxmEq5</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.Addition_Chains.html#axp_scheme">axp_scheme</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Critical.html#A_">A_</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<br/><br/><a id="definition_B"></a><h2>B </h2>
+<a href="hydras.OrdinalNotations.ON_Finite.html#bad">bad</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#BadSubstF2.substF">BadSubstF2.substF</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.MoreAck.BadSubst.html#BadSubst.substF">BadSubst.substF</a> [in <a href="hydras.MoreAck.BadSubst.html">hydras.MoreAck.BadSubst</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#bad_term">bad_term</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="additions.FirstSteps.html#Bad.correct_expt_function">Bad.correct_expt_function</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Monoid_instances.html#Bad.fact">Bad.fact</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad.Fchain_correct">Bad.Fchain_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Monoid_instances.html#Bad.int63_from_nat">Bad.int63_from_nat</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad4.Fplus">Bad4.Fplus</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Bad4.F23">Bad4.F23</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#battle_length">battle_length</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="Goedel.PRrepresentable.html#beta">beta</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#betaFormula">betaFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#beta_h">beta_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#beta1">beta1</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="additions.fib.html#bigarg">bigarg</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#bigO">bigO</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#big_h">big_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#big0">big0</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="additions.FirstSteps.html#binary_power_mult">binary_power_mult</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#binary_power_mult">binary_power_mult</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#binary_power_mult">binary_power_mult</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.fib.html#binary_power_mult">binary_power_mult</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Addition_Chains.html#binary_chain">binary_chain</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#bin_pow_scheme">bin_pow_scheme</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.fib.html#bits">bits</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#bodd">bodd</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#bool_decide">bool_decide</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#boundComputation">boundComputation</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearch">boundedSearch</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#boundedSearchHelp">boundedSearchHelp</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#bounded_by">bounded_by</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="gaia_hydras.GPaths.html#bounded_transitionS">bounded_transitionS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#bounded_transitionS">bounded_transitionS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="Goedel.goedel2.html#box">box</a> [in <a href="Goedel.goedel2.html">Goedel.goedel2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#but_last">but_last</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#B_termination">B_termination</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="definition_C"></a><h2>C </h2>
+<a href="hydras.Epsilon0.F_alpha.html#call_lt">call_lt</a> [in <a href="hydras.Epsilon0.F_alpha.html">hydras.Epsilon0.F_alpha</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#canon">canon</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="gaia_hydras.GCanon.html#canon">canon</a> [in <a href="gaia_hydras.GCanon.html">gaia_hydras.GCanon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Canon">Canon</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#canon">canon</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="gaia_hydras.GF_alpha.html#Canon_plus">Canon_plus</a> [in <a href="gaia_hydras.GF_alpha.html">gaia_hydras.GF_alpha</a>]<br/>
+<a href="gaia_hydras.GPaths.html#Canon_plus">Canon_plus</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#Canon_plus">Canon_plus</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.ex_middle">CantorOrdinal.ex_middle</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_fct">CantorOrdinal.limit_fct</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_of">CantorOrdinal.limit_of</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="gaia_hydras.nfwfgaia.html#CantorOrdinal.limit_v1">CantorOrdinal.limit_v1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
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+<a href="hydras.Epsilon0.T1.html#Ex4">Ex4</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Schutte.Schutte.html#Ex42">Ex42</a> [in <a href="hydras.Schutte.Schutte.html">hydras.Schutte.Schutte</a>]<br/>
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+<br/><br/><a id="definition_G"></a><h2>G </h2>
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+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_eqType">Gamma0.T2_eqType</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Gamma0.T2_eqMixin">Gamma0.T2_eqMixin</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#gamma1">gamma1</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#ge">ge</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#get_decomposition">get_decomposition</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.GPaths.html#gnaw">gnaw</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Gnaw">Gnaw</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnaw">gnaw</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#gnawS">gnawS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#GnawS">GnawS</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#gnawS">gnawS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#GRelation">GRelation</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#gstrict_lub">gstrict_lub</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Gamma0.T2.html#gtail">gtail</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.Ex42">G0.Ex42</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.le">G0.le</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.lt">G0.lt</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#G0.LT">G0.LT</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h">g2h</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#g2h">g2h</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#g2h_seq">g2h_seq</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<br/><br/><a id="definition_H"></a><h2>H </h2>
+<a href="additions.BinaryStrat.html#half">half</a> [in <a href="additions.BinaryStrat.html">additions.BinaryStrat</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hcons_mult">hcons_mult</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#height">height</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.BigBattle.html#hinit">hinit</a> [in <a href="hydras.Hydra.BigBattle.html">hydras.Hydra.BigBattle</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hsize">hsize</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hs2l">hs2l</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydrae_rect2">Hydrae_rect2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Hydra_rect2">Hydra_rect2</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hyd_mult">hyd_mult</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#hyper_exp2">hyper_exp2</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#hy_app">hy_app</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#h_forall">h_forall</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="gaia_hydras.GHprime.html#H'_">H'_</a> [in <a href="gaia_hydras.GHprime.html">gaia_hydras.GHprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_succ_fun">H'_succ_fun</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#H'_">H'_</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h0">h0</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h1">h1</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h2">h2</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g">h2g</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T2Bridge.html#h2g">h2g</a> [in <a href="gaia_hydras.T2Bridge.html">gaia_hydras.T2Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#h2g_seq">h2g_seq</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Hydra.KP_example.html#h3">h3</a> [in <a href="hydras.Hydra.KP_example.html">hydras.Hydra.KP_example</a>]<br/>
+<br/><br/><a id="definition_I"></a><h2>I </h2>
+<a href="additions.Monoid_instances.html#Id2">Id2</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Ackermann.fol.html#iffH">iffH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#ifThenElseH">ifThenElseH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#ignoreParams">ignoreParams</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#image">image</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#Included_s">Included_s</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#Inconsistent">Inconsistent</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#independent">independent</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="hydras.Ackermann.PA.html#InductionSchema">InductionSchema</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#Infinite">Infinite</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.Schutte.Correctness_E0.html#inject">inject</a> [in <a href="hydras.Schutte.Correctness_E0.html">hydras.Schutte.Correctness_E0</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpFormula">interpFormula</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpRels">interpRels</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTerm">interpTerm</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTerms">interpTerms</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Ackermann.model.html#interpTermsVector">interpTermsVector</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#interval">interval</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.solutions_exercises.FibonacciPR.html#inv">inv</a> [in <a href="hydras.solutions_exercises.FibonacciPR.html">hydras.solutions_exercises.FibonacciPR</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#inv_fun">inv_fun</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#In_freeVarSys">In_freeVarSys</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iota">iota</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#iota">iota</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Hydra.O2H.html#iotas">iotas</a> [in <a href="hydras.Hydra.O2H.html">hydras.Hydra.O2H</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#iota_from">iota_from</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#iota_fun">iota_fun</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.Bad.bottom">iota_demo.Bad.bottom</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.zero">iota_demo.zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#iota_demo.zero">iota_demo.zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#iota_ij">iota_ij</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_op">Iso_same_op</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_fun">Iso_same_fun</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#Iso_same_cst">Iso_same_cst</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#isqrt">isqrt</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.solutions_exercises.isqrt.html#isqrt_spec">isqrt_spec</a> [in <a href="hydras.solutions_exercises.isqrt.html">hydras.solutions_exercises.isqrt</a>]<br/>
+<a href="hydras.Schutte.Lub.html#is_lub">is_lub</a> [in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="hydras.Schutte.CNF.html#is_cnf_of">is_cnf_of</a> [in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_limit">is_limit</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_succ">is_succ</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#is_finite">is_finite</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.onType.html#is_successor_of">is_successor_of</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#iterate">iterate</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#iterate">iterate</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#i1">i1</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#i2">i2</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<br/><br/><a id="definition_K"></a><h2>K </h2>
+<a href="additions.Euclidean_Chains.html#Kchain">Kchain</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#Kchain_correct">Kchain_correct</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_correct">Kchain_correct</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kchain_correct_nat">Kchain_correct_nat</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFF">KFF</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFF">KFF</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#KFK">KFK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#KFK">KFK</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kkont">Kkont</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kkont_equiv">Kkont_equiv</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#Kkont_proper">Kkont_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#kont_type">kont_type</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#KP_arrowS">KP_arrowS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k17_7">k17_7</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#K2F">K2F</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k3_1">k3_1</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#k7_3">k7_3</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.AM.html#K99_24">K99_24</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<br/><br/><a id="definition_L"></a><h2>L </h2>
+<a href="hydras.Ackermann.fol.html#language_decidable">language_decidable</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Large">Large</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#large">large</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#large">large</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#Largeb">Largeb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeb">largeb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeS">LargeS</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeS">largeS</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="gaia_hydras.GLarge_Sets.html#largeS">largeS</a> [in <a href="gaia_hydras.GLarge_Sets.html">gaia_hydras.GLarge_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#LargeSb">LargeSb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#largeSb">largeSb</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#large_set_check">large_set_check</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#LE">LE</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Gamma0.T2.html#le">le</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_O.html#le">le</a> [in <a href="hydras.OrdinalNotations.ON_O.html">hydras.OrdinalNotations.ON_O</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_mult.html#le">le</a> [in <a href="hydras.OrdinalNotations.ON_mult.html">hydras.OrdinalNotations.ON_mult</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_plus.html#le">le</a> [in <a href="hydras.OrdinalNotations.ON_plus.html">hydras.OrdinalNotations.ON_plus</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#le">le</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Well_Orders.html#Le">Le</a> [in <a href="hydras.Schutte.Well_Orders.html">hydras.Schutte.Well_Orders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Least">Least</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
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+<a href="hydras.Prelude.MoreOrders.html#leq">leq</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
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+<a href="gaia_hydras.onType.html#limit_v2">limit_v2</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#lt_depth">lt_depth</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="hydras.rpo.list_set.html#Make.remove_not_common">Make.remove_not_common</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
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+<a href="hydras.rpo.term.html#Make.replace_at_pos_list">Make.replace_at_pos_list</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.replace_at_pos">Make.replace_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.rpo_term">Make.rpo_term</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
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+<a href="hydras.rpo.list_set.html#Make.singleton">Make.singleton</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.size">Make.size</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.size2">Make.size2</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.rpo.html#Make.size3">Make.size3</a> [in <a href="hydras.rpo.rpo.html">hydras.rpo.rpo</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.subset">Make.subset</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.substitution">Make.substitution</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subst_comp">Make.subst_comp</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.subterm_at_pos">Make.subterm_at_pos</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.symbol">Make.symbol</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.term.html#Make.Term_eq_dec.A">Make.Term_eq_dec.A</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec8">Make.term_rec8</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec7">Make.term_rec7</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec4">Make.term_rec4</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec3">Make.term_rec3</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.term_rec2">Make.term_rec2</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.list_set.html#Make.union">Make.union</a> [in <a href="hydras.rpo.list_set.html">hydras.rpo.list_set</a>]<br/>
+<a href="hydras.rpo.term.html#Make.variable">Make.variable</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.term.html#Make.well_formed_list">Make.well_formed_list</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
+<a href="hydras.rpo.term.html#Make.well_formed">Make.well_formed</a> [in <a href="hydras.rpo.term.html">hydras.rpo.term</a>]<br/>
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+<a href="hydras.rpo.more_list.html#map_without_repetition">map_without_repetition</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
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+<a href="hydras.Prelude.Sort_spec.html#mark">mark</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
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+<a href="hydras.Ackermann.fol.html#mem">mem</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="additions.Monoid_instances.html#mult_mod">mult_mod</a> [in <a href="additions.Monoid_instances.html">additions.Monoid_instances</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_fin_r">mult_fin_r</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="hydras.Schutte.Addition.html#mult_Sn">mult_Sn</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
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+<br/><br/><a id="definition_N"></a><h2>N </h2>
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+<a href="hydras.Ackermann.primRec.html#notZero">notZero</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.Ackermann.folProof.html#nVars">nVars</a> [in <a href="hydras.Ackermann.folProof.html">hydras.Ackermann.folProof</a>]<br/>
+<a href="additions.Compatibility.html#N_pow">N_pow</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.FirstSteps.html#N_bpow">N_bpow</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#N_bpow">N_bpow</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#N_bpow">N_bpow</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.Naive.html#N_mod.power">N_mod.power</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.Naive.html#N_mod.mult_mod">N_mod.mult_mod</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<a href="additions.More_on_positive.html#N2pos">N2pos</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<br/><br/><a id="definition_O"></a><h2>O </h2>
+<a href="additions.AM.html#OK">OK</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="additions.Euclidean_Chains.html#OK">OK</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_omega">omega_omega</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_tower">omega_tower</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#omega_term">omega_term</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_ex2">omega_ex2</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_ex1">omega_ex1</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets.html#omega_square_thrice_eqn">omega_square_thrice_eqn</a> [in <a href="hydras.Epsilon0.Large_Sets.html">hydras.Epsilon0.Large_Sets</a>]<br/>
+<a href="hydras.Schutte.AP.html#omega_tower">omega_tower</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#omega_omega_1_3">omega_omega_1_3</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Epsilon0.Large_Sets_Examples.html#omega_omega_1_4">omega_omega_1_4</a> [in <a href="hydras.Epsilon0.Large_Sets_Examples.html">hydras.Epsilon0.Large_Sets_Examples</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit_type">Omega_limit_type</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Prelude.MoreOrders.html#Omega_limit">Omega_limit</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#omega_limit">omega_limit</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.onType.html#om12">om12</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#om67">om67</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.Exports.tricho">ONDef.Exports.tricho</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#ONDef.on_struct">ONDef.on_struct</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
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+<a href="gaia_hydras.onType.html#onFiniteType">onFiniteType</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaMixin">onOmegaMixin</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.onType.html#onOmegaType">onOmegaType</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_op_ok">ON_op_ok</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_fun_ok">ON_fun_ok</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_cst_ok">ON_cst_ok</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_le">ON_le</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_lt">ON_lt</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_compare">ON_compare</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ON_t">ON_t</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
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+<a href="hydras.Ackermann.PA.html#open">open</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#oplus">oplus</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="gaia_hydras.GHessenberg.html#oplus">oplus</a> [in <a href="gaia_hydras.GHessenberg.html">gaia_hydras.GHessenberg</a>]<br/>
+<a href="additions.Addition_Chains.html#optimal">optimal</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Addition_Chains.html#optimal_generator">optimal_generator</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ord">ord</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#ordering_segment">ordering_segment</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.fol.html#orH">orH</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.primRec.html#orRel">orRel</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
+<a href="hydras.rpo.more_list.html#o_length">o_length</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Epsilon0.Hessenberg.html#o_finite_mult">o_finite_mult</a> [in <a href="hydras.Epsilon0.Hessenberg.html">hydras.Epsilon0.Hessenberg</a>]<br/>
+<a href="gaia_hydras.ON_gfinite.html#o_36_of_42">o_36_of_42</a> [in <a href="gaia_hydras.ON_gfinite.html">gaia_hydras.ON_gfinite</a>]<br/>
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+<a href="hydras.OrdinalNotations.Example_3PlusOmega.html#O3O">O3O</a> [in <a href="hydras.OrdinalNotations.Example_3PlusOmega.html">hydras.OrdinalNotations.Example_3PlusOmega</a>]<br/>
+<br/><br/><a id="definition_P"></a><h2>P </h2>
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+<a href="hydras.Ackermann.PA.html#PA">PA</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.folLogic3.html#PairwiseEqual">PairwiseEqual</a> [in <a href="hydras.Ackermann.folLogic3.html">hydras.Ackermann.folLogic3</a>]<br/>
+<a href="additions.Addition_Chains.html#parametric">parametric</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#path">path</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#pathS">pathS</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#pathS">pathS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_tob">path_tob</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#path_to">path_to</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA1">PA1</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA2">PA2</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA3">PA3</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA4">PA4</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA5">PA5</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA6">PA6</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Ackermann.PA.html#PA7">PA7</a> [in <a href="hydras.Ackermann.PA.html">hydras.Ackermann.PA</a>]<br/>
+<a href="hydras.Prelude.DecPreOrder_Instances.html#PB">PB</a> [in <a href="hydras.Prelude.DecPreOrder_Instances.html">hydras.Prelude.DecPreOrder_Instances</a>]<br/>
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+<a href="hydras.rpo.list_permut.html#Permut.list_permut">Permut.list_permut</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.rpo.list_permut.html#Permut.list_to_multiset">Permut.list_to_multiset</a> [in <a href="hydras.rpo.list_permut.html">hydras.rpo.list_permut</a>]<br/>
+<a href="hydras.Schutte.Critical.html#phi">phi</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#phi">phi</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
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+<a href="hydras.Gamma0.Gamma0.html#plus">plus</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Schutte.Addition.html#plus">plus</a> [in <a href="hydras.Schutte.Addition.html">hydras.Schutte.Addition</a>]<br/>
+<a href="additions.Compatibility.html#Pos_pow">Pos_pow</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.FirstSteps.html#Pos_bpow">Pos_bpow</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#Pos_bpow">Pos_bpow</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#Pos_bpow">Pos_bpow</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.More_on_positive.html#pos_eq_dec">pos_eq_dec</a> [in <a href="additions.More_on_positive.html">additions.More_on_positive</a>]<br/>
+<a href="additions.FirstSteps.html#power">power</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Pow_variant.html#power">power</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#power">power</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="additions.FirstSteps.html#power_t">power_t</a> [in <a href="additions.FirstSteps.html">additions.FirstSteps</a>]<br/>
+<a href="additions.Addition_Chains.html#power_R">power_R</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="additions.Pow_variant.html#pow_17">pow_17</a> [in <a href="additions.Pow_variant.html">additions.Pow_variant</a>]<br/>
+<a href="additions.Pow.html#pow_17">pow_17</a> [in <a href="additions.Pow.html">additions.Pow</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#pow3">pow3</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp">pp</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#ppE0">ppE0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp_size">pp_size</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#pp0">pp0</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Gamma0.T2.html#pred">pred</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
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+<a href="Goedel.PRrepresentable.html#primRecFormula">primRecFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecPiFormulaHelp">primRecPiFormulaHelp</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecsFormula">primRecsFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecSigmaFormula">primRecSigmaFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#primRecSigmaFormulaHelp">primRecSigmaFormulaHelp</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
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+<a href="hydras.Ackermann.primRec.html#projectionListPR">projectionListPR</a> [in <a href="hydras.Ackermann.primRec.html">hydras.Ackermann.primRec</a>]<br/>
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+<a href="hydras.Gamma0.T2.html#psi_term">psi_term</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#ptwise_le">ptwise_le</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
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+<a href="gaia_hydras.T1Bridge.html#refines0">refines0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines1">refines1</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#refines2">refines2</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inv">rel_inv</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_inj">rel_inj</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_onto">rel_onto</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_functional">rel_functional</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_codomain">rel_codomain</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#rel_domain">rel_domain</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Schutte.Countable.html#rel_enumerates">rel_enumerates</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Countable.html#rel_numbers">rel_numbers</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.rpo.more_list.html#remove">remove</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.rpo.more_list.html#remove_list">remove_list</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#rep">rep</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceFormulaTerm">ReplaceFormulaTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermsTerm">ReplaceTermsTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTerm">ReplaceTermTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Ackermann.codeSubFormula.html#ReplaceTermTermsTerm">ReplaceTermTermsTerm</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Representable">Representable</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#Representable">Representable</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentableAlternate">RepresentableAlternate</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf1">RepresentableHalf1</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHalf2">RepresentableHalf2</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentableHelp">RepresentableHelp</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Ackermann.expressible.html#RepresentableHelp">RepresentableHelp</a> [in <a href="hydras.Ackermann.expressible.html">hydras.Ackermann.expressible</a>]<br/>
+<a href="Goedel.PRrepresentable.html#RepresentablesHelp">RepresentablesHelp</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Representable_ext">Representable_ext</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="Goedel.rosser.html#RepresentsInSelf">RepresentsInSelf</a> [in <a href="Goedel.rosser.html">Goedel.rosser</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restrict">restrict</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.onType.html#restrict">restrict</a> [in <a href="gaia_hydras.onType.html">gaia_hydras.onType</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#restrict">restrict</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#restrict_build">restrict_build</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round">round</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_t">round_t</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_spec">round_spec</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_star">round_star</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_plus">round_plus</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#round_n">round_n</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_nat_elaguee">R_nat_elaguee</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.GRelations.html#R_inv">R_inv</a> [in <a href="hydras.Schutte.GRelations.html">hydras.Schutte.GRelations</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#R1">R1</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.PartialFun.html#r2i">r2i</a> [in <a href="hydras.Schutte.PartialFun.html">hydras.Schutte.PartialFun</a>]<br/>
+<a href="hydras.Hydra.Hydra_Lemmas.html#R2_ind2">R2_ind2</a> [in <a href="hydras.Hydra.Hydra_Lemmas.html">hydras.Hydra.Hydra_Lemmas</a>]<br/>
+<br/><br/><a id="definition_S"></a><h2>S </h2>
+<a href="hydras.solutions_exercises.Limit_Infinity.html#s">s</a> [in <a href="hydras.solutions_exercises.Limit_Infinity.html">hydras.solutions_exercises.Limit_Infinity</a>]<br/>
+<a href="hydras.solutions_exercises.T1_ltNotWf.html#s">s</a> [in <a href="hydras.solutions_exercises.T1_ltNotWf.html">hydras.solutions_exercises.T1_ltNotWf</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#segment">segment</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#Sentence">Sentence</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#Sentence">Sentence</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#Sentence">Sentence</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.solutions_exercises.MultisetWf.html#seq">seq</a> [in <a href="hydras.solutions_exercises.MultisetWf.html">hydras.solutions_exercises.MultisetWf</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#seq_mono">seq_mono</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Countable.html#seq_range">seq_range</a> [in <a href="hydras.Schutte.Countable.html">hydras.Schutte.Countable</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#set_eq">set_eq</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#shift">shift</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.Euclidean_Chains.html#signature_measure">signature_measure</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="additions.Euclidean_Chains.html#signature_exponent">signature_exponent</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#simple_last">simple_last</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="additions.fib.html#slowexp">slowexp</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="hydras.Hydra.Epsilon0_Needed_Generic.html#small_h">small_h</a> [in <a href="hydras.Hydra.Epsilon0_Needed_Generic.html">hydras.Hydra.Epsilon0_Needed_Generic</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#some">some</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#some_pos'">some_pos'</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#some_pos">some_pos</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Schutte.MoreEpsilonIota.html#some_pos">some_pos</a> [in <a href="hydras.Schutte.MoreEpsilonIota.html">hydras.Schutte.MoreEpsilonIota</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort">sort</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Schutte.CNF.html#sorted">sorted</a> [in <a href="hydras.Schutte.CNF.html">hydras.Schutte.CNF</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_spec">sort_spec</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_correct">sort_correct</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_rel">sort_rel</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#sort_fun_t">sort_fun_t</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.rpo.more_list.html#split_list">split_list</a> [in <a href="hydras.rpo.more_list.html">hydras.rpo.more_list</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#sp_merge_sort">sp_merge_sort</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#stable">stable</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Sort_spec.html#stable_test">stable_test</a> [in <a href="hydras.Prelude.Sort_spec.html">hydras.Prelude.Sort_spec</a>]<br/>
+<a href="hydras.Prelude.Merge_Sort.html#stable_merge_sort">stable_merge_sort</a> [in <a href="hydras.Prelude.Merge_Sort.html">hydras.Prelude.Merge_Sort</a>]<br/>
+<a href="additions.AM.html#stack">stack</a> [in <a href="additions.AM.html">additions.AM</a>]<br/>
+<a href="gaia_hydras.GPaths.html#standard_gnaw">standard_gnaw</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="gaia_hydras.GPaths.html#standard_path">standard_path</a> [in <a href="gaia_hydras.GPaths.html">gaia_hydras.GPaths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_path">standard_path</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_pathS">standard_pathS</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.Paths.html#standard_gnaw">standard_gnaw</a> [in <a href="hydras.Epsilon0.Paths.html">hydras.Epsilon0.Paths</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_maj">strict_lub_maj</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub_lub">strict_lub_lub</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#strict_lub">strict_lub</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.Prelude.Iterates.html#strict_mono">strict_mono</a> [in <a href="hydras.Prelude.Iterates.html">hydras.Prelude.Iterates</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#strict_mono">strict_mono</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Schutte.Critical.html#strongly_critical">strongly_critical</a> [in <a href="hydras.Schutte.Critical.html">hydras.Schutte.Critical</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllFormula">subAllFormula</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerm">subAllTerm</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.Ackermann.subAll.html#subAllTerms">subAllTerms</a> [in <a href="hydras.Ackermann.subAll.html">hydras.Ackermann.subAll</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_op">SubON_same_op</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_fun">SubON_same_fun</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#SubON_same_cst">SubON_same_cst</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="Goedel.fixPoint.html#subStar">subStar</a> [in <a href="Goedel.fixPoint.html">Goedel.fixPoint</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substF">substF</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaForall">substituteFormulaForall</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaHelp">substituteFormulaHelp</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaImp">substituteFormulaImp</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substituteFormulaNot">substituteFormulaNot</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substT">substT</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Ackermann.folProp.html#substTs">substTs</a> [in <a href="hydras.Ackermann.folProp.html">hydras.Ackermann.folProp</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#succ">succ</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
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+<a href="hydras.Gamma0.T2.html#succ">succ</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#succ">succ</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
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+<a href="hydras.Prelude.MoreOrders.html#Successor">Successor</a> [in <a href="hydras.Prelude.MoreOrders.html">hydras.Prelude.MoreOrders</a>]<br/>
+<a href="Goedel.PRrepresentable.html#succFormula">succFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#succ_spec">succ_spec</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
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+<a href="hydras.Schutte.Schutte_basics.html#sup">sup</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
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+<a href="hydras.Ackermann.LNN.html#SysPrf">SysPrf</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#SysPrf">SysPrf</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.LNN.html#System">System</a> [in <a href="hydras.Ackermann.LNN.html">hydras.Ackermann.LNN</a>]<br/>
+<a href="hydras.Ackermann.LNT.html#System">System</a> [in <a href="hydras.Ackermann.LNT.html">hydras.Ackermann.LNT</a>]<br/>
+<a href="hydras.Ackermann.fol.html#System">System</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
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+<a href="hydras.Gamma0.T2.html#t2_length_aux">t2_length_aux</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.T2.html#t2_length">t2_length</a> [in <a href="hydras.Gamma0.T2.html">hydras.Gamma0.T2</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_size">T2_size</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#T2_2_term">T2_2_term</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<br/><br/><a id="definition_U"></a><h2>U </h2>
+<a href="hydras.Schutte.Schutte_basics.html#Unbounded">Unbounded</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm9">Unnamed_thm9</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm8">Unnamed_thm8</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm7">Unnamed_thm7</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm6">Unnamed_thm6</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm5">Unnamed_thm5</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm5">Unnamed_thm5</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm4">Unnamed_thm4</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm3">Unnamed_thm3</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm2">Unnamed_thm2</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm1">Unnamed_thm1</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="gaia_hydras.T1Choice.html#Unnamed_thm">Unnamed_thm</a> [in <a href="gaia_hydras.T1Choice.html">gaia_hydras.T1Choice</a>]<br/>
+<a href="hydras.Prelude.More_Arith.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.Prelude.More_Arith.html">hydras.Prelude.More_Arith</a>]<br/>
+<a href="hydras.MoreAck.expressibleExamples.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.MoreAck.expressibleExamples.html">hydras.MoreAck.expressibleExamples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="hydras.MoreAck.LNN_Examples.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.MoreAck.LNN_Examples.html">hydras.MoreAck.LNN_Examples</a>]<br/>
+<a href="additions.Monoid_def.html#Unnamed_thm">Unnamed_thm</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Ackermann.fol.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.Ackermann.fol.html">hydras.Ackermann.fol</a>]<br/>
+<a href="hydras.Epsilon0.Canon.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.Epsilon0.Canon.html">hydras.Epsilon0.Canon</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm1">Unnamed_thm1</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.Prelude.LibHyps_Experiments.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.Prelude.LibHyps_Experiments.html">hydras.Prelude.LibHyps_Experiments</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm1">Unnamed_thm1</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="hydras.solutions_exercises.F_3.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.solutions_exercises.F_3.html">hydras.solutions_exercises.F_3</a>]<br/>
+<a href="additions.Addition_Chains.html#Unnamed_thm">Unnamed_thm</a> [in <a href="additions.Addition_Chains.html">additions.Addition_Chains</a>]<br/>
+<a href="hydras.Epsilon0.Hprime.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.Epsilon0.Hprime.html">hydras.Epsilon0.Hprime</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.FolExamples.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.MoreAck.FolExamples.html">hydras.MoreAck.FolExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm2">Unnamed_thm2</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm1">Unnamed_thm1</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm0">Unnamed_thm0</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.MoreAck.PrimRecExamples.html#Unnamed_thm">Unnamed_thm</a> [in <a href="hydras.MoreAck.PrimRecExamples.html">hydras.MoreAck.PrimRecExamples</a>]<br/>
+<a href="hydras.Prelude.MoreLists.html#unshift">unshift</a> [in <a href="hydras.Prelude.MoreLists.html">hydras.Prelude.MoreLists</a>]<br/>
+<a href="hydras.Ackermann.model.html#updateValue">updateValue</a> [in <a href="hydras.Ackermann.model.html">hydras.Ackermann.model</a>]<br/>
+<a href="hydras.Schutte.Lub.html#upper_bound">upper_bound</a> [in <a href="hydras.Schutte.Lub.html">hydras.Schutte.Lub</a>]<br/>
+<a href="Goedel.PRrepresentable.html#Usual.beta">Usual.beta</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<br/><br/><a id="definition_V"></a><h2>V </h2>
+<a href="hydras.Ackermann.codeSubFormula.html#varFormula">varFormula</a> [in <a href="hydras.Ackermann.codeSubFormula.html">hydras.Ackermann.codeSubFormula</a>]<br/>
+<a href="hydras.Gamma0.Gamma0.html#Vars.var">Vars.var</a> [in <a href="hydras.Gamma0.Gamma0.html">hydras.Gamma0.Gamma0</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vector_nth">vector_nth</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vector_triple_rect">vector_triple_rect</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vector_double_rect">vector_double_rect</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#vect_from_fun">vect_from_fun</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Vec2_proj">Vec2_proj</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<a href="hydras.Prelude.MoreVectors.html#Vid">Vid</a> [in <a href="hydras.Prelude.MoreVectors.html">hydras.Prelude.MoreVectors</a>]<br/>
+<br/><br/><a id="definition_W"></a><h2>W </h2>
+<a href="hydras.Ackermann.wConsistent.html#wConsistent">wConsistent</a> [in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedFormula">wellFormedFormula</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTerm">wellFormedTerm</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTerms">wellFormedTerms</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Ackermann.wellFormed.html#wellFormedTermTerms">wellFormedTermTerms</a> [in <a href="hydras.Ackermann.wellFormed.html">hydras.Ackermann.wellFormed</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#well_founded_restriction_rect">well_founded_restriction_rect</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="hydras.Prelude.Restriction.html#well_founded_restriction">well_founded_restriction</a> [in <a href="hydras.Prelude.Restriction.html">hydras.Prelude.Restriction</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#well_founded_P">well_founded_P</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.extension">Wfsum.extension</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f">Wfsum.f</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f_spec">Wfsum.f_spec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f1">Wfsum.f1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.f2">Wfsum.f2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wfsum.psum">Wfsum.psum</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f">Wf_ex3.f</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f2">Wf_ex3.f2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f1">Wf_ex3.f1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.f_spec">Wf_ex3.f_spec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex3.lte">Wf_ex3.lte</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f">Wf_ex.f</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f2">Wf_ex.f2</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f1">Wf_ex.f1</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="gaia_hydras.nfwfgaia.html#Wf_ex.f_spec">Wf_ex.f_spec</a> [in <a href="gaia_hydras.nfwfgaia.html">gaia_hydras.nfwfgaia</a>]<br/>
+<a href="hydras.Ackermann.wConsistent.html#wInconsistent">wInconsistent</a> [in <a href="hydras.Ackermann.wConsistent.html">hydras.Ackermann.wConsistent</a>]<br/>
+<br/><br/><a id="definition_X"></a><h2>X </h2>
+<a href="hydras.Ackermann.cPair.html#xPair">xPair</a> [in <a href="hydras.Ackermann.cPair.html">hydras.Ackermann.cPair</a>]<br/>
+<br/><br/><a id="definition_Z"></a><h2>Z </h2>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#zero">zero</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#zero">zero</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="Goedel.PRrepresentable.html#zeroFormula">zeroFormula</a> [in <a href="Goedel.PRrepresentable.html">Goedel.PRrepresentable</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Generic.html#ZeroLimitSucc_dec">ZeroLimitSucc_dec</a> [in <a href="hydras.OrdinalNotations.ON_Generic.html">hydras.OrdinalNotations.ON_Generic</a>]<br/>
+<a href="hydras.Epsilon0.T1.html#zero_limit_succ_dec">zero_limit_succ_dec</a> [in <a href="hydras.Epsilon0.T1.html">hydras.Epsilon0.T1</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#zero_limit_succ_dec">zero_limit_succ_dec</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega2.html#Zero_limit_succ_dec">Zero_limit_succ_dec</a> [in <a href="hydras.OrdinalNotations.ON_Omega2.html">hydras.OrdinalNotations.ON_Omega2</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html#Zero_limit_succ_dec">Zero_limit_succ_dec</a> [in <a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">hydras.OrdinalNotations.ON_Omega_plus_omega</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Finite.html#Zero_limit_succ_dec">Zero_limit_succ_dec</a> [in <a href="hydras.OrdinalNotations.ON_Finite.html">hydras.OrdinalNotations.ON_Finite</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Zero_limit_succ_dec">Zero_limit_succ_dec</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#Zero_Limit_Succ_dec">Zero_Limit_Succ_dec</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<a href="hydras.OrdinalNotations.ON_Omega.html#Zero_limit_succ_dec">Zero_limit_succ_dec</a> [in <a href="hydras.OrdinalNotations.ON_Omega.html">hydras.OrdinalNotations.ON_Omega</a>]<br/>
+<a href="additions.fib.html#Zfibt">Zfibt</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.fib.html#ZtoR">ZtoR</a> [in <a href="additions.fib.html">additions.fib</a>]<br/>
+<a href="additions.Compatibility.html#Z_pow">Z_pow</a> [in <a href="additions.Compatibility.html">additions.Compatibility</a>]<br/>
+<a href="additions.Naive.html#Z.power">Z.power</a> [in <a href="additions.Naive.html">additions.Naive</a>]<br/>
+<br/><br/><a id="definition__"></a><h2>_ </h2>
+<a href="hydras.Schutte.AP.html#_phi0">_phi0</a> [in <a href="hydras.Schutte.AP.html">hydras.Schutte.AP</a>]<br/>
+<a href="hydras.Schutte.Schutte_basics.html#_omega">_omega</a> [in <a href="hydras.Schutte.Schutte_basics.html">hydras.Schutte.Schutte_basics</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#_f">_f</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<a href="hydras.Schutte.Ordering_Functions.html#_A">_A</a> [in <a href="hydras.Schutte.Ordering_Functions.html">hydras.Schutte.Ordering_Functions</a>]<br/>
+<br/><br/><a id="definition_*"></a><h2>other </h2>
+<a href="gaia_hydras.HydraGaia_Examples.html#ab410a966ac148e9b78c65c6cdf301fd">α</a> [in <a href="gaia_hydras.HydraGaia_Examples.html">gaia_hydras.HydraGaia_Examples</a>]<br/>
+<a href="gaia_hydras.HydraGaia_Examples.html#dc5233cb1d950ecad15b1e9b2514f665">β</a> [in <a href="gaia_hydras.HydraGaia_Examples.html">gaia_hydras.HydraGaia_Examples</a>]<br/>
+<br/><br/><hr/>
+<h1>Record Index</h1>
+<a id="record_A"></a><h2>A </h2>
+<a href="additions.Monoid_def.html#Abelian_Monoid">Abelian_Monoid</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Abelian_EMonoid">Abelian_EMonoid</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="hydras.Prelude.STDPP_compat.html#Assoc">Assoc</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="record_B"></a><h2>B </h2>
+<a href="additions.Euclidean_Chains.html#Bad3.Fchain_proper">Bad3.Fchain_proper</a> [in <a href="additions.Euclidean_Chains.html">additions.Euclidean_Chains</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#Battle">Battle</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<a href="hydras.Hydra.Hydra_Definitions.html#BoundedVariant">BoundedVariant</a> [in <a href="hydras.Hydra.Hydra_Definitions.html">hydras.Hydra.Hydra_Definitions</a>]<br/>
+<br/><br/><a id="record_C"></a><h2>C </h2>
+<a href="hydras.Prelude.Comparable.html#Comparable">Comparable</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<a href="hydras.Prelude.Comparable.html#Compare">Compare</a> [in <a href="hydras.Prelude.Comparable.html">hydras.Prelude.Comparable</a>]<br/>
+<br/><br/><a id="record_D"></a><h2>D </h2>
+<a href="hydras.Prelude.STDPP_compat.html#Decision">Decision</a> [in <a href="hydras.Prelude.STDPP_compat.html">hydras.Prelude.STDPP_compat</a>]<br/>
+<br/><br/><a id="record_E"></a><h2>E </h2>
+<a href="additions.Monoid_def.html#EMonoid">EMonoid</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="additions.Monoid_def.html#Equiv">Equiv</a> [in <a href="additions.Monoid_def.html">additions.Monoid_def</a>]<br/>
+<a href="gaia_hydras.T1Bridge.html#E0">E0</a> [in <a href="gaia_hydras.T1Bridge.html">gaia_hydras.T1Bridge</a>]<br/>
+<a href="hydras.Epsilon0.E0.html#E0">E0</a> [in <a href="hydras.Epsilon0.E0.html">hydras.Epsilon0.E0</a>]<br/>
+<br/><br/><a id="record_F"></a><h2>F </h2>
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+</tr>
+<tr>
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+<td>(119 entries)</td>
+</tr>
+<tr>
+<td>Projection Index</td>
+<td><a href="index.html#projection_A">A</a></td>
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+<td>(110 entries)</td>
+</tr>
+<tr>
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+<td>(284 entries)</td>
+</tr>
+<tr>
+<td>Instance Index</td>
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+<td>(366 entries)</td>
+</tr>
+<tr>
+<td>Abbreviation Index</td>
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+<td>(182 entries)</td>
+</tr>
+<tr>
+<td>Definition Index</td>
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+<td>(1487 entries)</td>
+</tr>
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diff --git a/theories/html/toc.html b/theories/html/toc.html
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+++ b/theories/html/toc.html
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Table of contents</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<div id="toc">
+<h2><a href="hydras.OrdinalNotations.ON_Omega.html">Library hydras.OrdinalNotations.ON_Omega</a></h2>
+<h2><a href="hydras.OrdinalNotations.ON_Generic.html">Library hydras.OrdinalNotations.ON_Generic</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.OrdinalNotations.ON_Generic.html#lab1">Relative correctness of a constant or a function</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.OrdinalNotations.ON_Finite.html">Library hydras.OrdinalNotations.ON_Finite</a></h2>
+<h2><a href="hydras.OrdinalNotations.ON_mult.html">Library hydras.OrdinalNotations.ON_mult</a></h2>
+<ul class="doclist">
+<li><a href="hydras.OrdinalNotations.ON_mult.html#lab2">Definitions</a>
+
+</li>
+<li><a href="hydras.OrdinalNotations.ON_mult.html#lab3">Properties</a>
+
+</li>
+</ul>
+<h2><a href="hydras.OrdinalNotations.ON_Omega_plus_omega.html">Library hydras.OrdinalNotations.ON_Omega_plus_omega</a></h2>
+<h2><a href="hydras.OrdinalNotations.ON_plus.html">Library hydras.OrdinalNotations.ON_plus</a></h2>
+<h2><a href="hydras.OrdinalNotations.ON_O.html">Library hydras.OrdinalNotations.ON_O</a></h2>
+<h2><a href="hydras.OrdinalNotations.ON_Omega2.html">Library hydras.OrdinalNotations.ON_Omega2</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.OrdinalNotations.ON_Omega2.html#lab4">Additive principal ordinals</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.OrdinalNotations.Example_3PlusOmega.html">Library hydras.OrdinalNotations.Example_3PlusOmega</a></h2>
+<h2><a href="hydras.OrdinalNotations.OmegaOmega.html">Library hydras.OrdinalNotations.OmegaOmega</a></h2>
+<ul class="doclist">
+<li><a href="hydras.OrdinalNotations.OmegaOmega.html#lab5">An implementation of  &omega;<sup>&omega;</sup></a>
+
+</li>
+<li><a href="hydras.OrdinalNotations.OmegaOmega.html#lab6">Representation by lists of pairs of integers</a>
+
+</li>
+<li><a href="hydras.OrdinalNotations.OmegaOmega.html#lab7">well formed ordinal representation</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Prelude.ssrnat_extracts.html">Library hydras.Prelude.ssrnat_extracts</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Prelude.ssrnat_extracts.html#lab8">Notations adapted from Mathcomp's ssrnat</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Prelude.MoreDecidable.html">Library hydras.Prelude.MoreDecidable</a></h2>
+<h2><a href="hydras.Prelude.STDPP_compat.html">Library hydras.Prelude.STDPP_compat</a></h2>
+<h2><a href="hydras.Prelude.Comparable.html">Library hydras.Prelude.Comparable</a></h2>
+<h2><a href="hydras.Prelude.First_toggle.html">Library hydras.Prelude.First_toggle</a></h2>
+<h2><a href="hydras.Prelude.Simple_LexProd.html">Library hydras.Prelude.Simple_LexProd</a></h2>
+<h2><a href="hydras.Prelude.DecPreOrder.html">Library hydras.Prelude.DecPreOrder</a></h2>
+<h2><a href="hydras.Prelude.Exp2.html">Library hydras.Prelude.Exp2</a></h2>
+<h2><a href="hydras.Prelude.Sort_spec.html">Library hydras.Prelude.Sort_spec</a></h2>
+<h2><a href="hydras.Prelude.Merge_Sort.html">Library hydras.Prelude.Merge_Sort</a></h2>
+<h2><a href="hydras.Prelude.WfVariant.html">Library hydras.Prelude.WfVariant</a></h2>
+<h2><a href="hydras.Prelude.More_Arith.html">Library hydras.Prelude.More_Arith</a></h2>
+<h2><a href="hydras.Prelude.DecPreOrder_Instances.html">Library hydras.Prelude.DecPreOrder_Instances</a></h2>
+<h2><a href="hydras.Prelude.Fuel.html">Library hydras.Prelude.Fuel</a></h2>
+<h2><a href="hydras.Prelude.Iterates.html">Library hydras.Prelude.Iterates</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Prelude.Iterates.html#lab9">Abstract properties of arithmetic functions</a>
+
+</li>
+<li><a href="hydras.Prelude.Iterates.html#lab10">Abstract properties of iterate</a>
+
+</li>
+<li><a href="hydras.Prelude.Iterates.html#lab11">Second-order iterate</a>
+
+</li>
+<li><a href="hydras.Prelude.Iterates.html#lab12">Exponential and hyper exponential of base 2</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Prelude.OrdNotations.html">Library hydras.Prelude.OrdNotations</a></h2>
+<h2><a href="hydras.Prelude.MoreLists.html">Library hydras.Prelude.MoreLists</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Prelude.MoreLists.html#lab13">Sets of natural numbers as lists</a>
+<ul class="doclist">
+<li><a href="hydras.Prelude.MoreLists.html#lab14">Definitions</a>
+
+</li>
+<li><a href="hydras.Prelude.MoreLists.html#lab15">Lemmas</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Prelude.Restriction.html">Library hydras.Prelude.Restriction</a></h2>
+<h2><a href="hydras.Prelude.MoreOrders.html">Library hydras.Prelude.MoreOrders</a></h2>
+<h2><a href="hydras.Prelude.MoreVectors.html">Library hydras.Prelude.MoreVectors</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Prelude.MoreVectors.html#lab16">Maximum of a vector of nat</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Prelude.Compat815.html">Library hydras.Prelude.Compat815</a></h2>
+<h2><a href="hydras.Prelude.MoreLibHyps.html">Library hydras.Prelude.MoreLibHyps</a></h2>
+<h2><a href="hydras.Prelude.LibHyps_Experiments.html">Library hydras.Prelude.LibHyps_Experiments</a></h2>
+<h2><a href="hydras.Epsilon0.Hessenberg.html">Library hydras.Epsilon0.Hessenberg</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Hessenberg.html#lab17">Hessenberg sum of ordinals terms  (commutative and strictly monotonous)</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Large_Sets.html">Library hydras.Epsilon0.Large_Sets</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab18">minimal large sequences</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab19">Last step of a minimal path</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab20">Paths starting with a finite ordinal (<span class="inlinecode"><span class="id" title="var">fin</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>)</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab21">Properties of m-largeness and L-functions</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab22">About the length of mlarge intervals</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab23">Properties of <span class="inlinecode"><span class="id" title="var">L_spec</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab24">Finite ordinals</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab25"><span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">omega</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab26">Closed formula</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab27">Plain large sequences (not necessarily minimal)</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Large_Sets.html#lab28">large sequences (as in KS ) (may contain zeros)</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Epsilon0rpo.html">Library hydras.Epsilon0.Epsilon0rpo</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Epsilon0rpo.html#lab29">Module Type Variables.</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Epsilon0rpo.html#lab30"> </a>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Hprime.html">Library hydras.Epsilon0.Hprime</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Hprime.html#lab31">A hierarchy of rapidly growing functions</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Hprime.html#lab32">A variant of the Hardy Wainer hierarchy</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Hprime.html#lab33">Monotony of <span class="inlinecode"><span class="id" title="var">H'</span></span> w.r.t. its first argument</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.E0.html">Library hydras.Epsilon0.E0</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Epsilon0.E0.html#lab34">Lifting functions from <span class="inlinecode"><span class="id" title="var">T1</span></span> to <span class="inlinecode"><span class="id" title="var">E0</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.E0.html#lab35">Lemmas</a>
+
+</li>
+<li><a href="hydras.Epsilon0.E0.html#lab36">On equality on type <span class="inlinecode"><span class="id" title="var">E0</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.E0.html#lab37">Adaptation to <span class="inlinecode"><span class="id" title="var">E0</span></span> of lemmas about <span class="inlinecode"><span class="id" title="var">T1</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.E0.html#lab38">Rewriting lemmas</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.T1.html">Library hydras.Epsilon0.T1</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.T1.html#lab39">Definitions</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.T1.html#lab40">A type of terms (not necessarily in normal form)</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab41">A linear  strict order on <span class="inlinecode"><span class="id" title="var">T1</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab42">Properties of <span class="inlinecode"><span class="id" title="var">compare</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab43">Properties of Eq</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab44">Properties of <span class="inlinecode"><span class="id" title="var">lt</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab45">Properties of <span class="inlinecode"><span class="id" title="var">lt</span></span> inv</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab46">The predicate "to be in normal form"</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab47">Arithmetic functions</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.T1.html#lab48">Successor</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab49">Predecessor (partial function</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab50">Addition</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab51">multiplication</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab52">Substraction  (used as a helper for exponentiation)</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab53">exponentiation functions</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab54">Lemmas</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.T1.html#lab55">Successors, limits and zero</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab56">Second part on <span class="inlinecode"><span class="id" title="var">lt</span></span> and <span class="inlinecode"><span class="id" title="var">le</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab57">Properties of <span class="inlinecode"><span class="id" title="var">compare</span></span>: second part</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab58">Properties of <span class="inlinecode"><span class="id" title="var">max</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab59">Restriction of lt and le to terms in normal form</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab60">Well foundedness of <span class="inlinecode"><span class="id" title="var">LT</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.T1.html#lab61">Strong accessibility (inspired by Tait's proof)</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab62">A (last) structural induction</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab63">Properties of successor</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab64">properties of addition</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab65"><span class="inlinecode"><span class="id" title="var">plus</span></span> and <span class="inlinecode"><span class="id" title="var">LT</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab66">On additive principal ordinals</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab67">monotonicity of <span class="inlinecode"><span class="id" title="var">succ</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab68">monotonicity of <span class="inlinecode"><span class="id" title="var">phi0</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab69">Multiplication</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab70">About minus</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab71">Exponential</a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab72">Relations between <span class="inlinecode"><span class="id" title="var">cons</span></span>, <span class="inlinecode"><span class="id" title="var">phi0</span></span> and <span class="inlinecode">+</span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab73">An abstract syntax for ordinals in Cantor normal form</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.T1.html#lab74">Examples</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Paths.html">Library hydras.Epsilon0.Paths</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Epsilon0.Paths.html#lab75">relations  associated with canonical sequences</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab76">Paths inside epsilon_0</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab77">Accessibility (without traces)</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab78">following a path</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab79">Paths with constant index</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab80">standard paths</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab81">Bridge Lemmas</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab82">Composition/decomposition of paths</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab83">Properties of gnaw</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab84">Properties of acc_from</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab85">Paths with constant index: lemmas</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.Paths.html#lab86">Adaptation to <span class="inlinecode"><span class="id" title="var">E0</span></span></a>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Canon.html">Library hydras.Epsilon0.Canon</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Canon.html#lab87">Definitions</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Canon.html#lab88">Properties of canonical sequences</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.Canon.html#lab89">Canonical sequences and the order LT</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Canon.html#lab90">Canonical sequences of limit ordinals</a>
+
+</li>
+<li><a href="hydras.Epsilon0.Canon.html#lab91">Adaptation to E0 (well formed terms of type <span class="inlinecode"><span class="id" title="var">T1</span></span> )</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Large_Sets_Examples.html">Library hydras.Epsilon0.Large_Sets_Examples</a></h2>
+<h2><a href="hydras.Epsilon0.L_alpha.html">Library hydras.Epsilon0.L_alpha</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.L_alpha.html#lab92">Computing the length of paths from <span class="inlinecode"><span class="id" title="var">alpha</span></span> to <span class="inlinecode"><span class="id" title="var">zero</span></span></a>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.Epsilon0.html">Library hydras.Epsilon0.Epsilon0</a></h2>
+<h2><a href="hydras.Epsilon0.F_alpha.html">Library hydras.Epsilon0.F_alpha</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.F_alpha.html#lab93">The Wainer hierarchy of rapidly growing functions (variant)</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.F_alpha.html#lab94">Definition, using <span class="inlinecode"><span class="id" title="var">coq</span>-<span class="id" title="var">equations</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab95">We get the "usual" equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.F_alpha.html#lab96">Relations between <span class="inlinecode"><span class="id" title="var">F_star</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab97">Usual equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab98">First steps of the hierarchy</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab99">Comparison with the Hardy hierarchy</a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab100">A variant (Lob-Wainer hierarchy)</a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.F_alpha.html#lab101">We get the "usual" equations for <span class="inlinecode"><span class="id" title="var">F_</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Epsilon0.F_alpha.html#lab102">Relations between <span class="inlinecode"><span class="id" title="var">F_star</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_alpha.html#lab103">Usual equations for <span class="inlinecode"><span class="id" title="var">f_</span></span></a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Epsilon0.F_omega.html">Library hydras.Epsilon0.F_omega</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Epsilon0.F_omega.html#lab104">Rewriting lemmas</a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_omega.html#lab105">Comparison between <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> and <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_omega.html#lab106"><span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">omega</span></span> is not primitive recursive</a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_omega.html#lab107"><span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">alpha</span></span> is not primitive recursive, for <span class="inlinecode"><span class="id" title="var">omega</span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" title="var">alpha</span></span></a>
+
+</li>
+<li><a href="hydras.Epsilon0.F_omega.html#lab108">On the other hand, <span class="inlinecode"><span class="id" title="var">F_</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is PR for any <span class="inlinecode"><span class="id" title="var">n</span>:<span class="id" title="var">nat</span></span></a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Gamma0.T2.html">Library hydras.Gamma0.T2</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Gamma0.T2.html#lab109">Definitions</a>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Gamma0.T2.html#lab110">Examples</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Gamma0.T2.html#lab111">Length (as in Schutte)</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Gamma0.Gamma0.html">Library hydras.Gamma0.Gamma0</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Gamma0.Gamma0.html#lab112">A notation system for ordinals less than  Gamma0</a>
+<ul class="doclist">
+<li><a href="hydras.Gamma0.Gamma0.html#lab113">terms in normal form</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Gamma0.Gamma0.html#lab114">Well-foundedness (with rpo) (Evelyne Contejean)</a>
+<ul class="doclist">
+<li><a href="hydras.Gamma0.Gamma0.html#lab115">Module Type Variables.</a>
+
+</li>
+<li><a href="hydras.Gamma0.Gamma0.html#lab116">Every (representation of a) natural number is less than</a>
+
+</li>
+<li><a href="hydras.Gamma0.Gamma0.html#lab117">inclusion of the order <span class="inlinecode"><span class="id" title="var">lt</span></span> in the rpo</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Gamma0.Gamma0.html#lab118">the Veblen function phi</a>
+<ul class="doclist">
+<li><a href="hydras.Gamma0.Gamma0.html#lab119">Temporary compatibility  nf/nfb</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.decidable_set.html">Library hydras.rpo.decidable_set</a></h2>
+<h2><a href="hydras.rpo.dickson.html">Library hydras.rpo.dickson</a></h2>
+<ul class="doclist">
+<li><a href="hydras.rpo.dickson.html#lab120">Dickson Lemma: the multiset extension of a well-founded ordering is well-founded.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.dickson.html#lab121">Definition of the multiset extension of a relation.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.dickson.html#lab122">Accessibility lemmata.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.list_set.html">Library hydras.rpo.list_set</a></h2>
+<ul class="doclist">
+<li><a href="hydras.rpo.list_set.html#lab123">Sets built with lists</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.list_set.html#lab124">Definition of sets using lists.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.list_permut.html">Library hydras.rpo.list_permut</a></h2>
+<ul class="doclist">
+<li><a href="hydras.rpo.list_permut.html#lab125">Permutation over lists, and finite multisets.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.list_permut.html#lab126">Definition of permutation over lists.</a>
+
+</li>
+<li><a href="hydras.rpo.list_permut.html#lab127">Permutation is a equivalence relation.</a>
+
+</li>
+<li><a href="hydras.rpo.list_permut.html#lab128">Compatibility Properties.</a>
+
+</li>
+<li><a href="hydras.rpo.list_permut.html#lab129">Permutation for short lists.</a>
+
+</li>
+<li><a href="hydras.rpo.list_permut.html#lab130">Link with AC syntactic decomposition.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.more_list.html">Library hydras.rpo.more_list</a></h2>
+<ul class="doclist">
+<li><a href="hydras.rpo.more_list.html#lab131">Some additional properties for the Coq lists.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.more_list.html#lab132">Relations between length, map, append, In and nth.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab133">A measure on lists based on a measure on elements.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab134">Induction principles for list.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab135">How to remove an element in a list, whenever it is present.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab136">Iterators.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab137">more properties on the nth element.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab138">Association lists.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.more_list.html#lab139">find.</a>
+
+</li>
+<li><a href="hydras.rpo.more_list.html#lab140">number of occurences of the first element of a pair.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.closure.html">Library hydras.rpo.closure</a></h2>
+<h2><a href="hydras.rpo.rpo.html">Library hydras.rpo.rpo</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.rpo.rpo.html#lab141">Module Type Precedence,</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab142">Definition of a precedence.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab143">Module Type RPO,</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab144">Definition of RPO from a precedence on symbols.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab145">Definition of rpo.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab146">rpo is a preorder, and its reflexive closure is an ordering.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab147">Main theorem: when the precedence is well-founded, so is the rpo.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab148">RPO is compatible with the instanciation by a substitution.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab149">RPO is compatible with adding context.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab150">Definition of size-based well-founded orderings for induction.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab151">Definition of rpo.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab152">rpo is a preorder, and its reflexive closure is an ordering.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab153">Well-foundedness of rpo.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab154">Main theorem: when the precedence is well-founded, so is the rpo.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab155">RPO is compatible with the instanciation by a substitution.</a>
+
+</li>
+<li><a href="hydras.rpo.rpo.html#lab156">RPO is compatible with adding context.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.rpo.term.html">Library hydras.rpo.term</a></h2>
+<ul class="doclist">
+<li><a href="hydras.rpo.term.html#lab157">Term algebra defined as functor from a Module Signature and a Module Variable.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab158">Module Type Signature.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab159">Module Type Variables.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab160">Module Type Term built from a signature and a set of variables.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.term.html#lab161">Recursion on terms.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab162">Double recursion on terms.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab163">Equality on terms is decidable.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab164">Well-formedness of terms, according to the arity.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab165">Substitutions.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab166">Positions in a term.</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab167">A functor building a Term Module from a Signature and a set of Variables.</a>
+<ul class="doclist">
+<li><a href="hydras.rpo.term.html#lab168">Recursion on terms.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab169">Double recursion on terms.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab170">Equality on terms is decidable.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab171">Well-formedness of terms, according to the arity.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab172">Substitutions.</a>
+
+</li>
+<li><a href="hydras.rpo.term.html#lab173">Positions in a term.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.MoreEpsilonIota.html">Library hydras.Schutte.MoreEpsilonIota</a></h2>
+<h2><a href="hydras.Schutte.CNF.html">Library hydras.Schutte.CNF</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Schutte.CNF.html#lab174">Every countable ordinal has (at least) a Cantor normal form</a>
+
+</li>
+<li><a href="hydras.Schutte.CNF.html#lab175">Unicity of cnf</a>
+
+</li>
+<li><a href="hydras.Schutte.CNF.html#lab176">The main result</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Schutte.CNF.html#lab177">Cantor Normal Form and the ordinal epsilon0</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.AP.html">Library hydras.Schutte.AP</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Schutte.AP.html#lab178">Additive principal ordinals</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.AP.html#lab179">Main Definitions</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.AP.html#lab180">Additive principal ordinals</a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab181">Exponential of basis omega</a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab182">Omega-towers</a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab183">The limit ordinal <span class="inlinecode"><span class="id" title="var">epsilon0</span></span></a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab184">Proofs, proofs, proofs ...</a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab185">About additive principals</a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab186">Properties of <span class="inlinecode"><span class="id" title="var">phi0</span></span></a>
+
+</li>
+<li><a href="hydras.Schutte.AP.html#lab187">Properties of <span class="inlinecode"><span class="id" title="var">epsilon0</span></span></a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Schutte_basics.html">Library hydras.Schutte.Schutte_basics</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte_basics.html#lab188">Definitions</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab189">The three axioms by Schutte</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab190">First definitions</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte_basics.html#lab191">Limits</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab192">Basic properties</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte_basics.html#lab193">Properties of <span class="inlinecode"><span class="id" title="var">le</span></span> and <span class="inlinecode"><span class="id" title="var">lt</span></span></a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab194">Global properties</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab195">About zero</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab196">Properties of successor</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab197">Building limits</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab198">Sequences of ordinals</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab199">Properties of omega</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab200">About zero, is_succ and is_limit</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab201">About members</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte_basics.html#lab202">Extensional equalities on sets of ordinals</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Ordering_Functions.html">Library hydras.Schutte.Ordering_Functions</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Schutte.Ordering_Functions.html#lab203">Main definitions</a>
+
+</li>
+<li><a href="hydras.Schutte.Ordering_Functions.html#lab204">Properties of  segments</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Countable.html">Library hydras.Schutte.Countable</a></h2>
+<h2><a href="hydras.Schutte.Critical.html">Library hydras.Schutte.Critical</a></h2>
+<h2><a href="hydras.Schutte.Schutte.html">Library hydras.Schutte.Schutte</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte.html#lab205">Warning</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte.html#lab206">Schutte's Axioms</a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte.html#lab207">Classical logic and Hilbert style</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Schutte.html#lab208">From <span class="inlinecode"><span class="id" title="var">Coq.Logic.Classical</span></span></a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte.html#lab209">From <span class="inlinecode"><span class="id" title="var">Coq.Logic.Epsilon</span></span></a>
+
+</li>
+<li><a href="hydras.Schutte.Schutte.html#lab210">Needed for <span class="inlinecode"><span class="id" title="var">epsilon</span></span> to work properly.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Well_Orders.html">Library hydras.Schutte.Well_Orders</a></h2>
+<h2><a href="hydras.Schutte.Addition.html">Library hydras.Schutte.Addition</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Addition.html#lab211">Definitions</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Addition.html#lab212">addition, multiplication by a positive integer</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Schutte.Addition.html#lab213">Proofs, proofs, proofs</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.Addition.html#lab214">Basic properties of addition</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Correctness_E0.html">Library hydras.Schutte.Correctness_E0</a></h2>
+<h2><a href="hydras.Schutte.PartialFun.html">Library hydras.Schutte.PartialFun</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Schutte.PartialFun.html#lab215">Partial functions</a>
+<ul class="doclist">
+<li><a href="hydras.Schutte.PartialFun.html#lab216">Transformation of a functional relation into a function</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Schutte.Lub.html">Library hydras.Schutte.Lub</a></h2>
+<h2><a href="hydras.Schutte.GRelations.html">Library hydras.Schutte.GRelations</a></h2>
+<h2><a href="hydras.Hydra.Battle_length.html">Library hydras.Hydra.Battle_length</a></h2>
+<h2><a href="hydras.Hydra.Omega_Small.html">Library hydras.Hydra.Omega_Small</a></h2>
+<h2><a href="hydras.Hydra.BigBattle.html">Library hydras.Hydra.BigBattle</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Hydra.BigBattle.html#lab217">A long hydra battle</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Epsilon0_Needed_Generic.html">Library hydras.Hydra.Epsilon0_Needed_Generic</a></h2>
+<h2><a href="hydras.Hydra.Hydra_Extraction.html">Library hydras.Hydra.Hydra_Extraction</a></h2>
+<h2><a href="hydras.Hydra.KP_example.html">Library hydras.Hydra.KP_example</a></h2>
+<h2><a href="hydras.Hydra.Hydra_Termination.html">Library hydras.Hydra.Hydra_Termination</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Termination.html#lab218">Proof of termination of all hydra battles</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Termination.html#lab219">Converting any hydra into an ordinal less than</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Termination.html#lab220">Monotonicity lemmas for S0, R1, etc .</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Hydra_Theorems.html">Library hydras.Hydra.Hydra_Theorems</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Theorems.html#lab221">Liveness</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Theorems.html#lab222">Termination of free battles</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Theorems.html#lab223">Impossibility theorems</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Theorems.html#lab224">Battle length is not PR</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Omega2_Small.html">Library hydras.Hydra.Omega2_Small</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Omega2_Small.html#lab225">Proof of the inequality <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">small_h</span></span> <span class="inlinecode"><span class="id" title="var">o</span>&lt;</span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">big_h</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Omega2_Small.html#lab226">Proof of the inequality <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">big_h</span></span> <span class="inlinecode"><span class="id" title="var">o</span>≤</span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="var">small_h</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Omega2_Small.html#lab227">Let us decompose any inequality p o&lt; q into elementary steps</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Hydra_Definitions.html">Library hydras.Hydra.Hydra_Definitions</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab228">Data types</a>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab229">height (length of longest branch)</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab230">Abbreviations</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab231">Heads : A head is just a node without daughters</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab232">Hydra with 0, 1, 2 or 3 daughters</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab233">Adds n copies of the same hydra h at the right of  s</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab234">Hydra with n equal daughters</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab235">Managing sequences of hydras</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab236">Appending</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab237">Adds a  head to  the right of <span class="inlinecode"><span class="id" title="var">s</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab238">Adds i   heads to the right of <span class="inlinecode"><span class="id" title="var">s</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab239">adds <span class="inlinecode"><span class="id" title="var">i</span></span> heads to  the root of <span class="inlinecode"><span class="id" title="var">h</span></span> (to the right of its daughters)</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab240">Hydra Battles</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab241">Relation associated with a single round.</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab242">S0</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab243">R1</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab244">S1</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab245">R2</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab246"><span class="inlinecode"><span class="id" title="var">round_n</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab247">Binary relation associated with a battle round</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab248">transitive closures of round</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab249">Experimental tactics for interactive battles</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab250">removes the i-th daughter (should be a head)</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab251">Traces</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab252">Classes of battles</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab253">Uniform termination</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab254">Variants for proving termination</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab255">Strictly Bounded variants</a>
+<ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Definitions.html#lab256">Liveness</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Epsilon0_Needed_Free.html">Library hydras.Hydra.Epsilon0_Needed_Free</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Epsilon0_Needed_Free.html#lab257">Bounded variants</a>
+
+</li>
+<li><a href="hydras.Hydra.Epsilon0_Needed_Free.html#lab258">Proof of the inequality m small_h t1&lt; m big_h</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.Hydra_Examples.html">Library hydras.Hydra.Hydra_Examples</a></h2>
+<h2><a href="hydras.Hydra.Hydra_Lemmas.html">Library hydras.Hydra.Hydra_Lemmas</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Hydra_Lemmas.html#lab259">Properties of the  <span class="inlinecode"><span class="id" title="var">hydra</span></span> data type</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Lemmas.html#lab260">Properties of single rounds</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Lemmas.html#lab261">Properties of <span class="inlinecode"><span class="id" title="var">battle</span></span></a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Lemmas.html#lab262">Properties of standard and free battle classes</a>
+
+</li>
+<li><a href="hydras.Hydra.Hydra_Lemmas.html#lab263">Generic properties of round_plus</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Hydra.O2H.html">Library hydras.Hydra.O2H</a></h2>
+<h2><a href="hydras.Hydra.Epsilon0_Needed_Std.html">Library hydras.Hydra.Epsilon0_Needed_Std</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.Hydra.Epsilon0_Needed_Std.html#lab264">Proof of the inequality m small_h t1&lt; m big_h</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.codeNatToTerm.html">Library hydras.Ackermann.codeNatToTerm</a></h2>
+<h2><a href="hydras.Ackermann.code.html">Library hydras.Ackermann.code</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.code.html#lab265">Encoding terms, formulas and proofs</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.extEqualNat.html">Library hydras.Ackermann.extEqualNat</a></h2>
+<h2><a href="hydras.Ackermann.folLogic.html">Library hydras.Ackermann.folLogic</a></h2>
+<h2><a href="hydras.Ackermann.fol.html">Library hydras.Ackermann.fol</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.fol.html#lab266">First Order Formulas over a language</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.fol.html#lab267">Extensions of the basic language of formulas</a>
+
+</li>
+<li><a href="hydras.Ackermann.fol.html#lab268">Decidability of equality between terms, formulas, ...</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.fol.html#lab269">Depth Induction</a>
+
+</li>
+<li><a href="hydras.Ackermann.fol.html#lab270">Implicit arguments and notations</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.fol.html#lab271">Implicit arguments</a>
+
+</li>
+<li><a href="hydras.Ackermann.fol.html#lab272">The <span class="inlinecode"><span class="id" title="var">fol_scope</span></span> notation scope</a>
+
+</li>
+<li><a href="hydras.Ackermann.fol.html#lab273">Examples</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.LNN.html">Library hydras.Ackermann.LNN</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN.html#lab274">Instantiations of a few generic constructs</a>
+
+</li>
+<li><a href="hydras.Ackermann.LNN.html#lab275">Notations (Experimental and unstable)</a>
+
+</li>
+<li><a href="hydras.Ackermann.LNN.html#lab276">Basic and derived properties</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.NN2PA.html">Library hydras.Ackermann.NN2PA</a></h2>
+<h2><a href="hydras.Ackermann.PAtheory.html">Library hydras.Ackermann.PAtheory</a></h2>
+<h2><a href="hydras.Ackermann.prLogic.html">Library hydras.Ackermann.prLogic</a></h2>
+<h2><a href="hydras.Ackermann.wellFormed.html">Library hydras.Ackermann.wellFormed</a></h2>
+<h2><a href="hydras.Ackermann.checkPrf.html">Library hydras.Ackermann.checkPrf</a></h2>
+<h2><a href="hydras.Ackermann.codePA.html">Library hydras.Ackermann.codePA</a></h2>
+<h2><a href="hydras.Ackermann.cPair.html">Library hydras.Ackermann.cPair</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.cPair.html#lab277">Bijection from <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> to <span class="inlinecode"><span class="id" title="var">nat</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.cPair.html#lab278">Preliminary definitions</a>
+
+</li>
+<li><a href="hydras.Ackermann.cPair.html#lab279">Definition and properties of <span class="inlinecode"><span class="id" title="var">cPair</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.cPair.html#lab280">Triples</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.folProof.html">Library hydras.Ackermann.folProof</a></h2>
+<h2><a href="hydras.Ackermann.Languages.html">Library hydras.Ackermann.Languages</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.Languages.html#lab281">Language of Number Theory: <span class="inlinecode"><span class="id" title="var">LNT</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.Languages.html#lab282">Language of Natural Numbers: <span class="inlinecode"><span class="id" title="var">LNN</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.Languages.html#lab283">Goedel encoding for LNT</a>
+
+</li>
+<li><a href="hydras.Ackermann.Languages.html#lab284">Goedel encoding for LNN</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.LNT.html">Library hydras.Ackermann.LNT</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNT.html#lab285">Instantiations of a few generic constructs</a>
+
+</li>
+<li><a href="hydras.Ackermann.LNT.html#lab286">Notations for LNT formulas: experimental and unstable</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNT.html#lab287">Notations for printing computed formulas/terms with derived connectives</a>
+
+</li>
+<li><a href="hydras.Ackermann.LNT.html#lab288">Examples</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.LNT.html#lab289">Basic properties</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNT.html#lab290">List of free variables of a formula</a>
+
+</li>
+<li><a href="hydras.Ackermann.LNT.html#lab291">Basic and derived proof rules</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.NNtheory.html">Library hydras.Ackermann.NNtheory</a></h2>
+<h2><a href="hydras.Ackermann.PA.html">Library hydras.Ackermann.PA</a></h2>
+<h2><a href="hydras.Ackermann.subAll.html">Library hydras.Ackermann.subAll</a></h2>
+<h2><a href="hydras.Ackermann.codeFreeVar.html">Library hydras.Ackermann.codeFreeVar</a></h2>
+<h2><a href="hydras.Ackermann.codeSubFormula.html">Library hydras.Ackermann.codeSubFormula</a></h2>
+<h2><a href="hydras.Ackermann.Deduction.html">Library hydras.Ackermann.Deduction</a></h2>
+<h2><a href="hydras.Ackermann.folLogic2.html">Library hydras.Ackermann.folLogic2</a></h2>
+<h2><a href="hydras.Ackermann.folProp.html">Library hydras.Ackermann.folProp</a></h2>
+<h2><a href="hydras.Ackermann.ListExt.html">Library hydras.Ackermann.ListExt</a></h2>
+<h2><a href="hydras.Ackermann.misc.html">Library hydras.Ackermann.misc</a></h2>
+<h2><a href="hydras.Ackermann.NN.html">Library hydras.Ackermann.NN</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.NN.html#lab292">Axioms of <span class="inlinecode"><span class="id" title="var">NN</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.NN.html#lab293">Properties of the system <span class="inlinecode"><span class="id" title="var">NN</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.NN.html#lab294">Generic instantiation of axioms</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.subProp.html">Library hydras.Ackermann.subProp</a></h2>
+<h2><a href="hydras.Ackermann.codeList.html">Library hydras.Ackermann.codeList</a></h2>
+<h2><a href="hydras.Ackermann.codeSubTerm.html">Library hydras.Ackermann.codeSubTerm</a></h2>
+<h2><a href="hydras.Ackermann.expressible.html">Library hydras.Ackermann.expressible</a></h2>
+<h2><a href="hydras.Ackermann.folLogic3.html">Library hydras.Ackermann.folLogic3</a></h2>
+<h2><a href="hydras.Ackermann.folReplace.html">Library hydras.Ackermann.folReplace</a></h2>
+<h2><a href="hydras.Ackermann.LNN2LNT.html">Library hydras.Ackermann.LNN2LNT</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab295">Translation of terms</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab296">Inverse translation</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab297">Translation of formulas</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab298">Translation of <span class="inlinecode">(<span class="id" title="var">v</span>#0</span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">v</span>#1)%<span class="id" title="var">fol</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab299">Translation of <span class="inlinecode">(<span class="id" title="var">t</span></span> <span class="inlinecode">&lt;</span> <span class="inlinecode"><span class="id" title="var">t'</span>)%<span class="id" title="var">fol</span></span></a>
+
+</li>
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab300">Translation of any <span class="inlinecode"><span class="id" title="var">LNN</span></span>-formula</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab301">Helpful rewriting lemmas</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab302">Inverse translation</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab303">Commutation lemmas</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.LNN2LNT.html#lab304">Proof translation</a>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.model.html">Library hydras.Ackermann.model</a></h2>
+<h2><a href="hydras.Ackermann.PAconsistent.html">Library hydras.Ackermann.PAconsistent</a></h2>
+<h2><a href="hydras.Ackermann.primRec.html">Library hydras.Ackermann.primRec</a></h2>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.primRec.html#lab305">Definitions</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.primRec.html#lab306">Semantics</a>
+<ul class="doclist">
+<li><a href="hydras.Ackermann.primRec.html#lab307">Constants</a>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab308">Projections</a>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab309">Function composition</a>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab310">Primitive recursion</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab311">Predicates "to be primitive recursive"</a>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab312">Usual projections (in curried form) are primitive recursive</a>
+
+</li>
+<li><a href="hydras.Ackermann.primRec.html#lab313">Composition lemmas</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.Ackermann.wConsistent.html">Library hydras.Ackermann.wConsistent</a></h2>
+<h2><a href="hydras.Ackermann.NewNotations.html">Library hydras.Ackermann.NewNotations</a></h2>
+<h2><a href="hydras.MoreAck.Ack.html">Library hydras.MoreAck.Ack</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.MoreAck.Ack.html#lab314">Exercise</a>
+
+</li>
+<li><a href="hydras.MoreAck.Ack.html#lab315">Usual equations</a>
+
+</li>
+<li><a href="hydras.MoreAck.Ack.html#lab316">First values</a>
+
+</li>
+<li><a href="hydras.MoreAck.Ack.html#lab317">monotony properties</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.MoreAck.Ack.html#lab318">Bounding nested calls of <span class="inlinecode"><span class="id" title="var">Ack</span></span></a>
+
+</li>
+<li><a href="hydras.MoreAck.Ack.html#lab319"><span class="inlinecode"><span class="id" title="var">Ack</span></span> is (partially) strictly monotonous in its first argument</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.MoreAck.AckNotPR.html">Library hydras.MoreAck.AckNotPR</a></h2>
+<ul class="doclist">
+<li><a href="hydras.MoreAck.AckNotPR.html#lab320">Proof that Ack is not primitive recursive</a>
+<ul class="doclist">
+<li><a href="hydras.MoreAck.AckNotPR.html#lab321">Comparing an n-ary and a binary functions</a>
+
+</li>
+<li><a href="hydras.MoreAck.AckNotPR.html#lab322">Every primitive recursive function is majorized by <span class="inlinecode"><span class="id" title="var">Ack</span></span></a>
+<ul class="doclist">
+<li><a href="hydras.MoreAck.AckNotPR.html#lab323">Base cases</a>
+
+</li>
+<li><a href="hydras.MoreAck.AckNotPR.html#lab324">The general case is proved by  induction on x</a>
+
+</li>
+<li><a href="hydras.MoreAck.AckNotPR.html#lab325">Now, let us assume that <span class="inlinecode"><span class="id" title="var">Ack</span></span> is PR</a>
+
+</li>
+<li><a href="hydras.MoreAck.AckNotPR.html#lab326">Any function which dominates (diagonalized) <span class="inlinecode"><span class="id" title="var">Ack</span></span> fails to be PR</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.MoreAck.AckNotPR.html#lab327">Nevertheless, for any <span class="inlinecode"><span class="id" title="var">n</span></span>, <span class="inlinecode"><span class="id" title="var">Ack</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is primitive recursive.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.MoreAck.FolExamples.html">Library hydras.MoreAck.FolExamples</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.MoreAck.FolExamples.html#lab328">Preliminary lemmas</a>
+
+</li>
+<li><a href="hydras.MoreAck.FolExamples.html#lab329">Universal quantifier</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.MoreAck.Iterate_compat.html">Library hydras.MoreAck.Iterate_compat</a></h2>
+<h2><a href="hydras.MoreAck.PrimRecExamples.html">Library hydras.MoreAck.PrimRecExamples</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.MoreAck.PrimRecExamples.html#lab330">Examples of terms of type <span class="inlinecode"><span class="id" title="var">PrimRec</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> and their interpretation</a>
+
+</li>
+<li><a href="hydras.MoreAck.PrimRecExamples.html#lab331">Understanding some constructions ...</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="hydras.MoreAck.PrimRecExamples.html#lab332">First proofs of isPR statements</a>
+
+</li>
+</ul>
+<h2><a href="hydras.MoreAck.LNN_Examples.html">Library hydras.MoreAck.LNN_Examples</a></h2>
+<h2><a href="hydras.MoreAck.BadSubst.html">Library hydras.MoreAck.BadSubst</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.MoreAck.BadSubst.html#lab333">Preliminary lemmas</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.MoreAck.expressibleExamples.html">Library hydras.MoreAck.expressibleExamples</a></h2>
+<h2><a href="hydras.solutions_exercises.MinPR.html">Library hydras.solutions_exercises.MinPR</a></h2>
+<h2><a href="hydras.solutions_exercises.MinPR2.html">Library hydras.solutions_exercises.MinPR2</a></h2>
+<h2><a href="hydras.solutions_exercises.FibonacciPR.html">Library hydras.solutions_exercises.FibonacciPR</a></h2>
+<h2><a href="hydras.solutions_exercises.MorePRExamples.html">Library hydras.solutions_exercises.MorePRExamples</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="hydras.solutions_exercises.MorePRExamples.html#lab334">Solution to an exercise</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.solutions_exercises.isqrt.html">Library hydras.solutions_exercises.isqrt</a></h2>
+<h2><a href="hydras.solutions_exercises.T1_ltNotWf.html">Library hydras.solutions_exercises.T1_ltNotWf</a></h2>
+<h2><a href="hydras.solutions_exercises.predSuccUnicity.html">Library hydras.solutions_exercises.predSuccUnicity</a></h2>
+<h2><a href="hydras.solutions_exercises.lt_succ_le.html">Library hydras.solutions_exercises.lt_succ_le</a></h2>
+<h2><a href="hydras.solutions_exercises.Limit_Infinity.html">Library hydras.solutions_exercises.Limit_Infinity</a></h2>
+<h2><a href="hydras.solutions_exercises.ge_omega_iff.html">Library hydras.solutions_exercises.ge_omega_iff</a></h2>
+<h2><a href="hydras.solutions_exercises.schutte_cnf_counter_example.html">Library hydras.solutions_exercises.schutte_cnf_counter_example</a></h2>
+<h2><a href="hydras.solutions_exercises.F_3.html">Library hydras.solutions_exercises.F_3</a></h2>
+<ul class="doclist">
+<li><a href="hydras.solutions_exercises.F_3.html#lab335">Proof that F_alpha  (S n) &gt; exp2 (F alpha n) for alpha &gt;= 3 and</a>
+<ul class="doclist">
+<li><a href="hydras.solutions_exercises.F_3.html#lab336">Base case</a>
+
+</li>
+<li><a href="hydras.solutions_exercises.F_3.html#lab337">Limit case</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="hydras.solutions_exercises.is_F_monotonous.html">Library hydras.solutions_exercises.is_F_monotonous</a></h2>
+<h2><a href="hydras.solutions_exercises.MultisetWf.html">Library hydras.solutions_exercises.MultisetWf</a></h2>
+<h2><a href="hydras.solutions_exercises.OnCodeList.html">Library hydras.solutions_exercises.OnCodeList</a></h2>
+<h2><a href="additions.AM.html">Library additions.AM</a></h2>
+<ul class="doclist">
+<li><a href="additions.AM.html#lab338">Euclidean chain generation</a>
+
+</li>
+</ul>
+<h2><a href="additions.Addition_Chains.html">Library additions.Addition_Chains</a></h2>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab339">Addition Chains</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab340">Computations composed of multiplications over type  A</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab341">Monadic Notation for computations</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab342">Definition</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab343">Our Favorite example</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab344">Chain length (number of mutiplications)</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab345">Execution of chains</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab346">Chain correctness</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab347">Correctness Proof by Reflection</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab348">Correctness and parametricity</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab349">computation of  compared to computation of </a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab350">Correctness Theorem</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab351">Remark</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab352">Back to 87</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab353">Correct by construction chains</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab354">Definition</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab355">Some chain generators</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab356">Chains associated to the well-known binary exponentiation algorithm</a>
+<ul class="doclist">
+<li><a href="additions.Addition_Chains.html#lab357">computation of </a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab358">Correctness of the binary method</a>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab359">The binary method is not optimal</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Addition_Chains.html#lab360">How to prove chain correctness in general ?</a>
+
+</li>
+</ul>
+<h2><a href="additions.BinaryStrat.html">Library additions.BinaryStrat</a></h2>
+<h2><a href="additions.Compatibility.html">Library additions.Compatibility</a></h2>
+<ul class="doclist">
+<li><a href="additions.Compatibility.html#lab361">Compatibility with StdLib exponentiation functions</a>
+<ul class="doclist">
+<li><a href="additions.Compatibility.html#lab362">really logarithmic versions of N.pow, Pos.pow and Z.pow</a>
+
+</li>
+<li><a href="additions.Compatibility.html#lab363">Equivalence between StdLib definitions and ours</a>
+
+</li>
+<li><a href="additions.Compatibility.html#lab364">Tests</a>
+<ul class="doclist">
+<li><a href="additions.Compatibility.html#lab365">Adapting lemmas from Standard Library</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Demo.html">Library additions.Demo</a></h2>
+<h2><a href="additions.Demo_power.html">Library additions.Demo_power</a></h2>
+<h2><a href="additions.Dichotomy.html">Library additions.Dichotomy</a></h2>
+<h2><a href="additions.Euclidean_Chains.html">Library additions.Euclidean_Chains</a></h2>
+<ul class="doclist">
+<li><a href="additions.Euclidean_Chains.html#lab366">Euclidean chains</a>
+<ul class="doclist">
+<li><a href="additions.Euclidean_Chains.html#lab367">Introduction</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Euclidean_Chains.html#lab368">CPS chain construction</a>
+<ul class="doclist">
+<li><a href="additions.Euclidean_Chains.html#lab369">Remark</a>
+
+</li>
+<li><a href="additions.Euclidean_Chains.html#lab370">More chain combinators</a>
+<ul class="doclist">
+<li><a href="additions.Euclidean_Chains.html#lab371">Automatic generation of correct euclidean chains</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Euclidean_Chains.html#lab372">Full correctness</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.fib.html">Library additions.fib</a></h2>
+<h2><a href="additions.Fib2.html">Library additions.Fib2</a></h2>
+<h2><a href="additions.FirstSteps.html">Library additions.FirstSteps</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="additions.FirstSteps.html#lab373">Examples</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.FirstSteps.html#lab374">A wrong definition of correctness</a>
+
+</li>
+</ul>
+<h2><a href="additions.Monoid_def.html">Library additions.Monoid_def</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="additions.Monoid_def.html#lab375">The <span class="inlinecode"><span class="id" title="var">Monoid</span></span> type class (with Operational Type Classes)</a>
+<ul class="doclist">
+<li><a href="additions.Monoid_def.html#lab376">Exercice</a>
+
+</li>
+<li><a href="additions.Monoid_def.html#lab377">Monoids and Equivalence Relations</a>
+
+</li>
+<li><a href="additions.Monoid_def.html#lab378">Coercion from Monoid to EMonoid</a>
+
+</li>
+<li><a href="additions.Monoid_def.html#lab379">Commutative Monoids</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Monoid_instances.html">Library additions.Monoid_instances</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="additions.Monoid_instances.html#lab380">Some useful instances of Monoid classes</a>
+<ul class="doclist">
+<li><a href="additions.Monoid_instances.html#lab381">Multiplicative monoid on <span class="inlinecode"><span class="id" title="var">Z</span></span></a>
+
+</li>
+<li><a href="additions.Monoid_instances.html#lab382">Multiplicative monoid on <span class="inlinecode"><span class="id" title="var">nat</span></span></a>
+
+</li>
+<li><a href="additions.Monoid_instances.html#lab383">Additive monoid on <span class="inlinecode"><span class="id" title="var">nat</span></span></a>
+
+</li>
+<li><a href="additions.Monoid_instances.html#lab384">Multiplicative monoid on 63-bits integers</a>
+
+</li>
+<li><a href="additions.Monoid_instances.html#lab385">Monoid of 2x2 matrices</a>
+
+</li>
+<li><a href="additions.Monoid_instances.html#lab386">Integers modulo m</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.More_on_positive.html">Library additions.More_on_positive</a></h2>
+<ul class="doclist">
+<li><a href="additions.More_on_positive.html#lab387">Basic lemmas about positive and N</a>
+<ul class="doclist">
+<li><a href="additions.More_on_positive.html#lab388">Relationship with <span class="inlinecode"><span class="id" title="var">nat</span></span> and <span class="inlinecode"><span class="id" title="var">N</span></span></a>
+
+</li>
+<li><a href="additions.More_on_positive.html#lab389">Surjection from <span class="inlinecode"><span class="id" title="var">N</span></span> into <span class="inlinecode"><span class="id" title="var">positive</span></span></a>
+
+</li>
+<li><a href="additions.More_on_positive.html#lab390">Lemmas on Euclidean division</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Naive.html">Library additions.Naive</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="additions.Naive.html#lab391">Naïve  and Monomorphic  Functions for Computing Powers.</a>
+<ul class="doclist">
+<li><a href="additions.Naive.html#lab392">Exponentiation of integers</a>
+
+</li>
+<li><a href="additions.Naive.html#lab393">Some basic properties of <span class="inlinecode"><span class="id" title="var">Z.power</span></span></a>
+
+</li>
+<li><a href="additions.Naive.html#lab394">Exponentiation modulo <span class="inlinecode"><span class="id" title="var">m</span></span></a>
+
+</li>
+<li><a href="additions.Naive.html#lab395">Exponentiation of  matrices</a>
+<ul class="doclist">
+<li><a href="additions.Naive.html#lab396"> </a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Pow.html">Library additions.Pow</a></h2>
+<ul class="doclist">
+<li><a href="additions.Pow.html#lab397">Exponentiation   in a Monoid</a>
+<ul class="doclist">
+<li><a href="additions.Pow.html#lab398">Two functions for computing powers</a>
+<ul class="doclist">
+<li><a href="additions.Pow.html#lab399">The "naive" reference function</a>
+
+</li>
+<li><a href="additions.Pow.html#lab400">The binary exponentiation function (exponents in  <span class="inlinecode"><span class="id" title="var">positive</span></span>)</a>
+
+</li>
+<li><a href="additions.Pow.html#lab401"> </a>
+
+</li>
+<li><a href="additions.Pow.html#lab402"> </a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Pow.html#lab403">Properties of the power function</a>
+
+</li>
+<li><a href="additions.Pow.html#lab404">Correctness of the binary algorithm</a>
+
+</li>
+<li><a href="additions.Pow.html#lab405">Remark</a>
+
+</li>
+<li><a href="additions.Pow.html#lab406">Properties of Abelian Monoids</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Pow_variant.html">Library additions.Pow_variant</a></h2>
+<ul class="doclist">
+<li><a href="additions.Pow_variant.html#lab407">Exponentiation   in a Monoid</a>
+<ul class="doclist">
+<li><a href="additions.Pow_variant.html#lab408">Two functions for computing powers</a>
+<ul class="doclist">
+<li><a href="additions.Pow_variant.html#lab409">The "naive" reference function</a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab410">The binary exponentiation function (exponents in  <span class="inlinecode"><span class="id" title="var">positive</span></span>)</a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab411"> </a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab412"> </a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab413">Properties of the power function</a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab414">Correctness of the binary algorithm</a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab415">Remark</a>
+
+</li>
+<li><a href="additions.Pow_variant.html#lab416">Properties of Abelian Monoids</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="additions.Strategies.html">Library additions.Strategies</a></h2>
+<h2><a href="additions.Trace_exercise.html">Library additions.Trace_exercise</a></h2>
+<h2><a href="additions.Wf_transparent.html">Library additions.Wf_transparent</a></h2>
+<h2><a href="gaia_hydras.T1Bridge.html">Library gaia_hydras.T1Bridge</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.T1Bridge.html#lab417">Bridge between Hydra-battle's and Gaia's <span class="inlinecode"><span class="id" title="var">T1</span></span>  (Experimental)</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab418">refinement of constants, functions, etc.</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.T1Bridge.html#lab419">Refinements of usual constants</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab420">unary functions and predicates</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab421">Equality and comparison</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab422">Ordinal addition</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab423">Ordinal multiplication</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab424">Ordinal terms in normal form</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab425">Well formed ordinals as a data type</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab426">Make E0 an ordinal notation</a>
+
+</li>
+<li><a href="gaia_hydras.T1Bridge.html#lab427">Abstract properties of arithmetic functions (with SSreflect inequalities)</a>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GaiaToHydra.html">Library gaia_hydras.GaiaToHydra</a></h2>
+<h2><a href="gaia_hydras.nfwfgaia.html">Library gaia_hydras.nfwfgaia</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab428">Ordinals in Pure Coq</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab429">prelude</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab430">Example 1</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab431">Example 2</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab432">Example 3</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab433">More on accessiblity</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab434">The type T1</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab435">Equality</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab436">Order on T1</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab437">Normal form</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab438">Successor</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab439">Addition</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab440">Limits</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab441">Definition</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab442">The normal form</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab443">Realizing the limit</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab444">Normal functions</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab445">multiplication</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab446">Preparation of the exponention</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab447">Normality of multiplication</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab448">Exponentiation</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab449">Existence and uniqueness of the Cantor Normal Form</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab450">Basic Properties</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab451">The type T2</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab452">Definition and Equality</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab453">Order</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab454">Normal form</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab455">Successor and predecessor</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab456">Addition</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab457">The function phi</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab458">Ackermann</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.nfwfgaia.html#lab459">Comparison</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab460">Normal form</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab461">Successor Predecessor</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab462">Addition</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab463">limit</a>
+
+</li>
+<li><a href="gaia_hydras.nfwfgaia.html#lab464">additive principal</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GCanon.html">Library gaia_hydras.GCanon</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.GCanon.html#lab465">Adaptation of <span class="inlinecode"><span class="id" title="var">canon</span></span> to type E0</a>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GF_alpha.html">Library gaia_hydras.GF_alpha</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.GF_alpha.html#lab466">Rapidly growing functions</a>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GL_alpha.html">Library gaia_hydras.GL_alpha</a></h2>
+<h2><a href="gaia_hydras.GHprime.html">Library gaia_hydras.GHprime</a></h2>
+<ul class="doclist">
+<li><ul class="doclist">
+<li><a href="gaia_hydras.GHprime.html#lab467">Equations for <span class="inlinecode"><span class="id" title="var">H'_</span></span></a>
+
+</li>
+<li><a href="gaia_hydras.GHprime.html#lab468">Examples</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.ON_gfinite.html">Library gaia_hydras.ON_gfinite</a></h2>
+<h2><a href="gaia_hydras.GPaths.html">Library gaia_hydras.GPaths</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.GPaths.html#lab469">Adaptation to E0</a>
+
+</li>
+<li><a href="gaia_hydras.GPaths.html#lab470">Examples</a>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GLarge_Sets.html">Library gaia_hydras.GLarge_Sets</a></h2>
+<h2><a href="gaia_hydras.GHessenberg.html">Library gaia_hydras.GHessenberg</a></h2>
+<h2><a href="gaia_hydras.HydraGaia_Examples.html">Library gaia_hydras.HydraGaia_Examples</a></h2>
+<h2><a href="gaia_hydras.GHydra.html">Library gaia_hydras.GHydra</a></h2>
+<h2><a href="gaia_hydras.T2Bridge.html">Library gaia_hydras.T2Bridge</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.T2Bridge.html#lab471">Bridge between Hydra-battle's and Gaia's <span class="inlinecode"><span class="id" title="var">T1</span></span>  (Experimental)</a>
+
+</li>
+</ul>
+<h2><a href="gaia_hydras.GPrelude.html">Library gaia_hydras.GPrelude</a></h2>
+<h2><a href="gaia_hydras.T1Choice.html">Library gaia_hydras.T1Choice</a></h2>
+<h2><a href="gaia_hydras.onType.html">Library gaia_hydras.onType</a></h2>
+<ul class="doclist">
+<li><a href="gaia_hydras.onType.html#lab472">Preliminaries</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.onType.html#lab473">Complements on orderType</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="gaia_hydras.onType.html#lab474">First instances of ON</a>
+<ul class="doclist">
+<li><a href="gaia_hydras.onType.html#lab475">The Ordinal notation for 'I_i</a>
+
+</li>
+<li><a href="gaia_hydras.onType.html#lab476">An ordinal notation for omega</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="Goedel.rosser.html">Library Goedel.rosser</a></h2>
+<h2><a href="Goedel.goedel1.html">Library Goedel.goedel1</a></h2>
+<h2><a href="Goedel.PRrepresentable.html">Library Goedel.PRrepresentable</a></h2>
+<h2><a href="Goedel.fixPoint.html">Library Goedel.fixPoint</a></h2>
+<h2><a href="Goedel.codeSysPrf.html">Library Goedel.codeSysPrf</a></h2>
+<h2><a href="Goedel.rosserPA.html">Library Goedel.rosserPA</a></h2>
+<h2><a href="Goedel.goedel2.html">Library Goedel.goedel2</a></h2>
+</div>
+<hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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